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# Strong parametric dispersive shifts in a statically decoupled multi-qubit
cavity QED system

T. Noh<EMAIL_ADDRESS>Associate of the National Institute of Standards
and Technology, Boulder, Colorado 80305, USA  Department of Physics and
Applied Physics, University of Massachusetts, Lowell, MA 01854, USA  Z. Xiao
Department of Physics and Applied Physics, University of Massachusetts,
Lowell, MA 01854, USA  K. Cicak  National Institute of Standards and
Technology, 325 Broadway St MS686.05, Boulder, Colorado 80305, USA  X. Y. Jin
Associate of the National Institute of Standards and Technology, Boulder,
Colorado 80305, USA  Department of Physics, University of Colorado, Boulder,
Colorado 80309, USA  E. Doucet  Department of Physics and Applied Physics,
University of Massachusetts, Lowell, MA 01854, USA  J. Teufel  National
Institute of Standards and Technology, 325 Broadway St MS686.05, Boulder,
Colorado 80305, USA  J. Aumentado  National Institute of Standards and
Technology, 325 Broadway St MS686.05, Boulder, Colorado 80305, USA  L. C. G.
Govia  Quantum Engineering and Computing, Raytheon BBN Technologies,
Cambridge, Massachusetts 02138, USA  L. Ranzani  Quantum Engineering and
Computing, Raytheon BBN Technologies, Cambridge, Massachusetts 02138, USA  A.
Kamal  Department of Physics and Applied Physics, University of Massachusetts,
Lowell, MA 01854, USA  R. W. Simmonds<EMAIL_ADDRESS>National
Institute of Standards and Technology, 325 Broadway St MS686.05, Boulder,
Colorado 80305, USA

###### Abstract

Cavity quantum electrodynamics (QED) with in-situ tunable interactions is
important for developing novel systems for quantum simulation and
computing.Rauschenbeutel  _et al._ (1999); Allman  _et al._ (2010); Srinivasan
_et al._ (2011); Potts  _et al._ (2016); Vaidya  _et al._ (2018); Suleymanzade
_et al._ (2020) The ability to tune the dispersive shifts of a cavity QED
system provides more functionality for performing either quantum measurements
or logical manipulations. Here, we couple two transmon qubits to a lumped-
element cavity through a shared dc-SQUID.Zakka-Bajjani  _et al._ (2011); Lu
_et al._ (2017) Our design balances the mutual capacitive and inductive
circuit components so that both qubits are highly decoupled from the cavityLu
_et al._ (2017), offering protection from decoherence processesWhittaker  _et
al._ (2014); Zhang  _et al._ (2017). We show that by parametrically
drivingZakka-Bajjani  _et al._ (2011); Allman  _et al._ (2014); Lu  _et al._
(2017) the SQUID with an oscillating flux it is possible to independently tune
the interactions between either of the qubits and the cavity dynamically. The
strength and detuning of this cavity-QED interaction can be fully controlled
through the choice of the parametric pump frequency and amplitude. As a
practical demonstration, we perform pulsed parametric dispersive readout of
both qubits while statically decoupled from the cavity. The dispersive
frequency shifts of the cavity mode follow the expected magnitude and sign
based on a simple theory that is supported by a more thorough theoretical
investigationXiao  _et al._ (2021). This parametric approach creates a new
tunable cavity QED framework for developing quantum information systems with
various future applications, such as entanglement and error correction via
multi-qubit parity readoutRoch  _et al._ (2014); Riste  _et al._ (2013);
Andersen  _et al._ (2019), state and entanglement stabilizationLu  _et al._
(2017); Kimchi-Schwartz  _et al._ (2016); Doucet  _et al._ (2020), and
parametric logical gatesReagor  _et al._ (2018); Noguchi  _et al._ (2018).

Cavity QED has become a backbone for some of the leading quantum information
processing systems. In particular, systems based on superconducting circuits,
so called circuit QED systemsBlais  _et al._ (2004); Koch  _et al._ (2007),
provide a promising platform for realizing a practical quantum information
processing unit. These systems usually rely on a static coupling between the
qubit and cavity. Schuster  _et al._ (2005); Houck  _et al._ (2008). In order
to overcome the drawbacks in these systems, such as frequency crowding,
additional decoherence, and always-on interactions, systems have been
developed that use flux tunable elements Allman  _et al._ (2010); Srinivasan
_et al._ (2011); Allman  _et al._ (2014); Whittaker  _et al._ (2014); Zhang
_et al._ (2017) to set the coupling strength between a qubit and cavity via a
static magnetic field. Tunable couplers have led to improved coherence and
fast, high-fidelity two-qubit entangling gates Chen  _et al._ (2014) and qubit
measurements have been performed while avoiding decoherence caused by unwanted
interactions with the cavity. Whittaker  _et al._ (2014); Zhang  _et al._
(2017)

Figure 1: (Color online) (a) Optical micrograph of a tunable circuit QED
system. (b) Schematic diagram of the system including the three elements:
cavity mode (C), Left transmon (L), and Right transmon (R). The parametric
flux drive is applied through a shared dc-SQUID coupler. (c) Periodic cavity
spectrum. (d) Frequency of the three resonant modes, C, R, and L as a function
of the flux bias $\phi=\Phi/\Phi_{0}$. (e) Energy scale diagram showing the
relationship between defined quantities, see the main text for the details.

One difficulty with systems that rely on a tunable static flux is that the
frequency of all the coupled elements are usually strongly dependent on the
applied flux.Allman  _et al._ (2010); Srinivasan  _et al._ (2011); Allman  _et
al._ (2014); Whittaker  _et al._ (2014); Zhang  _et al._ (2017) Therefore, the
tuning of the cavity QED parameters also involves strong manipulation of the
frequency of the cavity QED elements. This can be a disadvantage as each
element experiences a phase shift when their resonant frequency is adjusted.
When performing quantum operations, these shifts must usually be accounted for
or corrected. In addition, when many elements share the same frequency space,
flux tuning can lead to unwanted frequency collisions and unwanted coupled
interactions, complicating the system operation.

Parametric interactions represent an alternative coupling mechanism with many
benefits. Zakka-Bajjani  _et al._ (2011); Allman  _et al._ (2014); Chen  _et
al._ (2014); Roushan  _et al._ (2017) When used in cavity QED-type systems
where the parametric pump is resonant with the difference frequency between
the qubit and cavity, one can perform rapid swap operations between the
coupled systems with rates that are directly proportional to the amplitude of
the pump drive. Zakka-Bajjani  _et al._ (2011); Allman  _et al._ (2014) These
type of interactions can be important for cavity photon state manipulation,
photon-photon interference, and performing gate operations. Zakka-Bajjani  _et
al._ (2011); Nguyen  _et al._ (2012); Allman  _et al._ (2014); Rosenblum  _et
al._ (2018). However, parametric operations within the dispersive regime,
where the coupled systems do not exchange energy, can also be very useful.
Three-dimensional cavity QED systems have taken advantage of four-wave mixing
through a single Josephson junction of a capacitively coupled
transmonRosenblum  _et al._ (2018) to provide for dynamically adjustable
dispersive shifts. Rosenblum  _et al._ (2012) Parametric coupling between
Qubit-cavity systems can also useful for stabilizing the state of a single
transmon qubit Lu  _et al._ (2017) or the entanglement of multi-qubit systems
Doucet  _et al._ (2020); et al. (2021a).

In this work, we describe a two-transmon qubit cavity QED system whereby the
interactions with a single cavity mode coupled to an input/output microwave
feedline are tuned through the use of a parametric flux drive applied to a
shared dc-SQUID coupler. We investigate in detail the dispersive frequency
shifts on the cavity mode induced by inserting a single photon in one of the
qubits as a function of the parametric pump’s drive frequency and amplitude.
We find that these shifts can be well described by a simple model that is a
parametric generalization of the standard dispersive formulas introduced for
circuit QED with a transmon. Koch  _et al._ (2007) Moreover, we observe
experimentally some interesting new features at higher pump amplitudes that
can be explained by higher order corrections from the absorption of multiple
photons between the transmon’s energy levels. Xiao  _et al._ (2021) In
addition, this system also allows parametric interactions between the two
qubits, independent of the cavity, for applying parametric entangling
gatesReagor  _et al._ (2018); Noguchi  _et al._ (2018) or between all three
systems for the stabilization of arbitrarily entangled two-qubit statesKimchi-
Schwartz  _et al._ (2016); Doucet  _et al._ (2020), as will be described in
future worket al. (2021b, c, a).

With a nearly 0.5 GHz separation between the qubits, the parametric dispersive
shifts of the cavity due to either qubit can be independently controlled by
selectively applying a specific pump frequency. Both the amplitude and sign of
these shifts can be controlled by adjusting the pump amplitude and frequency.
Remarkably, we can dynamically access both positive and negative relative
detunings allowing us access to all three distinct dispersive regions Koch
_et al._ (2007); Xiao  _et al._ (2021), beyond the usual ac Stark effect
expected for a two-level atom. Therefore, along with the commonly used
negative dispersive shifts, we can also generate large, positive shifts that
hitherto only occurred in the “straddling regime” of static circuit QED Koch
_et al._ (2007), where the cavity frequency must sit between the first two
qubit transition frequencies, $\omega_{ge}$ and $\omega_{ef}$. In this work,
by dynamically driving the coupler, a “parametric straddling regime” has been
discovered even with the cavity frequency nearly 3 GHz above both qubits. This
results from the fact that the parametric generalization of the qubit-cavity
detuning $\Delta_{p}$, the parametric coupling strength $g_{p}$, and either
transmon anharmonicity $\alpha$ satisfy the necessary straddling regime
conditions, making it fully accessible. Koch  _et al._ (2007); Xiao  _et al._
(2021)

As a practical demonstration of parametrically controlled dispersive shifts,
we show that we can pulse “on” these shifts in-time in order to measure either
qubit, or apply them simultaneously to perform a joint readout of the two-
qubit states Roch  _et al._ (2014); Riste  _et al._ (2013); Andersen  _et al._
(2019). One unique design feature of our parametric strategy is that each
qubit can be protected Srinivasan  _et al._ (2011); Zhang  _et al._ (2017)
from the cavity during its logical operations. Only when it is time to measure
the qubit do we turn on the parametric interactions generating large
dispersive shifts to enable qubit state readout. To this end, we characterize
the residual static dispersive shift between the qubits and the cavity as a
function of the flux in the SQUID coupler. Furthermore, by injecting photons
into the cavity, we can witness the elimination of additional qubit dephasing
even in the presence of photon shot noise in the cavity. Zhang  _et al._
(2017) These results set the stage for future work. Because these parametric
dispersive shifts can be applied simultaneously, with adjustable strength and
sign, and add linearly, we can explore the construction of multi-qubit parity
measurements, useful for measuring error syndromes and performing quantum
error correction. Roch  _et al._ (2014); Riste  _et al._ (2013); Andersen  _et
al._ (2019)

Our circuit, as shown in Fig. 1(a)–(b), has been constructed to largely cancel
static coupling between both qubits and the cavity Allman  _et al._ (2010,
2014); Lu  _et al._ (2017), so that parametric couplings dominate the
interactions (see Supplemental Material (SM)). Each frequency is partially
defined by the three top capacitor plates ($C$, $L$, and $R$) for the three
lumped-element resonators. The left ($L$) and the right ($R$) transmons each
have nominally the same Josephson junction with critical current $I_{o}\approx
36$ nA that defines the Josephson energy $E_{J}=I_{o}\Phi_{o}/2\pi$. The
central cavity ($C$) uses a series array of 7 larger junctions (each with
$I_{oC}\approx 15I_{o}$) to form a nearly linear inductance $L_{C}$. The dc-
SQUID coupler, formed by a small loop that includes two Josephson junctions
with critical currents $I_{o1}=I_{o2}\approx 15I_{o}$, shares current between
all three elements with the large common electrode, or bottom plate $B$. We
have purposefully chosen the orientation and distance between the three top
capacitor plates in order for their mutual capacitive coupling to cancel the
mutual inductive coupling of the SQUID at a particular static flux bias
$\phi_{c}=\Phi_{c}/\Phi_{o}$, which we refer to as the “cancellation flux”,
where $\Phi_{o}=h/2e$ is the magnetic flux quantum. At this flux, the static
coupling between all three elements should in-principle be zero, leaving them
uncoupled and thus independent. Of course, achieving this condition perfectly
is challenging (see SM).

Figure 2: (Color online) (a) A typical full cavity spectrum while sweeping the
pump frequency $\omega_{p}$. The upper and lower panel shows the phase of the
cavity signal with the R transmon in state $|g\rangle_{R}$ and
$|e\rangle_{R}$, respectively. (b) The parametrically induced dispersive
shifts as a function of the pump detuning frequency,
$\Delta_{pR}=\omega_{p}-|\Delta_{R}|$, at various calibrated pump amplitudes:
50 (red), 100 (orange), 150 (yellow), 200 (green), 250 (blue), and 300 mV
(purple), respectively. The solid lines are fits for the data from the theory.
See the text for the details. (c) Parametric $g_{pR}$ as a function of pump
amplitude.

Standard circuit QED theory Blais  _et al._ (2004); Koch  _et al._ (2007)
describes the interaction between an individual transmon $k=L,R$, with
anharmonicity $\alpha_{k}<0$, and a cavity $C$ as (with $\hbar=1$),

$H=\sum_{k=\\{L,R\\}}\omega_{k}q_{k}^{\dagger}q_{k}+\frac{\alpha_{k}}{2}(q_{k}^{\dagger}q_{k})^{2}+\omega_{C}c^{\dagger}c+H_{Ik},$
(1)

with the interaction Hamiltonian is given by,

$H_{Ik}=g_{sk}(q_{k}+q_{k}^{\dagger})(c+c^{\dagger}),$ (2)

where $q_{k}^{\dagger}$ ($q_{k}$) is the creation (annihilation) operator of
the $k$th transmon mode, $c^{\dagger}$ ($c$) corresponds to the same for the
cavity mode, $\omega_{k}$ and $\omega_{C}$ are the frequencies of the $k$th
transmon and cavity, respectively, and $g_{sk}\equiv g_{sk}(\phi)$ is the
SQUID-tunable, total static coupling rate Allman  _et al._ (2010, 2014);
Whittaker  _et al._ (2014); Lu  _et al._ (2017) between the $k$th transmon and
the cavity. We will define the states $|g\rangle_{k}$, $|e\rangle_{k}$ and
$|f\rangle_{k}$ as the first three eigenstates of the $k$th transmon,
respectively, while the cavity states are referred to as $|n\rangle$ for
photon numbers $n=0,1,2\cdots$.

In the dispersive regime of cavity QED Blais  _et al._ (2004); Koch  _et al._
(2007); Boissonneault  _et al._ (2009), when $g_{sk}\ll|\Delta_{k}|$ where
$\Delta_{k}=\omega_{k}-\omega_{C}$ is the detuning between the $k$th transmon
and $g_{sk}$ is the static coupling strength, the interaction Hamiltonian in
Eq. (2) simply becomes $H_{Ik}=2\chi_{sk}q_{k}^{\dagger}q_{k}c^{\dagger}c$. In
this regime, the qubit and cavity both experience a total dispersive frequency
shift $2\chi_{sk}$ depending on the energy state of the other element of the
system. In our case, the flux-tunable static dispersive strength
$\chi_{sk}(\phi)=(g_{sk}^{2}(\phi)/\Delta_{k})(\alpha_{k}/(\alpha_{k}+\Delta_{k}))$.
Koch  _et al._ (2007) Here, the separation between the cavity transition
frequencies when the $k$th qubit is in $|g\rangle_{k}$ and $|e\rangle_{k}$ is
$2\chi_{sk}$. At a flux bias far from the cancellation flux, the residual
static coupling $g_{sk}$ is predominantly capacitive (when $\phi<\phi_{c}$) or
inductive (when $\phi>\phi_{c}$). We can easily perform qubit spectroscopy
when $\chi_{sk}$ is relatively large, however the spectrum fades away on
either side of the cancellation flux as $g_{sk}$ becomes significantly reduced
(see Fig. 1(d)).

The frequencies of all the resonant modes are periodic as a function of $\phi$
(as shown for the cavity in Fig. 1(c)). In Fig. 1(d), we shown all the spectra
over nearly a half-a-period in $\phi$. At $\phi=0$, the maximum qubit
frequencies are about 5.9 GHz ($L$) and 6.4 GHz ($R$), while the maximum
cavity frequency is about 9.4 GHz. The cavity is coupled to a co-planar
waveguide feedline for the input and output of microwave photons leading to a
typical total decay rate of $\kappa/2\pi\approx 10$ MHz, which is flux
dependent (see SM). The transmon anharmonicities are also flux dependent (see
SM), with average values of $\alpha_{L}/2\pi\approx-180$ MHz and
$\alpha_{R}/2\pi\approx-220$ MHz.

We can access a new, rich set of dynamics by moving beyond the static
interactions typical of standard cavity QED systems, by driving the tunable
coupler with a sinusoidal flux modulation in addition to a static flux bias,
$\phi(t)=\phi_{m}\sin(\omega_{p}t)+\phi_{s}$. This parametric pump drive then
modifies the interaction strength in Eq. (2), leading to the following
coupling terms written in an interaction frame w.r.t. the free Hamiltonian
Allman  _et al._ (2014),

$H_{pk}=-i\sum_{k=\\{L,R\\}}g_{pk}\left(e^{i\omega_{p}t}-e^{-i\omega_{p}t}\right)(q_{k}c^{\dagger}e^{i\Delta_{k}t}+{\rm
h.c.}),$ (3)

where we have ignored the sum frequency terms for the regime under
consideration, i.e. $\omega_{p}$ near the difference frequency $|\Delta_{k}|$.
In general, the pump frequency can be detuned by $\Delta_{pk}$,
$\omega_{p}=|\Delta_{k}|+\Delta_{pk}$, in order to sweep between
parametrically-induced resonant-type ($\Delta_{pk}\ll g_{pk}$) and dispersive-
type ($\Delta_{pk}\gg g_{pk}$) interactions, corresponding to the
$|g\rangle_{k}\rightarrow|e\rangle_{k}$ transition and the cavity mode. This
can been seen by measuring the full spectrum of the cavity mode for a fixed
pump amplitude when $\omega_{p}\approx|\Delta_{R}|$, as shown in the upper
panel of Fig. 2(a). Notice the avoided crossing is clearly visible with a
coupling strength, $2g_{pR}/2\pi\approx 10$ MHz. This data was taken when
starting from $|g\rangle_{R}$. If, after a $\pi$-pulse, we start from
$|e\rangle_{R}$ and take the same spectrum in the presence of the pump tone,
we can see (lower panel of Fig. 2(a)) another avoided crossing from the
$|e\rangle_{R}\rightarrow|f\rangle_{R}$ transition becoming resonant with the
cavity mode. Notice that this splitting is centered about
$\Delta_{pR}=-\alpha_{R}>0$. This behavior is also seen for higher level
transmon transitions Xiao  _et al._ (2021), which agrees with Eq. 4, and these
results have been repeated successfully for the $L$ transmon as well (see SM).

In order to fully characterize the additional dispersive shifts imparted to
the cavity by the presence of the parametric pump, we measure the full cavity
spectrum for various calibrated pump amplitudes (see SM) while sweeping
$\omega_{p}$. We take two data sets for each pump amplitude, one starting from
$|g\rangle_{R}$ and one starting from $|e\rangle_{R}$, like the data sets
shown in Fig. 2(a)). By comparing the two results, we can extract the full
dispersive shifts as shown in Fig. 2(b). For the smallest pump amplitude and
largest $\Delta_{pR}$, notice that the dispersive shift is nearly zero. This
is because we have chosen $\phi\approx\phi_{c}$, as discussed below. In
general, the parametric dispersive shifts simply add to the static shifts Xiao
_et al._ (2021) (see SM for data at more bias points).

Figure 3: (Color online) (a) The static dispersive shift $2\chi_{s}/2\pi$ as a
function of the R transmon frequency. For red open squares, the dispersive
shift was measured by comparing the cavity response with the R transmon in
either state $|g\rangle_{R}$ or $|e\rangle_{R}$. For blue filled circles, the
dispersive shift was extracted by measuring additional dephasing $\Gamma_{\phi
R}$ while driving the cavity to a weak coherent state, see the main text for
the details. The solid line represents a polynomial fit to the data. (b)
Additional dephasing as a function of the average photon number $\bar{n}$ at
three different flux biases where the R tranmon frequencies are 6.215 (red
square), 6.231 (yellow circle), and 6.282 GHz (green triangle), respectively.
Solid lines are a linear fit for the data.

In order to gain physical intuition, the parametric interactions can be
analyzed in a rotating frame defined w.r.t. the pump frequency $\omega_{p}$.
For an arbitrary pump detuning $\Delta_{pk}$, this necessitates including
contributions from both positive and negative frequency components, as well as
those corresponding to the sum and difference frequencies. Xiao  _et al._
(2021) However, for sufficiently small $\Delta_{pk}$, in this rotating frame,
the form of Eq. (3) becomes analogous to the static dispersive case with the
identification $g_{sk}\rightarrow g_{pk}$ and
$\Delta_{k}\rightarrow\Delta_{pk}$. Koch  _et al._ (2007) We can even extend
this picture to include higher transmon levels (see SM),

$\chi_{pk}=\frac{g_{pk}^{2}}{\Delta_{pk}}\left[\frac{\alpha_{k}}{(\alpha_{k}+\Delta_{pk})}-\frac{\Delta_{pk}}{(2\alpha_{k}+\Delta_{pk})}-\frac{\Delta_{pk}}{(3\alpha_{k}+\Delta_{pk})}\cdots\right],$
(4)

This simplified expression describes the data quite well and the extracted
values for $g_{pR}$ are plotted in Fig. 2(c), which clearly shows a linear
dependence on the pump amplitude. Moreover, at a fixed amplitude,
$g_{pR}\propto\sqrt{(d\omega_{R}/d\phi)(d\omega_{C}/d\phi)}$ agrees with
predictions Zakka-Bajjani  _et al._ (2011); Allman  _et al._ (2014) based on
the slope of the qubit and cavity at the cancellation flux $\phi_{c}$ (see
SM). At larger powers, however, we find a very interesting peak in the data
that occurs at $\Delta_{pR}=-2\alpha_{R}$. This corresponds to a pump-mediated
two-photon transition between the first- and third-excited states of the
transmon Xiao  _et al._ (2021). Such features are described by quartic-order
contributions to dispersive shift $\chi_{pk}$ and exhibit an even lineshape in
pump detuning, thus providing a means to distinguish them from the lower-order
quadratic features that lead to an odd lineshape (see Fig. 2(b)).

As a simple demonstration of the usefulness of programmable dispersive shifts
in this new parametric circuit-QED framework, we next describe qubit
measurements near the cancellation flux. One drawback when working with static
cavity QED systems is that a persistent dispersive interaction can lead to two
decoherence processes, enhanced energy relaxation Houck  _et al._ (2008),
reducing the qubit relaxation time $T_{1}$, and additional dephasing Schuster
_et al._ (2005), reducing the qubit dephasing time $T_{2}$. Although the qubit
and the cavity are detuned from each other, the qubit’s decay rate $1/T_{1}$
is enhanced by an additional Purcell factor of $(g/\Delta)^{2}\kappa$ due to
radiative decay through the coupled cavity mode. Whereas qubit dephasing
occurs because the dispersive interaction also modifies the qubit’s frequency
depending on the photon occupancy within the cavity. The additional dephasing
is given by the rate
$\Gamma_{n}=8\bar{n}\kappa\chi_{s}^{2}/(\kappa^{2}+4\chi_{s}^{2})$, where
$\bar{n}$ is the average number of (coherent state) photons in the cavity.
Schuster  _et al._ (2005); Gambetta  _et al._ (2006) These decoherence
mechanisms can be reduced in circuit QED systems by using Purcell filters Reed
_et al._ (2010) (increases $T_{1}$ only) or eliminating the total qubit-cavity
coupling Allman  _et al._ (2010); Srinivasan  _et al._ (2011); Allman  _et
al._ (2014); Zhang  _et al._ (2017) (increases both $T_{1}$ and $T_{2}$). One
difficulty with the second approach is that once $g_{sk}\approx 0$, it is not
possible to measure the qubit dispersively, unless the qubit has hidden
degrees of freedom still coupled to the cavity Zhang  _et al._ (2017) or you
rapidly tune-up $g_{sk}(\phi)$ through a flux shift pulse in time for the
measurement Whittaker  _et al._ (2014).

Alternatively, our circuit allows us to minimize $g_{sk}$ by setting
$\phi=\phi_{c}$, while still enabling dispersive qubit readout by applying a
parametric pump tone with a given amplitude and frequency $\omega_{p}$, as
described above (see Fig. 2). Ideally, when $\chi_{s}\propto
g_{sk}^{2}\rightarrow 0$, both qubits are completely decoupled statically from
the cavity. In Fig. 3(a), we show the static dispersive shift $\chi_{sR}$ (and
$g_{sR}$, see SM) as a function of the R transmon frequency. For the open
square, we directly measured the dispersive shift by comparing the cavity
response with the R transmon in either state $|g\rangle_{R}$ or
$|e\rangle_{R}$. The solid line comes from a polynomial fit that matches our
predictions based on a circuit model (see SM). In Fig. 3(b), we also measured
additional dephasing $\Gamma_{\phi R}$ of the R transmon while driving the
cavity with a weak coherent state with an average photon number $\bar{n}$ for
each flux bias. Schuster  _et al._ (2005); Zhang  _et al._ (2017) Crucially,
the measurement of the R transmon near the cancellation flux required using a
parametric tone to perform the dispersive readout. We see that $\Gamma_{\phi
R}$ scales linearly with $\bar{n}$ as expected Schuster  _et al._ (2005);
Gambetta  _et al._ (2006) and the slope significantly diminishes as the bias
approaches the cancellation flux. From this data, and knowing $\kappa$, we can
extract $\chi_{sR}$ (and $g_{sR}$) for each bias (see SM). This result is
plotted as filled circles in Fig. 3(a) and agrees well with the direct
measurements. These results were also performed on the L transmon with similar
results (see SM). The minimum value for $\chi_{sR}/2\pi$ ($\chi_{sL}/2\pi$)
was $300$ kHz ($150$ kHz), which was limited by the nonlinearity of the SQUID
coupler itself (see SM).

Figure 4: (Color online) (a) The phase coherence time $T^{*}_{2}$ as a
function of the R transmon frequency when $\bar{n}\approx 0$. The solid black
line is a fit based on the dominant dephasing coming from flux noise. (b)
Comparison of $T^{*}_{2}$ in the presence of coherent cavity photons with
$\bar{n}=$ 0.5 (square), 1 (up-triangle), and 2 (down-triangle). The solid
lines are predictions based the measurement, see the main text for the
details.

Finally, we can demonstrate that the qubits are protected from photon shot
noise in the cavity by comparing the additional dephasing just discussed
($\bar{n}\neq 0$) with the background dephasing when $\bar{n}\approx 0$. Zhang
_et al._ (2017) In Fig. 4(a), we plot the phase coherence time $T^{*}_{2}$ of
the R qubit obtained by measuring Ramsey oscillations as a function of its
frequency. The solid line is a fit to the data assuming that the dominant
source of background dephasing is coming from flux noise in the coupler, which
is proportional to the slope $d{\omega_{R}(\phi)}/d\phi$ of the modulation
curve of R transmon frequency with flux (see Fig. 1). In Fig. 4(b), we compare
this result with $T^{*}_{2}$ of the R transmon in the presence of photon shot
noise for $\bar{n}=$0, 0.5, 1, and 2. Here, the solid lines are predictions
based on the directly measured values for $\chi_{sR}$ as a function of flux
and the average value of $\kappa$ over whole range of flux considered in the
measurement. As we can see, the R transmon is protected from shot noise in the
cavity near $\phi=\phi_{c}$, where the dephasing is dominated by flux noise in
the coupler. Similar results were found for the L transmon as well (See SM).

In conclusion, we have developed a unique tunable cavity QED system that
allows for in-situ dynamic control over the interaction between a qubit and a
cavity by applying a parametric tone. We demonstrated that this can be
accomplished with a two-qubit system, that can also allow for a fully
adjustable joint two-qubit measurement. We operated our two-transmon circuit
QED system in the dispersive regime and explored the dynamic control of
parametrically induced dispersive cavity shifts from either transmon. By
varying the sign and size of these shifts with the amplitude and frequency of
the parametric pump, we have unlocked a “parametric straddling regime” and
verified a simple theoretical model that generalizes this behavior in the
rotating frame. As a practical application, we pulsed the parametric pump
simultaneously with a readout tone, and with a proper choice for the amplitude
and frequency, we performed dispersive qubit readout on both transmons. Due to
our unique circuit design that minimizes static coupling to the cavity, the
transmons were protected during local logical operations from photon shot
noise in the cavity. We believe that this new tunable cavity QED framework
will open up a novel paradigm for controlling light-matter interactions by
providing a unique functionality along with qualitatively new features that
are not supported by static circuit-QED setups.

This work was partially performed under the following financial assistance
award 70NANB18H006 from U.S. Department of Commerce, National Institute of
Standards and Technology. T.N., Z.X.,E.D.,L.R.,L.G. and A.K. received support
from the Department of Energy under grant DE-SC0019461.

## References

  * Rauschenbeutel  _et al._ (1999) A. Rauschenbeutel, A. G. Nogues, S. Osnaghi, P. Bertet, M. Brune, J. M. Raimond, and S. Haroche, Phys. Rev. Lett. 83, 5166 (1999).
  * Allman  _et al._ (2010) M. Allman, F. Altomare, J. Whittaker, K. Cicak, D. Li, A. Sirois, J. Strong, J. Teufel, and R. Simmonds, Phys. Rev. Lett. 104, 177004 (2010).
  * Srinivasan  _et al._ (2011) S. J. Srinivasan, A. J. Hoffman, J. M. Gambetta, and A. A. Houck, Phys. Rev. Lett. 106, 083601 (2011).
  * Potts  _et al._ (2016) C. A. Potts, A. Melnyk, H. Ramp, M. H. Bitarafan, D. Vick, L. J. LeBlanc, J. P. Davis, and R. G. DeCorby, Appl. Phys. Lett. 108, 041103 (2016).
  * Vaidya  _et al._ (2018) V. D. Vaidya, Y. Guo, R. M. Kroeze, K. E. Ballantine, A. J. Kollár, J. Keeling, and B. L. Lev, Phys. Rev. X 8, 011002 (2018).
  * Suleymanzade  _et al._ (2020) A. Suleymanzade, A. Anferov, M. Stone, R. K. Naik, A. Oriani, J. Simon, and D. Schuster, Appl. Phys. Lett. 116, 104001 (2020).
  * Zakka-Bajjani  _et al._ (2011) E. Zakka-Bajjani, F. Nguyen, M. Lee, L. R. Vale, R. W. Simmonds, and J. Aumentado, Nature Physics 7, 599 (2011).
  * Lu  _et al._ (2017) Y. Lu, S. Chakram, N. Leung, N. Earnest, R. K. Naik, Z. Huang, P. Groszkowski, E. Kapit, J. Koch, and D. I. Schuster, Phys. Rev. Lett. 119, 150502 (2017).
  * Whittaker  _et al._ (2014) J. D. Whittaker, F. C. S. da Silva, M. S. Allman, F. Lecocq, K. Cicak, A. J. Sirois, J. D. Teufel, J. Aumentado, and R. W. Simmonds, Phys. Rev. B 90, 024513 (2014).
  * Zhang  _et al._ (2017) G. Zhang, Y. Liu, J. J. Raftery, and A. A. Houck, npj Quantum Information 3, 1 (2017).
  * Allman  _et al._ (2014) M. Allman, J. Whittaker, M. Castellanos-Beltran, K. Cicak, F. da Silva, M. DeFeo, F. Lecocq, A. Sirois, J. Teufel, J. Aumentado, and R. Simmonds, Phys. Rev. Lett. 112, 123601 (2014).
  * Xiao  _et al._ (2021) Z. Xiao, E. Doucet, T. Noh, R. W. Simmonds, L. Ranzani, L. C. G. Govia, and A. Kamal, Arxiv: 2103.09260 (2021).
  * Roch  _et al._ (2014) N. Roch, M. E. Schwartz, F. Motzoi, C. Macklin, R. Vijay, A. W. Eddins, A. N. Korotkov, K. B. Whaley, M. Sarovar, and I. Siddiqi, Phys. Rev. Lett. 112, 170501 (2014).
  * Riste  _et al._ (2013) D. Riste, M. Dukalski, C. A. Watson, G. de Lange, M. J. Tiggelman, Y. M. Blanter, K. W. Lehnert, R. N. Schouten, and L. DiCarlo, Nature 502, 350 (2013).
  * Andersen  _et al._ (2019) C. K. Andersen, A. Remm, S. Lazar, S. Krinner, J. Heinsoo, J.-C. Besse, M. Gabureac, A. Wallraff, and C. Eichler, npj Quantum Information 5, 69 (2019).
  * Kimchi-Schwartz  _et al._ (2016) M. E. Kimchi-Schwartz, L. Martin, E. Flurin, C. Aron, M. Kulkarni, H. E. Tureci, and I. Siddiqi, Phys. Rev. Lett. 116, 240503 (2016).
  * Doucet  _et al._ (2020) E. Doucet, F. Reiter, L. Ranzani, and A. Kamal, Phys. Rev. Research 2, 023370 (2020).
  * Reagor  _et al._ (2018) M. Reagor, C. B. Osborn, N. Tezak, A. Staley, G. Prawiroatmodjo, M. Scheer, N. Alidoust, E. A. Sete, N. Didier, M. P. da Silva, E. Acala, J. Angeles, A. Bestwick, M. Block, B. Bloom, A. Bradley, C. Bui, S. Caldwell, L. Capelluto, R. Chilcott, J. Cordova, G. Crossman, M. Curtis, S. Deshpande, T. E. Bouayadi, D. Girshovich, S. Hong, A. Hudson, P. Karalekas, K. Kuang, M. Lenihan, R. Manenti, T. Manning, J. Marshall, Y. Mohan, W. O’Brien, J. Otterbach, A. Papageorge, J.-P. Paquette, M. Pelstring, A. Polloreno, V. Rawat, C. A. Ryan, R. Renzas, N. Rubin, D. Russel, M. Rust, D. Scarabelli, M. Selvanayagam, R. Sinclair, R. Smith, M. Suska, T.-W. To, M. Vahidpour, N. Vodrahalli, T. Whyland, K. Yadav, W. Zeng, and C. T. Rigetti, Science Advances 4, 2 (2018).
  * Noguchi  _et al._ (2018) A. Noguchi, A. Osada, S. Masuda, S. Kono, K. Heya, S. P. Wolski, H. Takahashi, T. Sugiyama, D. Lachance-Quirion, and Y. Nakamura, Phys. Rev. A 102, 062408 (2018).
  * Blais  _et al._ (2004) A. Blais, R.-S. Huang, A. Wallraff, S. M. Girvin, and R. J. Schoelkopf, Phys. Rev. A 69, 062320 (2004).
  * Koch  _et al._ (2007) J. Koch, T. M. Yu, J. Gambetta, A. A. Houck, D. I. Schuster, J. Majer, A. Blais, M. H. Devoret, S. M. Girvin, and R. J. Schoelkopf, Phys. Rev. A 76, 042319 (2007).
  * Schuster  _et al._ (2005) D. I. Schuster, A. Wallraff, A. Blais, L. Frunzio, R.-S. Huang, J. Majer, S. Girvin, and R. J. Schoelkopf, Phys. Rev. Lett. 94, 123602 (2005).
  * Houck  _et al._ (2008) A. A. Houck, J. A. Schreier, B. R. Johnson, J. M. Chow, J. Koch, J. M. Gambetta, D. I. Schuster, L. Frunzio, M. H. Devoret, S. M. Girvin, and R. J. Schoelkopf, Phys. Rev. Lett. 101, 080502 (2008).
  * Chen  _et al._ (2014) Y. Chen, C. Neill, P. Roushan, N. Leung, M. Fang, R. Barends, B. C. J. Kelly, B. C. Z. Chen, A. Dunsworth, E. Jeffrey, A. Megrant, J. Y. Mutus, P. J. J. O’Malley, C. M. Quintana, D. Sank, A. Vainsencher, J. Wenner, T. C. White, M. R. Geller, A. N. Cleland, and J. M. Martinis, Phys. Rev. Lett. 113, 220502 (2014).
  * Roushan  _et al._ (2017) P. Roushan, C. Neill, A. Megrant, Y. Chen, R. Babbush, R. Barends, B. Campbell, Z. Chen, B. Chiaro, A. Dunsworth, A. Fowler, E. Jeffrey, J. Kelly, E. Lucero, J. Mutus, P. J. J. O’Malley, M. Neeley, C. Quintana, D. Sank, A. Vainsencher, J.Wenner, T. White, E. Kapit, H. Neven, and J. Martinis, Nature Physics 13, 146 (2017).
  * Nguyen  _et al._ (2012) F. Nguyen, E. Zakka-Bajjani, R. W. Simmonds, and J. Aumentado, Phys. Rev. Lett. 108, 163602 (2012).
  * Rosenblum  _et al._ (2018) S. Rosenblum, Y. Gao, P. Reinhold, C. Wang, C. Axline, L. Frunzio, S. Girvin, L. Jiang, M. Mirrahimi, M. Devoret, and R. Schoelkopf, Nature Communications 9, 652 (2018).
  * Rosenblum  _et al._ (2012) S. Rosenblum, P. Reinhold, M. Mirrahimi, L. Jiang, L. Frunzio, and R. Schoelkopf, Science 361, 266 (2012).
  * et al. (2021a) T. B. et al., Arxiv: in preparation (2021a).
  * et al. (2021b) X. Y. J. et al., Arxiv: in preparation (2021b).
  * et al. (2021c) X. Y. J. et al., Arxiv: in preparation (2021c).
  * Boissonneault  _et al._ (2009) M. Boissonneault, J. M. Gambetta, and A. Blais, Phys. Rev. A 79, 013819 (2009).
  * Gambetta  _et al._ (2006) J. Gambetta, A. Blais, D. I. Schuster, A. Wallraff, J. M. L. Frunzio, M. H. Devoret, S. M. Girvin, and R. J. Schoelkopf, Phys. Rev. A 74, 042318 (2006).
  * Reed  _et al._ (2010) M. D. Reed, B. R. Johnson, A. A. Houck, L. DiCarlo, J. M. Chow, D. I. Schuster, L. Frunzio, and R. J. Schoelkopf, Phys. Rev. Lett. 96, 203110 (2010).


# Monochromatic Sums and Products over $\mathbb{Q}$

Ryan Alweiss
Department of Pure Mathematics and Mathematical Statistics and Trinity
College, University of Cambridge. Email<EMAIL_ADDRESS>Research supported
by an NSF Mathematical Sciences Postdoctoral Fellowship.

###### Abstract

Hindman’s finite sums theorem states that in any finite coloring of the
naturals, there is an infinite sequence all of whose finite subset sums are
the same color. In 1979, Hindman showed that there is a finite coloring of the
naturals so that no infinite sequence has all of its pairwise sums and
pairwise products the same color. Hindman conjectured that for any $n$, a
finite coloring of the naturals contains $n$ numbers all of whose subset sums
and subset products are the same color. In this paper we prove the version of
this statement where we color the rationals instead of the integers. In other
words, we show that the pattern $\\{\sum_{i\in S}x_{i},\prod_{i\in
S}x_{i}\\}$, where $S$ ranges over all nonempty subsets of $[n]$, is partition
regular over the rationals.

##  1 Introduction

One of the oldest results in Ramsey theory is Schur’s theorem [22], from 1916.
In order to prove that Fermat’s last theorem is false in the integers mod $p$,
Schur proved that in any finite coloring of $\mathbb{N}$, there are some $x$
and $y$ so that $x,y,x+y$ all receive the same color, i.e. that the pattern
$\\{x,y,x+y\\}$ is _partition regular_. Just over a decade later, van der
Waerden proved that in a finite coloring of $\mathbb{N}$, there are
arbitrarily long monochromatic arithmetic progressions. A few short years
later, Rado [21] proved a far-reaching generalization of the theorems of Schur
and van der Waerden, fully characterizing which linear systems of equations
are partition regular. One important corollary of Rado’s theorem is the so-
called Folkman’s theorem, which generalizes Schur’s theorem. Folkman’s theorem
states that in any finite coloring of $\mathbb{N}$, for any $n\geq 2$ there
are $x_{1},\cdots,x_{n}$ so that $\sum_{i\in S}x_{i}$ are all the same color,
for any nonempty subset $S$ of $[n]$. Note that $n=2$ case is just Schur’s
result. In 1974, Hindman [11] proved an infinitary version of Folkman’s
theorem. Hindman’s Theorem states that in any finite coloring of $\mathbb{N}$,
some color class contains an _infinite_ sequence and all its finite subset
sums. In some sense, this is the best possible. The infinitary versions of the
generalizations of Folkman’s theorem that follow from Rado’s theorem are
false. For instance, one cannot also ask for terms of the form $2x_{i}+x_{j}$
to be the same color (see [8], [17]). While the partition regularity of linear
equations is well understood, even in the infinitary case, very little is
known about the partition regularity of polynomial equations in general (see
e.g. [3], [2]). The general problem of determining whether or not such a
system is partition regular is not even known to be decidable.

However, there is one class of polynomial equations for which partition
regularity is very well understood. By considering powers of $2$, one can
easily characterize partition regularity for linear systems in $\log(x_{i})$.
For instance, one can quickly deduce the product version of Folkman’s theorem.
In any finite coloring of $\mathbb{N}$, for any $n\geq 2$ there are
$x_{1},\cdots,x_{n}$ so that $\prod_{i\in S}x_{i}$ are all the same color, for
any nonempty subset $S$ of $[n]$.

Combining addition and multiplication, which is needed to make progress on the
partition regularity of polynomials in general, is far more difficult. Perhaps
the most natural questions of this form concern the simplest common
generalization of the partition regularity of linear equations and their
multiplicative versions.

Over $\mathbb{N}$, Hindman showed [14] that one cannot even ask for an
infinite sequence all of whose _pairwise_ sums and products are monochromatic.
Thus, the natural common generalization of Hindman’s Theorem and its
multiplicative version is false. This has implications in the theory of
ultrafilters; from this it follows that there is no ultrafilter
$p\in\beta\mathbb{N}$ with $p+p=p\cdot p$ ([18], Corollary 17.17). Recently,
Hindman, Ivan, and Leader gave a new construction [15] of a coloring without
such an infinite sequence and made substantial progress toward disproving the
infinitary version of the same statement over $\mathbb{Q}$.

In the 1970’s, Hindman [13] asked about “the natural finite version of the
main sums and products problem”, i.e. the natural common generalization of the
additive and multiplicative forms of Folkman’s theorem. He has repeated this
conjecture on a number of occasions (see e.g. Question 17.18 of [18]).

######  Conjecture 1.1.

For any $n\geq 2$, if $\mathbb{N}$ is colored in finitely many colors, there
exist $x_{1},\cdots,x_{n}$ such that all the numbers $\sum_{i\in S}x_{i}$ and
$\prod_{i\in S}x_{i}$, for nonempty $S\subset[n]$, are the same color.

Hindman [12] has also conjectured the weaker form of this, where we are
coloring $\mathbb{Q}$ instead of $\mathbb{N}$.

######  Conjecture 1.2.

For any $n\geq 2$, if $\mathbb{Q}$ is colored in finitely many colors, there
exist $x_{1},\cdots,x_{n}$ such that all the numbers $\sum_{i\in S}x_{i}$ and
$\prod_{i\in S}x_{i}$, for nonempty $S\subset[n]$, are the same color.

1.1 is one of the most important and longstanding conjectures in partition
regularity, and very little is known about it. In particular, what Hindman
[13] calls “the simplest special case” of $n=2$, i.e. the partition regularity
of $\\{x,y,x+y,xy\\}$ over $\mathbb{N}$, has been highlighted several times
since Hindman made some numerical computations for it in his original paper,
and is still open (see e.g. [9], [16]). Moreira [20] made substantial progress
on this problem, showing that $\\{x,x+y,xy\\}$ is partition regular over
$\mathbb{N}$. Over $\mathbb{Q}$, the special case $n=2$ was settled recently
in an exciting work of Bowen and Sabok [6], who proved that $\\{x,y,x+y,xy\\}$
is partition regular over $\mathbb{Q}$. Before [6], there had also been
previous work on the partition regularity of $\\{x,y,x+y,xy\\}$ over fields
([7], [10], [23]). By a standard compactness argument (see [6]), the partition
regularity of a polynomial pattern over arbitrary fields follows from its
partition regularity over $\mathbb{Q}$, and so [6] subsumes all of this
previous work. For $n>2$, however, 1.2 was open. Over both the setting of
fields like $\mathbb{Q}$ and the setting of $\mathbb{N}$, the existing results
were all very specialized to the case $n=2$, and there were no nontrivial
results about what happens in the general case.

In this paper, we completely settle 1.2.

######  Theorem 1.3.

For any $n\geq 2$, if $\mathbb{Q}$ is colored in finitely many colors, we can
find some $x_{1},\cdots,x_{n}\neq 0$ such that all the numbers $\sum_{i\in
S}x_{i}$ and $\prod_{i\in S}x_{i}$, for nonempty $S\subset[n]$ are the same
color.

In addition to proving a much more general result, our proof has several key
differences from the result of Bowen and Sabok. Unlike Bowen and Sabok’s, our
proof does not use the result of Moreira as a black box. Instead, it is
similar in spirit to the author’s proof of Moreira’s result [1]. Furthermore,
in contrast to their proof, and like [1], our proof obtains explicit bounds.
Additionally, like in [1] and unlike the works of Moreira and Bowen-Sabok, we
use the polynomial van der Waerden theorem. We believe that something like
this is necessary to resolve 1.1, as explained in Conjecture $4.1$ of [1].

##  2 Preliminaries

Throughout this paper, the notation $M^{\prime}\gg M$ means that $M^{\prime}$
is a sufficiently large function of $M$. This notation will be useful for us,
because we will often need a large “reservoir” to ensure that we can apply
standard Ramsey theoretic results. For convenience, we define the following
notations:

######  Definition 2.1.

The size $s(x)$ of a rational number $x=\frac{a}{b}$ with $\gcd(a,b)=1$ is
$\max(|a|,|b|)$.

######  Definition 2.2.

A _good polynomial_ $P(x_{0};x_{1},\cdots,x_{n})$ is a rational linear
combination of $x_{0},\ldots,x_{n}$ with nonzero $x_{0}$ coefficient. The
_size_ $s(P)$ of a good polynomial is the maximum size of its coefficients and
the _leading term_ of a good polynomial is the coefficient $c_{0}$ of $x_{0}$.
For a good polynomial $P$, we also define the notation
$P(x_{0}):=P(x_{0},0,\cdots,0)=c_{0}x_{0}$.

The size condition is a technical condition to make our proof finitary. In
order to prevent division by $0$, we could work over $\mathbb{Q}^{+}$. In that
setting, the same proof we present here would go through, with the only change
being that we would consider only polynomial expressions with nonnegative
coefficients. In general, partition regularity over
$\mathbb{Q}\setminus\\{0\\}$ is equivalent to partition regularity over
$\mathbb{Q}^{+}$. Simply double the number of colors, coloring each $-x$ a
“negative shade” of the color of $x$. In the exact same way, partition
regularity over $\mathbb{Z}$ is equivalent to partition regularity over
$\mathbb{N}$. For the sake of elegance and to be consistent with notations in
the existing literature, we choose to present the proof over $\mathbb{Q}$,
glossing over this minor technical detail and implicitly assuming variables
are nonzero when necessary; there will always be more than enough freedom to
pick them so that this is the case.

One key ingredient in our proof is the _polynomial van der Waerden theorem_ ,
originally due to Bergelson and Leibman [5] (see also Walters [25] for a
combinatorial proof).

######  Theorem 2.3 (Polynomial van der Waerden).

Given a finite coloring of $\mathbb{Z}^{\ell}$, and some integer valued
polynomials $p_{1},\cdots,p_{k}$ with $p_{i}(0)=0$, for any vectors
$v_{1},\cdots,v_{k}$ there exist $n\in\mathbb{Z}$ and $u\in\mathbb{Z}^{\ell}$
so that $u+p_{i}(n)v_{i}$ for $i\leq k$ are all the same color.

The _multidimensional polynomial van der Waerden theorem_ (which follows for
instance from [5], Theorem $B$) also guarantees that
$u+P(p_{1}(n)v_{1},\cdots,p_{k}(n)v_{n})$ are the same color, for any linear
combination $P$ of $p_{1}(n)v_{1},\cdots,p_{k}(n)v_{k}$ with $s(P)\leq M$. An
easy scaling argument shows the same conclusion holds in $\mathbb{Q}^{\ell}$
or $\mathbb{Q^{+}}^{\ell}$.

######  Theorem 2.4 (Multidimensional Polynomial van der Waerden).

Given a finite coloring of $\mathbb{Q}^{\ell}$, and some integer valued
polynomials $p_{1},\cdots,p_{k}$ with $p_{i}(0)=0$, for any vectors
$v_{1},\cdots,v_{k}$ there exist $n\in\mathbb{Z}$ and $u\in\mathbb{Q}^{\ell}$
so that $u+\sum c_{i}p_{i}(n)v_{i}$ for $i\leq k$ are all the same color, as
the $c_{i}$’s range over all rational numbers with numerators and denominators
at most $M$.

##  3 The $n=3$ case of the main lemma

Before stating and proving our main lemma, we will prove the $n=3$ case, which
captures all of the main ideas.

######  Proposition 3.1.

For any finite coloring of $\mathbb{Q}$ and any size bound $M$, there exist
$a,b,c\in\mathbb{Q}$ so that for any good $P$ with $s(P)\leq M$:

  1. 1.

The color of $P(a;b,c,bc)$ is the same as that of $P(a)$.

  2. 2.

The color of $P(ac;b)$ is the same as that of $P(ac)$.

  3. 3.

The color of $P(b;c)$ is the same as that of $P(b)$.

###### Proof.

Let $\chi_{1}$ denote the initial coloring of $\mathbb{Q}$. We start by
finding $b,c$ so that the third condition holds. Consider the auxiliary
product coloring

$\chi_{2}(x)=(\chi_{1}(Kx))_{K\in\mathbb{Q},s(K)\leq M_{0}}$

for some huge $M_{0}\gg M_{1}\gg M_{2}\gg M_{3}=M$ that we choose with
foresight. We may take $M$ to be a sufficiently large function of the number
of colors used in the initial coloring, and we maintain this choice throughout
the remainder of the paper. This suppresses dependencies on the number of
colors in the original coloring.

By van der Waerden, we can find $b,c$ so that $\chi_{2}(b+P(c))=\chi_{2}(b)$
for all linear combinations $P$ with $s(P)\leq M_{0}$. Thus in the initial
coloring, if $M_{0}\gg M_{1}$, then the color of $P(b;c)$ will be the same as
that of $P(b)$, if $s(P)\leq M_{1}$. This is because as long as $M_{0}\gg
M_{1}$ (and in fact as long as $M_{1}\leq\sqrt{M_{0}}$), then if
$P(b;c)=k_{1}b+k_{2}c$ has $s(P)\leq M_{1}$, then $\frac{k_{2}}{k_{1}}$ and
$k_{1}$ will have size at most $M_{0}$. Hence,
$\frac{k_{1}b+k_{2}c}{k_{1}}=b+\frac{k_{2}}{k_{1}}c$ will have the same color
in $\chi_{2}$ as $b=\frac{k_{1}b}{k_{1}}$. Since $k_{1}$ also has size at most
$M_{0}$, by the construction of $\chi_{2}$, we have that $k_{1}b+k_{2}c$ has
the same color as $k_{2}b$ in the initial coloring.

We now aim to satisfy the first two conditions, which we will do by finding an
appropriate $a$, scaling $b$ and $c$ in the process. In order to satisfy the
second condition, we will need to “shift” $ac$ by $b$ (i.e. rename $a$ so that
$ac$ is replaced by $ac+b$), and to do so we will have to shift $a$ by
$\frac{b}{c}$. Similarly when we shift $a$ by $b,c,bc$, we end up shifting
$ac$ by $bc,c^{2},bc^{2}$. Thus we will actually end up showing something a
little stronger.

  1. 1.

The color of $P(a;\frac{b}{c},b,c,bc)$ is the same as that of $P(a)$.

  2. 2.

The color of $P(ac;b,bc,c^{2},bc^{2})$ is the same as that of $P(ac)$.

  3. 3.

The color of $P(b;c)$ is the same as that of $P(b)$.

Next, we will find $a$ that satisfies the first condition. We will apply
multidimensional polynomial van der Waerden (Theorem 2.4) in $\mathbb{Z}^{4}$.
Consider the coloring $\chi_{3}$ of $\mathbb{Z}^{4}$ given by

$\chi_{3}(q_{1},q_{2},q_{3},q_{4})=\mathcal{X}_{2}(a+q_{1}\frac{b}{c}+q_{2}b+q_{3}c+q_{4}bc)$

where $\mathcal{X}_{2}$ is defined just as $\chi_{2}$ is, with a size
parameter $M^{\prime}$ in place of $M_{0}$ so that $M_{2}\ll M^{\prime}\ll
M_{1}$. The use of this different size parameter is just so that the product
coloring does not use too many different colors.

With $v_{i}$ being the $i$-th canonical basis vector in $\mathbb{Z}^{4}$ and
$(p_{1},p_{2},p_{3},p_{4})=(n,n^{2},n,n^{3})$, multidimensional polynomial van
der Waerden gives us some $u=(u_{1},u_{2},u_{3},u_{4})\in\mathbb{Z}^{4}$ and
some $k\in\mathbb{N}$ such that the numbers

$u_{1}\frac{b}{c}+u_{2}b+u_{3}c+u_{4}bc+\ell_{1}k\frac{b}{c}+\ell_{2}k^{2}b+\ell_{3}kc+\ell_{4}k^{3}bc,$

for $\ell_{i}$ with $s(\ell_{i})\leq M_{2}\ll M^{\prime}\ll M_{1}$, all
receive the same color under $\mathcal{X}_{2}$. Replacing $b,c$ with
$k^{2}b,kc$ respectively, we have that the color of

$K(a+\ell_{1}\frac{b}{c}+\ell_{2}b+\ell_{3}c+\ell_{4}bc)$

is the same as the color of $Ka$ as long as $s(K)\leq M_{2}$. This renaming of
$b,c$ (by $\frac{b}{k^{2}},\frac{c}{k}$) preserves the third condition, with
$M_{1}$ replaced by $M_{2}\ll M^{\prime}\ll M_{1}$.

It remains to satisfy the second condition. We will again apply Theorem 2.4 in
$\mathbb{Z}^{4}$ to an appropriate auxiliary coloring. This time, we use the
coloring

$\chi_{4}(q_{1},q_{2},q_{3},q_{4})=\mathfrak{X}_{2}(c(a+v_{1}\frac{b}{c}+v_{2}b+v_{3}c+v_{4}bc))$

where again $\mathfrak{X}_{2}$ is an appropriate product coloring.

Applying Theorem 2.4 with
$(p_{1},p_{2},p_{3},p_{4})=(n^{2},n^{3},n^{2},n^{4})$ and $v_{i}$ again being
the $i$-th canonical basis vector yields some
$u=(u_{1},u_{2},u_{3},u_{4})\in\mathbb{Z}^{4}$ and some $k\in\mathbb{N}$ (both
very small compared to $M_{2}$) such that the numbers

$c(a+u_{1}\frac{b}{c}+u_{2}b+u_{3}c+u_{4}bc+\ell_{1}k^{2}\frac{b}{c}+\ell_{2}k^{3}b+\ell_{3}k^{2}c+\ell_{4}k^{4}bc),$

all receive the same color under $\mathfrak{X}_{2}$, when the $\ell_{i}$’s
have size at most $M_{3}$. Replace $a$ with

$a+u_{1}\frac{b}{c}+u_{2}b+u_{3}c+u_{4}bc,$

so that the numbers

$c(a+\ell_{1}k^{2}\frac{b}{c}+\ell_{2}k^{3}b+\ell_{3}k^{2}c+\ell_{4}k^{4}bc)$

are all the same color under $\chi_{2}$. Finally, replace $(a,b,c)$ with
$(\frac{a}{k},k^{2}b,kc)$ so that the numbers

$c(a+\ell_{1}\frac{b}{c}+\ell_{2}b+\ell_{3}c+\ell_{4}bc)$

all receive the same color. Because $k\ll M_{2}$, this preserves the first and
third conditions. ∎

##  4 Statement and proof of the main lemma

In this section, we state and prove the main lemma in its full generality.

######  Proposition 4.1.

For any finite coloring of $\mathbb{Q}$, any $M$, and any integer $n\geq 2$,
there are numbers $x_{n},x_{n-1},x_{n-2},\cdots,x_{1}\in\mathbb{Q}$ so that
for each $i\in[n]$ and $T\subseteq[i-1]$ (possibly empty), if $P$ is good with
$s(P)\leq M$ then the color of $P(x_{i}\prod_{j\in T}x_{j};\prod_{j\in
S}x_{j})$ is the same as that of $P(x_{i}\prod_{j\in T}x_{j})$, where $S$
ranges over all nonempty subsets of $[i-1]$ with all elements larger than all
elements of $T$.

By $P(x_{i}\prod_{j\in T}x_{j};\prod_{j\in S}x_{j})$ as $S$ ranges, we mean
$P(x_{i}\prod_{j\in T}x_{j};\prod_{j\in S_{1}}x_{j},\prod_{j\in
S_{2}}x_{j},\cdots)$ over all possible choices $S_{1},S_{2},\cdots$ of $S$.

The $n=3$ case (which generalizes the $n=2$ case) was proved in the previous
section with $x_{3}=a,x_{2}=b,x_{1}=c$.

###### Proof.

We use induction on $n$ to prove the proposition in general. Say we have
proved it for $x_{n-1},x_{n-2},\cdots,x_{2},x_{1}$.

The first order of business is to establish the proposition for the case where
$i=n$ and $T=\emptyset$, i.e. for $P(x_{n};\prod_{j\in S}x_{j})$, where $S$
ranges over all nonempty subsets of $[n-1]$.

With foresight to the term $P(x_{i}\prod_{j\in T}x_{j};\prod_{j\in S}x_{j})$,
we must in fact establish the conclusion of the proposition for expressions of
the form

$P\left(x_{n};\frac{\prod_{j\in S}x_{j}}{\prod_{j\in T}x_{j}}\right)$

where $S,T$ range so that all elements of $S$ are less than $n$ and greater
than all elements of $T$, and $S$ is nonempty.

Because of this, we must also control all terms of the form

$P\left(x_{n}\prod_{j\in T^{\prime}}x_{j};\frac{\prod_{j\in S}x_{j}\prod_{j\in
T^{\prime}}x_{j}}{\prod_{j\in T}x_{j}}\right)$

for nonempty $S$, where all elements of $S$ are less than $n$ and greater than
all elements of $T$, and $T^{\prime}$ is some subset of $[n-1]$.

First, we focus on the case where $T^{\prime}$ is empty. In the usual
auxiliary coloring (with some size parameter $M$), use Theorem 2.4 to pick
$x_{n}$ to be a linear combination of $\frac{\prod_{j\in S}x_{j}}{\prod_{j\in
T}x_{j}}$ so that the color of

$P\left(x_{n};k^{\sum_{j\in S}y_{j}-\sum_{j\in T}y_{j}}\frac{\prod_{j\in
S}x_{j}}{\prod_{j\in T}x_{j}}\right)$

is the same as that of $P(x_{n})$, for some integer $k$ that depends on the
(smaller) choice of $M$ and this stage, and some sufficiently fast-growing
sequence $y_{j}$ (for instance, $y_{j}=2^{j}$ suffices). Crucially, the
exponent $\sum_{j\in S}y_{j}-\sum_{j\in T}y_{j}$ will be positive.

Replace $x_{j}$ by $x_{j}k^{y_{j}}$ and scale so that the color of

$P\left(x_{n};\frac{\prod_{j\in S}x_{j}}{\prod_{j\in T}x_{j}}\right)$

is the same as that of $P(x_{n})$.

With this out of the way, we now handle the other choices of $T^{\prime}$ one
by one, in an arbitrary order. For some choices $T_{1},T_{2}$ of $T^{\prime}$
where the $T_{1}$ term has been dealt with, to ensure that

$P\left(x_{n}\prod_{j\in T_{2}}x_{j};\frac{\prod_{j\in S}x_{j}\prod_{j\in
T_{2}}x_{j}}{\prod_{j\in T}x_{j}}\right)$

all the same color, we shift $x_{n}$ by an appropriate linear combination of
$\frac{\prod_{j\in S}x_{j}}{\prod_{j\in T}x_{j}}$ (i.e. replace $x_{n}$ by the
sum of $x_{n}$ and this combination) so that we can replace $x_{j}$ by
$x_{j}k^{y_{j}}$ for appropriate $k$ and $x_{j}$ using Theorem 2.4. This may
scale the first term $x_{n}\prod_{j\in T_{2}}x_{j}$, but to deal with that we
may simply divide $x_{n}$ by the appropriate power of $k$. Such shifting and
scaling will only modestly reduce the size bound in the earlier terms

$P\left(x_{n}\prod_{j\in T_{1}}x_{j};\frac{\prod_{j\in S}x_{j}\prod_{j\in
T_{1}}x_{j}}{\prod_{j\in T}x_{j}}\right).$

In this way, we can handle the remaining terms $P(x_{i}\prod_{j\in
T}x_{j};\prod_{j\in S}x_{j})$ in any order.∎

##  5 Proof of main result

We will prove the main result in four steps.

Step I: By Proposition 4.1, for arbitrary $n$, we can find
$x_{n},\cdots,x_{1}$ so that $x_{i}\prod_{j\in T}x_{j}+P(\prod_{j\in S}x_{j})$
is the same color as $x_{i}\prod_{j\in T}x_{j}$, whenever $P$ has coefficients
$0$ and $1$, and $S$ ranges over nonempty subsets of $[n]$ with all elements
smaller than $i$ and larger than all elements of $T$ (with $i,T$ as in
Proposition 4.1).

Step II: By Ramsey’s theorem, we can pass to a subset of the $x_{i}$ so that
the color of $x_{i_{1}}x_{i_{2}}\cdots x_{i_{k}}$ depends only on $k$ as long
as $k\leq N$ for some large $N$ (for instance by doing this first for $k=1$,
then $k=2$, and so on). Renaming variables, we can find $x_{m},\cdots,x_{1}$
so that for all $k\leq N$, there is a color $f(k)$ so that $x_{i}\prod_{j\in
T}x_{j}+P(\prod_{j\in S}x_{j})$ has color $k=|T|+1$, where $S$ ranges over
nonempty subsets of $[m]$ with all elements smaller than $i$ and larger than
the smallest element of $T$, and $P$ is any linear combination with
coefficients $0$ and $1$. We can make $m$ arbitrary large, by taking
arbitrarily large $n$ in the lemma. In particular, the color of every monomial
$x_{i}\prod_{j\in T}x_{j}$ will only depend on its degree as long as the
degree is not too large. Again, by starting with a large $n$, we may make $N$
an arbitrarily large function of the number of colors in the initial coloring.

Step III: By Folkman’s Theorem, if $N$ is large enough, we may find
$a_{1},\cdots,a_{r}$ so that the numbers $\sum_{i\in S}a_{i}$ are all the same
color for any nonempty $S\subset[r]$ (where $r\ll m\ll n$, and $r$ goes to
infinity with $n$). In the present case, for any $a\in[N]$ we can color $a$ by
the product of $a$ polynomials, and so we can find such $a_{i}$ so that all
monomials of degree $\sum_{i\in S}a_{i}$ will be the same color (say, red), as
long as $\sum_{i\in S}a_{i}\leq N$.

Step IV: Now, pick sets $S_{i}$ of size $a_{r}-1$ for $1\leq i\leq r$, so that
everything in $S_{i+1}$ is larger than everything in $S_{i}$, and everything
in $S_{r}$ is less than $\frac{n}{2}$. For $1\leq i\leq r$, let
$X_{i}=x_{n-i}\prod_{j\in S_{i}}x_{j}$. Any product of a subset $S$ of the
$X_{i}$ will be a monomial of degree $\sum_{i\in S}a_{i}$ and so will be red
by our application of Folkman’s theorem. Any sum of a subset
$S=\\{X_{i_{1}},\cdots,X_{i_{s}}\\}$ of the $X_{i}$, for
$i_{1}>i_{2}>\cdots>i_{s}$ will be of the form

$x_{n-i_{s}}\prod_{j\in S_{i_{s}}}x_{j}+P(x_{n-i_{t}}\prod_{j\in
S_{i_{t}}}x_{j})$

where $t\in[s-1]$ ranges, and $P$ is a polynomial with coefficients $0$ and
$1$. Hence, it will be red by our assumption. Renaming $X_{i}$ as $x_{i}$
finishes the proof.

##  6 Discussions and open problems

We first note that our proof can be extended to give some more patterns. The
argument in this paper straightforwardly also gets $\sum_{i\leq m}\prod_{j\in
S_{i}}x_{j}$ to be monochromatic, as long as all of the elements of $S_{i}$
are less than all of the elements of $S_{i+1}$. Furthermore, Bowen-Sabok [6]
ask about patterns like $\\{x+P(y),y,xy\\}$ where $P$ ranges over a finite set
of monomials. This can also be handled with the argument here. If we consider
expressions like $P(x_{i}\prod_{j\in T}x_{j};\prod_{j\in S}x_{j},\prod_{j\in
S}x_{j}^{2})$ and not just $P(x_{i}\prod_{j\in T}x_{j};\prod_{j\in S}x_{j})$,
the partition regularity of $\\{x,x+y,x+y^{2},y,xy\\}$ over $\mathbb{Q}$
follows. By considering slightly more general $(S,T)$ with $\max_{j\in
S}j>\max_{j\in T}j$, Hunter [19] proved the partition regularity over
$\mathbb{Q}$ of sums of distinct products, i.e. $\sum_{S}\prod_{i\in S}x_{i}$
so that no $i$ appears in more than one $S$. It is also worth noting that in
an earlier paper [1], the author proved that $\\{x,x+y,xy\\}$ was partition
regular over $\mathbb{N}$ with effective bounds, essentially by considering a
more restricted family of $(S,T)$ than the ones we consider here, so that the
argument goes through over the integers.

Interestingly, our proof here implicitly also proves that expressions like
$\\{DX,DX+D^{2}\\}$ are partition regular over $\mathbb{Q}$, and also over
$\mathbb{N}$. To our knowledge, this is an original result, although the proof
is simple. We include a self-contained explicit proof here.

###### Proof.

Fix two integers $x,d$, both divisible by $N!$ for a sufficiently large $N$.
Given a coloring $f$ of the integers, define a new coloring $g$ of the
integers by $g(C)=f(dx+Cd^{2})$. By Theorem 2.4, with the polynomial
$p(n)=n^{2}$, we can find not-that-large $k$ and $c$ so that
$g(k)=g(k+c^{2})$. By the definition of $g$, this means that
$f(dx+kd^{2})=f(dx+(k+c^{2})d^{2})=f(dx+kd^{2}+(cd)^{2})$. Letting
$x/c+kd/c=X$ and $cd=D$, we get $DX,DX+D^{2}$ are the same color. ∎

A very similar argument shows that
$\\{Q(d)(X+P_{1}(d)),\cdots,Q(d)(X+P_{k}(d))\\}$ is partition regular for any
monic polynomial $Q$ and any family of $P_{i}$ with $P_{i}(0)=0$. One
interesting question, which we suspect would help prove 1.1, is whether we can
extend this argument to deal with $Q(d)(X+P_{i}(d))$ where $Q(0)=0$ but $Q$ is
not necessarily monic.

######  Question 6.1.

Let $Q,P_{1},\cdots,P_{k}$ be polynomials with $Q(0)=0$ and $P_{i}(0)=0$ for
all $1\leq i\leq k$. Is the pattern
$\\{Q(d)(x+P_{1}(d)),\cdots,Q(d)(x+P_{k}(d))\\}$ necessarily partition
regular?

Another interesting question is whether we can have a version of Theorem 2.4
where we allow negative exponents on $d$. We feel that the following statement
should be true.

######  Conjecture 6.2.

The pattern $\\{x,x+d,x+\frac{1}{d},x+d+\frac{1}{d}\\}$ is partition regular
over $\mathbb{Q}$.

Even without the $x+d+\frac{1}{d}$ term, this is not known.

######  Conjecture 6.3.

The pattern $\\{x,x+d,x+\frac{1}{d}\\}$ is partition regular over
$\mathbb{Q}$.

There is hope to use the ideas in this paper to tackle 1.1, where even the
simple case $n=2$ is open, and indeed Conjecture 4.1 of [1] essentially states
that we must use them. We restate Conjecture 4.1 of [1] here, for
completeness. We need the following notation, which also appears in [1].

######  Definition 6.4.

Let $P_{d}$ be the set of polynomials of a countable set of variables
$x_{1},\cdots,x_{n}$ with nonnegative integer coefficients, constant term, and
degree at most $d$ in each variable.

Conjecture $4.1$ of [1] states the following.

######  Conjecture 6.5.

For all $d\in\mathbb{N}$, the pattern $\\{x,y,x+y,xy\\}$ is not partition
regular over $P_{d}$.

Even the weaker version, that 1.1 is false over $P_{d}$, is not known.

######  Conjecture 6.6.

There exists $n$ so that for all $d\in\mathbb{N}$, the pattern $\\{\sum_{i\in
S}x_{i},\prod_{i\in S}x_{i}\\}$, where $S$ ranges over nonempty subsets of
$[n]$, is not partition regular over $P_{d}$.

The main issue with applying the arguments from this paper straightforwardly
to solve 1.1 is that we cannot simply “shift” $ac$ by $b$; this is because $b$
generally will not be divisible by $c$. In other words, to control the color
of $P(ac;b)$, we would have to control the color of $P(a;\frac{b}{c})$.
However, if $b,b+c,b+2c$ are the same color, we cannot guarantee $c\mid b$.
The $2$-coloring where $3^{x}y$ for $y\equiv 1\mod 3$ is red and $3^{x}y$ for
$y\equiv 2\mod 3$ is blue does not have three consecutive multiples of any
integer $c$ in the same color. Still, we believe it may be possible to get
around this obstacle by considering more general polynomial expressions such
as the aforementioned $Q(d)(x+P_{i}(d))$ for general $Q$ with
$Q(0)=P_{i}(0)=0$.

##  7 Acknowledgments

Thanks to Noga Alon, Matthew Bowen, Matija Bucic, Timothy Gowers, Neil
Hindman, Zach Hunter, Maria Ivan, Noah Kravitz, Imre Leader, Shachar Lovett,
and Marcin Sabok for helpful comments.

## References

  * [1] R. Alweiss, Monochromatic Sums and Products of Polynomials, available online at https://arxiv.org/abs/2211.00766
  * [2] J.M. Barrett, M. Lupini, and J. Moreira, On Rado Conditions for Nonlinear Diophantine Equations. European Journal of Combinatorics, 94 (2021), 103277.
  * [3] V. Bergelson, Ergodic Ramsey Theory-an update. Volume 228 of London Math. Soc. Lecture Note Ser, (1996), 1–61.
  * [4] V. Bergelson, N. Hindman and I. Leader, Additive and multiplicative Ramsey theory in the reals and the rationals. Journal of Combinatorial Theory, Series A, 85 (1999), 41–68.
  * [5] V. Bergelson and A. Leibman, Polynomial extensions of van der Waerden’s and Szemerédi’s theorems. J. Amer. Math. Soc., 9(3) (1996), 725–753.
  * [6] M. Bowen and M. Sabok, Monochromatic Sums and Products in the Rationals, available online at https://arxiv.org/abs/2210.12290
  * [7] J. Cilleruelo, Combinatorial problems in finite fields and sidon sets, Combinatorica 32 (2012), no. 5, 497–511.
  * [8] W. Deuber, N. Hindman, I. Leader, and H. Lefmann, Infinite partition regular matrices, Combinatorica 15 (1995), 333–355.
  * [9] B. Green, Some Open Problems. Manuscript, available on request.
  * [10] B. Green and T. Sanders, Monochromatic sums and products, Discrete Analysis (2016), 613.
  * [11] N. Hindman, Finite sums from sequences within cells of a partition of N. Journal of Combinatorial Theory, Series A, 17 (1974), 1—11.
  * [12] N. Hindman, Personal Communication.
  * [13] N. Hindman, Partitions and sums and products of integers, Trans. Amer. Math. Soc. 247 (1979), 227–245.
  * [14] N. Hindman, Partitions and pairwise sums and products. Journal of Combinatorial Theory, Series A, 37 (1984), 46-60.
  * [15] N. Hindman, M. Ivan, and I. Leader, Some New Results on Monochromatic Sums and Products in the Rationals, New York Journal of Mathematics, Volume 29 (2023), 301 – 322
  * [16] N. Hindman, I. Leader, and D. Strauss, Open Problems in Partition Regularity. Comb., Prob., Comput., 12(5-6), (2003), 571–583.
  * [17] N. Hindman, I. Leader, and D. Strauss, Maximality of Infinite Partition Regular Matrices, Transactions of the AMS, 355(3), (2002), 1213–1235
  * [18] N. Hindman and D. Strauss, Algebra in the Stone-Cech compactification, De Gruyter Textbook, Walter de Gruyter & Co., Berlin, (2012), Theory and applications, Second revised and extended edition.
  * [19] Z. Hunter, On a Method of Alweiss, available online at https://arxiv.org/abs/2308.10749
  * [20] J. Moreira, Monochromatic sums and products in N, Ann. Math. (3) 185, (2017), 1069–1090.
  * [21] R. Rado, Studien zur kombinatorik. Math. Zeit., 36, (1933), 242–280.
  * [22] I. Schur, Über die kongruenz $x^{m}+y^{m}=z^{m}$ (mod $p$). Jahresbericht der Deutschen Math Verein., 25 (1916), 114–117.
  * [23] I. D. Shkredov, On monochromatic solutions of some nonlinear equations in $\mathbb{Z}/p\mathbb{Z}$. Mat. Zametki, 88(4), (2010), 625–634.
  * [24] B.L. van der Waerden, Beweis einer Baudetschen Vermutung. Nieuw. Arch. Wisk. 15 (1927), 212–216.
  * [25] M. Walters, Combinatorial proofs of the polynomial van der Waerden theorem and the polynomial Hales–Jewett theorem. Journal of the London Math. Soc., 61(1), (2000), 1–12.


# Composition operators on function spaces on the halfplane: spectra and
semigroups

I. Chalendar  Isabelle Chalendar, Université Gustave Eiffel, LAMA, (UMR 8050),
UPEM, UPEC, CNRS, F-77454, Marne-la-Vallée (France)  isabelle.chalendar@univ-
eiffel.fr and  J.R. Partington  Jonathan R. Partington, School of Mathematics,
University of Leeds, Leeds LS2 9JT, UK<EMAIL_ADDRESS>
###### Abstract.

This paper considers composition operators on Zen spaces (a class of weighted
Bergman spaces of the right half-plane related to weighted function spaces on
the positive half-line by means of the Laplace transform). Generalizations are
given to work of Kucik on norms and essential norms, to work of Schroderus on
(essential) spectra, and to work by Arvanitidis and the authors on semigroups
of composition operators. The results are illustrated by consideration of the
Hardy–Bergman space; that is, the intersection of the Hardy and Bergman
Hilbert spaces on the half-plane.

MSC class: 30H10, 30H20, 47B33, 47D03

Keywords: Composition operator, Hardy space, Bergman space, spectrum,
essential spectrum, operator semigroup

##  1\. Introduction and background material

The subject of this paper involves properties of composition operators on
holomorphic function spaces on the right half-plane ${\mathbb{C}}_{+}$, both
as individual operators and as elements of one-parameter semigroups. One
difficulty, even in the case of the Hardy space $H^{2}({\mathbb{C}}_{+})$, is
that not all composition operators on the spaces are bounded, although in many
cases the bounded operators have been characterised, as we explain below. The
literature on semigroups of composition operators is not as extensive as it is
for the disc, although it has been an object of study since the work of
Berkson and Porta [5].

In this note we shall concentrate on the so-called Zen spaces (weighted
Hardy–Bergman spaces), which form a large class of spaces with applications in
systems and control theory [11, 12]. For these information on the bounded
composition operators is also available, thanks to Kucik [13].

The remainder of this section presents necessary background material. In
Section 2 we provide new results on the norm and spectral radius of
composition operators. Then, in Section 3, we consider the spectral theory of
composition operators on the half-plane with linear fractional symbols,
extending several results of Schroderus [15]. Finally, in Section 4, we
consider semigroups of composition operators on Zen spaces, providing
generalizations of results of Arvanitidis [3].

###  1.1. Zen spaces

Kucik [13] considered composition operators on Zen spaces $A^{2}_{\nu}$, which
are isometrically Laplace transforms of weighted Hardy spaces
$L^{2}_{w}(0,\infty)=L^{2}(0,\infty,w(t)dt)$. For general background see [6],
but we provide the basic facts now.

Let $\nu$ be a positive regular Borel measure on $[0,\infty)$ satisfying the
doubling condition

$R:=\sup_{t>0}\frac{\nu[0,2t)}{\nu[0,t)}<\infty.$

The Zen space $A^{2}_{\nu}$ is defined to consist of all analytic functions
$F$ on ${\mathbb{C}}_{+}$ such that the norm, given by

$\|F\|^{2}=\sup_{\epsilon>0}\int_{\overline{{\mathbb{C}}_{+}}}|F(s+\epsilon)|^{2}\,d\nu(x)\,dy$

is finite, where we write $s=x+iy$ for $x\geq 0$ and $y\in{\mathbb{R}}$.

The best-known examples here are:

  1. (1)

For $\nu=\delta_{0}$, a Dirac mass at $0$, we obtain the Hardy space
$H^{2}({\mathbb{C}}_{+})$;

  2. (2)

For $\nu$ equal to Lebesgue measure we obtain the Bergman space
$A^{2}({\mathbb{C}}_{+})$.

Often we shall have $\nu(\\{0\\})=0$, in which case $\|F\|^{2}$ can be written
simply as

$\int_{\overline{{\mathbb{C}}_{+}}}|F(s)|^{2}\,d\nu(x)\,dy.$

######  Theorem 1.1.

[11] Suppose that $w$ is given as a weighted Laplace transform

$w(t)=2\pi\int_{0}^{\infty}e^{-2rt}\,d\nu(r),\qquad(t>0).$

Then the Laplace transform provides an isometric map

$\mathcal{L}:L^{2}(0,\infty,w(t)dt)\to A^{2}_{\nu}.$

The result also holds for Hilbert-space valued functions [1].

The following result generalizes the Elliott–Jury theorem in [8] and the
Elliott–Wynn theorem in [9].

######  Theorem 1.2.

[13] The composition operator $C_{\varphi}$ is bounded on $A^{2}_{\nu}$ if and
only if $\varphi$ has a finite nonzero angular derivative
$L=\angle\lim_{z\to\infty}z/\varphi(z)$ at $\infty$. In that case

(1) $L\inf_{t>0}\frac{w(t)}{w(Lt)}\leq\|C_{\varphi}\|^{2}\leq
L\sup_{t>0}\frac{w(t)}{w(Lt)}.$

This gives correctly $\|C_{\varphi}\|=\sqrt{L}$ in the Hardy case and
$\|C_{\varphi}\|=L$ in the Bergman case.

###  1.2. Semigroups of composition operators

Berkson and Porta [5] give the following criterion for an analytic function
$G$ to generate a one-parameter semigroup of analytic mappings of the right
half-plane ${\mathbb{C}}_{+}$ to itself; that is, solutions to

(2) $\frac{\partial\varphi_{t}(z)}{\partial
t}=G(\varphi_{t}(z)),\qquad\varphi_{0}(z)=z,$

in terms of the following condition:

(3) $x\frac{\partial(\mathop{\rm Re}\nolimits G)}{\partial x}\leq\mathop{\rm
Re}\nolimits G\qquad\hbox{on}\quad{\mathbb{C}}_{+},$

where $x=\mathop{\rm Re}\nolimits z$. The associated composition operators
$C_{\varphi_{t}}$ are bounded on $H^{2}({\mathbb{C}}_{+})$ if and only if the
non-tangential limit

$L_{t}:=\angle\lim_{z\to\infty}z/\varphi_{t}(z)$

exists and is non-zero. It will be positive, and then
$\|C_{\varphi_{t}}\|=L_{t}^{1/2}$ (see [8, 14]).

Arvanitidis [3] showed that a necessary and sufficient condition for
boundedness of the composition operators is that
$\delta:=\angle\lim_{z\to\infty}G(z)/z$ exists. In this case
$\|C_{\varphi_{t}}\|=e^{-\delta t/2}$ and the semigroup is quasicontractive.

The following result is taken from [4].

######  Theorem 1.3.

For an operator $A$ given by $Af=Gf^{\prime}$ on $D(A)\subseteq
H^{2}({\mathbb{C}}_{+})$, the following are equivalent:
(i) $A$ generates a quasi-contractive $C_{0}$-semigroup of bounded composition
operators on $H^{2}({\mathbb{C}}_{+})$;
(ii) Condition (3) holds and $\angle\lim_{z\to\infty}G(z)/z$ exists.

##  2\. The essential norm and spectral radius

We know from [13, Lem. 4] that in every $A^{2}_{\nu}$ space the normalized
reproducing kernels $k_{n}/\|k_{n}\|$ tend weakly to $0$ as $n\to\infty$. Now
we can adapt the proof that there is no compact composition operator as
follows.

Using the isometry between $L^{2}(0,\infty;w(t)\,dt)$ and $A^{2}_{\nu}$ we may
pull back the reproducing kernel of $A^{2}_{\nu}$ at a point
$\lambda\in{\mathbb{C}}_{+}$ by the inverse Laplace transform to obtain
$k_{\lambda}(t):=e^{-\overline{\lambda}t}/w(t)$ and its norm is $\langle
k_{\lambda},k_{\lambda}\rangle^{1/2}$ or

(4) $\left(\int_{0}^{\infty}\frac{e^{-2\mathop{\rm Re}\nolimits\lambda
t}}{w(t)}\,dt\right)^{1/2}.$

######  Theorem 2.1.

Let $C_{\varphi}$ be a bounded composition operator on the Zen space
$A^{2}_{\nu}$ corresponding to a weight $w$ on $(0,\infty)$. Let $L$ denote
the finite nonzero angular derivative $L=\angle\lim_{z\to\infty}z/\varphi(z)$
at $\infty$. Then

(5)
$L\inf_{t>0}\frac{w(t)}{w(Lt)}\leq\|C_{\varphi}\|_{e}^{2}\leq\|C_{\varphi}\|^{2}\leq
L\sup_{t>0}\frac{w(t)}{w(Lt)}.$

###### Proof.

For every $\delta>0$ there is a compact operator $Q$ on $A^{2}_{\nu}$ such
that

$\displaystyle\|C_{\varphi}\|_{e}+\delta$ $\displaystyle\geq\|C_{\varphi}-Q\|$
$\displaystyle\geq\limsup_{n\to\infty}\|(C_{\varphi}-Q)^{*}k_{n}\|/\|k_{n}\|$
$\displaystyle=\limsup_{n\to\infty}\|k_{\varphi(n)}\|/\|k_{n}\|$

since $Q^{*}k_{n}/\|k_{n}\|\to 0$ in norm. Since this is true for all
$\delta>0$ we have

$\|C_{\varphi}\|^{2}_{e}\geq\limsup_{n\to\infty}\left(\int_{0}^{\infty}\frac{e^{-2\mathop{\rm
Re}\nolimits\varphi(n)t}}{w(t)}\,dt\right)/\left(\int_{0}^{\infty}\frac{e^{-2nt}}{w(t)}\,dt\right)$

by (4). Hence, for every $0<M<L$ we have

$\displaystyle\|C_{\varphi}\|^{2}_{e}$
$\displaystyle\geq\limsup_{n\to\infty}\left(\int_{0}^{\infty}\frac{e^{-2nt/M}}{w(t)}\,dt\right)/\left(\int_{0}^{\infty}\frac{e^{-2nt}}{w(t)}\,dt\right)$
$\displaystyle=\limsup_{n\to\infty}\left(\int_{0}^{\infty}\frac{e^{-2un}}{w(Mu)}M\,du\right)/\left(\int_{0}^{\infty}\frac{e^{-2nt}}{w(t)}\,dt\right)$
$\displaystyle\geq M\inf_{u>0}\frac{w(u)}{w(Mu)}.$

Now, using Theorem 1.2, we have the estimate for the essential norm given in
(5). ∎

For the case of the Hardy and weighted Bergman spaces we therefore recover the
formula $\|C_{\varphi}\|=\|C_{\varphi}\|_{e}$ from [8, 9].

Note that the left and right hand sides of (1) and (5) may certainly differ,
and as an example we shall consider the Hardy–Bergman space
$H^{2}({\mathbb{C}}_{+})\cap A^{2}({\mathbb{C}}_{+})$. On the disc the Hardy
space $H^{2}({\mathbb{D}})$ is contained in the Bergman space
$A^{2}({\mathbb{D}})$ but no such inclusion either way exists on the half-
plane, as we shall now explain. We may give $H^{2}({\mathbb{C}}_{+})\cap
A^{2}({\mathbb{C}}_{+})$ a Hilbert space structure with the norm defined by

$\|f\|^{2}_{H^{2}({\mathbb{C}}_{+})\cap
A^{2}({\mathbb{C}}_{+})}=\|f\|^{2}_{H^{2}({\mathbb{C}}_{+})}+\|f\|^{2}_{A^{2}({\mathbb{C}}_{+})}.$

Since the Hardy space is isomorphic by means of the Laplace transform to
$L^{2}(0,\infty)$ and the Bergman space to $L^{2}(0,\infty,\,dt/t)$ we see
that neither space is contained in the other and that the Hardy–Bergman space
$H^{2}({\mathbb{C}}_{+})\cap A^{2}({\mathbb{C}}_{+})$ corresponds to
$L^{2}(0,\infty,w(t)\,dt)$ with the weight $w(t)=1+1/t$.

In this case $w(t)/w(Lt)=\dfrac{t+1}{t+L}$, so that the sup and inf are $1$
and $1/L$ in some order.

For the spectral radius $\rho(C_{\varphi})$ we may apply (1) to get an
estimate which is once more tight in the case of the Hardy and weighted
Bergman spaces.

######  Corollary 2.2.

For a bounded composition $C_{\varphi}$ on a space $A^{2}_{\nu}$ corresponding
to a weight $w$, the spectral radius $\rho(C_{\varphi})$ satisfies

$L\limsup_{n\to\infty}\left(\inf_{t>0}\frac{w(t)}{w(L^{n}t)}\right)^{1/n}\leq\|\rho(C_{\varphi})\|^{2}\leq
L\liminf_{n\to\infty}\left(\sup_{t>0}\frac{w(t)}{w(L^{n}t)}\right)^{1/n},$

where $L=\angle\lim_{z\to\infty}z/\varphi(z)$.

###### Proof.

For the iterate $\varphi^{(n)}$ the corresponding angular derivative
$L_{n}=\angle\lim_{z\to\infty}z/\varphi^{(n)}(z)$ is simply $L^{n}$, being the
product of $n$ terms each tending to $L$. Thus

$L^{n}\inf_{t>0}\frac{w(t)}{w(L^{n}t)}\leq\|C_{\varphi}^{n}\|^{2}\leq
L^{n}\sup_{t>0}\frac{w(t)}{w(L^{n}t)},$

from which the result follows by the standard spectral radius formula. ∎

##  3\. Spectral theory

For the spectral theory of composition operators on the half-plane, little
seems to be known, although Schroderus [15] has determined the spectrum and
essential spectrum in the case of linear fractional mappings $\varphi$ for
Hardy and weighted Bergman spaces. The associated composition operator
$C_{\varphi}$ is bounded if and only if $\varphi$ is a parabolic or hyperbolic
mapping fixing $\infty$; that is, $\varphi(s)=\mu s+s_{0}$ with $\mu>0$ and
$\mathop{\rm Re}\nolimits s_{0}\geq 0$ (we shall discuss this in detail
below). The same applies in general Zen spaces, by Theorem 1.2.

Schroderus restricts herself to Hardy spaces and weighted Bergman spaces on
the upper half-plane. Transforming to the right half-plane these are Zen
spaces $A^{2}_{\nu}$ with measures $d\nu(x)=x^{\alpha}\,dx$ for $\alpha>-1$
(in the following the Hardy space is formally identified with the case
$\alpha=-1$). Her results in this particular context are:

######  Theorem 3.1.

[15, Thm. 1.1, Thm 1.2]

  1. (1)

In the parabolic case $\varphi(s)=s+s_{0}$, where $\mathop{\rm Re}\nolimits
s_{0}\geq 0$ and $s_{0}\neq 0$, the spectrum and essential spectrum of
$C_{\varphi}$ coincide and equal
(a) ${\mathbb{T}}$ if $s_{0}\in i{\mathbb{R}}$;
(b) $\\{e^{-s_{0}t}:t\geq 0\\}\cup\\{0\\}$ if $s_{0}\in{\mathbb{C}}_{+}$.

  2. (2)

In the hyperbolic case $\varphi(s)=\mu s+s_{0}$, where
$\mu\in(0,1)\cup(1,\infty)$ and $\mathop{\rm Re}\nolimits s_{0}\geq 0$, the
spectrum and essential spectrum of $C_{\varphi}$ coincide and equal
(a) $\\{\lambda\in{\mathbb{C}}:|\lambda|=\mu^{-(\alpha+2)/2}\\}$ if $s_{0}\in
i{\mathbb{R}}$;
(b) $\\{\lambda\in{\mathbb{C}}:|\lambda|\leq\mu^{-(\alpha+2)/2}\\}$ if
$s_{0}\in{\mathbb{C}}_{+}$.

###  3.1. The parabolic case

For a general Zen space, part (1) of Theorem 3.1 is easy to prove.

######  Proposition 3.2.

Let $\varphi(s)=s+s_{0}$ where $\mathop{\rm Re}\nolimits s_{0}\geq 0$ and
$s_{0}\neq 0$. Then the spectrum and essential spectrum of $C_{\varphi}$ on
$A^{2}_{\nu}$ coincide and equal
(a) ${\mathbb{T}}$ if $s_{0}\in i{\mathbb{R}}$;
(b) $\\{e^{-s_{0}t}:t\geq 0\\}\cup\\{0\\}$ if $s_{0}\in{\mathbb{C}}_{+}$.

###### Proof.

The composition operator is seen to be unitarily equivalent to the
multiplication operator on $L^{2}(0,\infty,w(t)dt)$ given by multiplication by
the function $t\mapsto e^{-s_{0}t}$. Thus it is a normal operator: its
spectrum and essential spectrum equal the closure of $\\{e^{-s_{0}t}:t\geq
0\\}$, as in (a) and (b). ∎

###  3.2. The hyperbolic case

In this subsection we take $\varphi(s)=\mu s+s_{0}$, where
$\mu\in(0,1)\cup(1,\infty)$ and $\mathop{\rm Re}\nolimits s_{0}\geq 0$.

Schroderus observed that we have the following simplifications:

For $s_{0}=iy$, with $y\in{\mathbb{R}}$, the operator $C_{\varphi}$ is similar
to $C_{\psi}$ with $\psi(s)=\mu s$. Indeed, if $\rho(s)=s+iy/(\mu-1)$ then

$\rho^{-1}\circ\psi\circ\rho(s)=\mu(s+iy/(\mu-1))-iy/(\mu-1)=\mu s+iy.$

Similarly, for $s_{0}=x+iy$ with $x>0$ and $y\in{\mathbb{R}}$, let
$\psi(s)=\mu s+x$ and take the same $\rho$. Then

$\rho^{-1}\circ\psi\circ\rho(s)=\mu(s+iy/(\mu-1))+x-iy/(\mu-1)=\mu s+x+iy.$

Thus, since $C_{\rho}$ is a unitary map on every $A^{2}_{\nu}$, we need only
consider the spectrum of $C_{\varphi}$ when $\varphi(s)=\mu s+x$ for
$\mu\in(0,1)\cup(1,\infty)$ and $x\geq 0$.

In case (a) we can estimate the spectral radius of $C_{\varphi}$ and
$C^{-1}_{\varphi}$, although this tells us only that the spectrum is contained
in an annulus (which may be a circle). In case (b) we have that the spectrum
is contained in a disc.

We showed in [6, Prop. 3.5] that the norm of the weighted composition operator
$C_{\psi}$ corresponding to $\psi(s)=\mu s$ with $\mu>0$ is exactly
$\displaystyle\left(\frac{1}{\mu}\sup_{t>0}\frac{w(\mu
t)}{w(t)}\right)^{1/2}$. The method of [6] also shows that for all $x\geq 0$
and $\varphi(s)=\mu s+x$ the composition operator has norm equal to the norm
of the mapping $f\mapsto g$ in $L^{2}(0,\infty;w(t)\,dt)$, where

$g(t)=\frac{1}{\mu}e^{-xt/\mu}f(t/\mu)$

since

$\displaystyle\int_{0}^{\infty}e^{-st}g(t)\,dt$
$\displaystyle=\frac{1}{\mu}\int_{0}^{\infty}e^{-st}e^{-xt/\mu}f(t/\mu)\,dt$
$\displaystyle=\int_{0}^{\infty}e^{-\mu s\tau}e^{-x\tau}f(\tau)\,d\tau$

and

$\|g\|^{2}=\frac{1}{\mu^{2}}\int_{0}^{\infty}e^{-2xt/\mu}|f(t/\mu)|^{2}w(t)\,dt=\frac{1}{\mu}\int_{0}^{\infty}e^{-2x\tau}|f(\tau)|^{2}w(\mu\tau)\,d\tau,$

whence

$\|C_{\varphi}\|^{2}=\frac{1}{\mu}\sup_{t>0}\frac{e^{-2xt}w(\mu t)}{w(t)}.$

Now by looking at iterates we can obtain an explicit formula for the spectral
radius of $C_{\varphi}$ as the $n$th iterate of $\varphi$ is given by
$\varphi_{n}(s)=\mu^{n}s+x_{n}$, where $x_{n}=\dfrac{\mu^{n}-1}{\mu-1}x$, and
this has the same form as $\varphi$.

That is,

(6)
$\rho(C_{\varphi})=\frac{1}{\sqrt{\mu}}\lim_{n\to\infty}\left(\sup_{t>0}\frac{e^{-2x_{n}t}w(\mu^{n}t)}{w(t)}\right)^{1/2n}.$

In the time domain (that is, using the inverse Laplace transform), the
operator $C_{\varphi}$ corresponds to the operator $B_{\varphi}$ on
$L^{2}(0,\infty,w(t)dt)$ with

$B_{\varphi}f(t)=af(bt)e^{-ct}$

such that we have

$\displaystyle\int_{0}^{\infty}af(bt)e^{-ct}e^{-st}dt$
$\displaystyle=\int_{0}^{\infty}af(u)e^{-cu/b}e^{-su/b}du/b$
$\displaystyle=(a/b)(\mathcal{L}f)(s/b+c/b)=(\mathcal{L}f)(\mu s+x),$

and so $a=b=1/\mu$ and $c=x/\mu$.

With $Af(t)=\frac{1}{\mu}f(t/\mu)e^{-xt/\mu}$ on $L^{2}(0,\infty,w(t)dt)$ we
find that

$\int_{0}^{\infty}\frac{1}{\mu}f(t/\mu)e^{-xt/\mu}\overline{g(t)}w(t)\,dt=\int_{0}^{\infty}f(u)e^{-xu}\overline{g(\mu
u)}\frac{w(\mu u)}{w(u)}w(u)\,du$

so that $\displaystyle A^{*}g(u)=g(\mu u)e^{-xu}\frac{w(\mu u)}{w(u)}$.

####  3.2.1. The case $x=0$

It is enough to consider the spectrum of $C_{\varphi}$ for $\varphi(s)=\mu s$
with $\mu>1$ since the case $0<\mu<1$ may be studied by taking inverses.

######  Theorem 3.3.

For the composition operator $C_{\varphi}$ on $A^{2}_{\nu}$ where
$\varphi(s)=\mu s$ with $\mu>1$ and $\nu$ determining the weight $w$ on
$(0,\infty)$ we have

$\sigma(C_{\varphi})\subseteq\\{z\in{\mathbb{C}}:r\leq|z|\leq R\\},$

where

$r=\frac{1}{\sqrt{\mu}}\lim_{n\to\infty}\left(\inf_{t>0}\frac{w(\mu^{n}t)}{w(t)}\right)^{1/(2n)}$

and

$R=\frac{1}{\sqrt{\mu}}\lim_{n\to\infty}\left(\sup_{t>0}\frac{w(\mu^{n}t)}{w(t)}\right)^{1/(2n)}.$

###### Proof.

This follows from (6). ∎

With $\displaystyle A^{*}f(t)=f(\mu t)\frac{w(\mu t)}{w(t)}$, we have
eigenvalues of $A^{*}$ given by $\mu^{\alpha}$, with eigenvectors
$f(t)=t^{\alpha}/w(t)$, provided that these eigenvectors lie in the space
$L^{2}(0,\infty,w(t)dt)$. Indeed the eigenvalues then have infinite
multiplicity, since $f\chi_{E}$ is an eigenvector for any measurable
$E\subset{\mathbb{R}}_{+}$ such that $\mu E=E$, the notation $\chi_{E}$
denoting the characteristic (indicator) function of a set $E$. Clearly there
are infinitely many distinct subsets $E\subset{\mathbb{R}}$ such that $\mu
E=E$. For example, if $\mu>1$ we may take

$E=\bigcup_{n=-\infty}^{\infty}\chi_{(\mu^{n},\mu^{n}(1+\delta))}$

for any $\delta$ with $0<\delta<\mu-1$. The same is true for $0<\mu<1$ working
with $1/\mu$ instead of $\mu$: this will be required later.

Thus we have the following lower bound for the essential spectrum
$\sigma_{e}(C_{\varphi})$:

######  Theorem 3.4.

For the composition operator $C_{\varphi}$ on $A^{2}_{\nu}$ where
$\varphi(s)=\mu s$ with $\mu>1$ and $\nu$ determining the weight $w$ on
$(0,\infty)$ it holds that $\sigma_{e}(C_{\varphi})$ contains all $\alpha$
such that

(7) $\int_{0}^{\infty}|t^{\alpha}|^{2}/w(t)<\infty.$

###### Proof.

The eigenvectors of $f(t)=t^{\alpha}/w(t)$ of the unitarily equivalent
operator $A^{*}$ lie in the space $L^{2}(0,\infty,w(t)dt)$ if and only if (7)
holds. ∎

Note that the $\alpha$ occurring in Theorem 3.4 form an annulus (if the set is
non-empty), and in some cases (e.g. the Hardy–Bergman space below) this
enables us to find the spectrum exactly.

####  3.2.2. The case $x>0$

Here the calculations are necessarily more difficult, but we do have the
spectral radius formula (6) to give a disc in which the spectrum is contained.
In many cases all the points in the interior are eigenvalues, either of $A$ or
$A^{*}$. Indeed, they are again eigenvalues of infinite multiplicity, and
hence in the essential spectrum, as we see in the following two propositions,

######  Proposition 3.5.

For $\mu>1$, if $\alpha\in{\mathbb{C}}$ and $\beta=x/(\mu-1)>0$ are such that
the function $f:t\mapsto t^{\alpha}e^{-\beta t}$ lies in
$L^{2}(0,\infty,w(t)dt)$, then $A$ has the eigenvalue $1/\mu^{\alpha+1}$ and
$f\chi_{E}$ is an eigenvector for any measurable $E\subset{\mathbb{R}}_{+}$
such that $\mu E=E$.

###### Proof.

This is an easy calculation since $Af(t)=\frac{1}{\mu}f(t/\mu)e^{-xt/\mu}$. ∎

######  Proposition 3.6.

For $0<\mu<1$ if $\alpha\in{\mathbb{C}}$ and $\beta=x/(1-\mu)>0$ are such that
the function $f(t)=t^{\alpha}e^{-\beta t}/w(t)$ lies in
$L^{2}(0,\infty,w(t)dt)$, then $A^{*}$ has the eigenvalue $\mu^{\alpha}$ and
$f\chi_{E}$ is an eigenvector for any measurable $E\subset{\mathbb{R}}_{+}$
such that $\mu E=E$.

###### Proof.

This is similarly straightforward, since

$A^{*}f(t)=f(\mu t)e^{-xt}{w(\mu t)}/{w(t)}.$

∎

One technique from [15] is not available in general: for the Hardy and
standard weighted Bergman spaces, the weighted composition operators with
$\mu$ and $1/\mu$ are related by taking adjoints. This is not true unless
$w(t)$ is a power of $t$.

However, as we shall see in the next section the information here is enough to
allow us to determine the exact spectrum in an important new example (as well
as the cases covered by Schroderus).

###  3.3. The Hardy–Bergman space

As a significant example of a situation not covered by older results, let us
look again at the Hardy–Bergman space $H^{2}({\mathbb{C}}_{+})\cap
A^{2}({\mathbb{C}}_{+})$. To determine the spectrum we may take any equivalent
norm, so we take $w(t)=1+1/t$ as before.

######  Theorem 3.7.

Let $X=H^{2}({\mathbb{C}}_{+})\cap A^{2}({\mathbb{C}}_{+})$ be the Hardy-
Bergman space and $C_{\varphi}\in{\mathcal{L}}(X)$.

  1. a)

If $\varphi(s)=\mu s$ with $\mu>0$, then $\sigma(C_{\varphi})$ is the annulus

$\sigma(C_{\varphi})=\left\\{z\in{\mathbb{C}}:\min\left\\{\frac{1}{\mu},\frac{1}{\sqrt{\mu}}\right\\}\leq|z|\leq\max\left\\{\frac{1}{\mu},\frac{1}{\sqrt{\mu}}\right\\}\right\\}.$

  2. b)

If $\varphi(s)=\mu s+(x+iy)$ with $\mu,x>0$ and $y\in{\mathbb{R}}$, then
$\sigma(C_{\varphi})$ is the closed disc

$\sigma(C_{\varphi})=\\{z\in{\mathbb{C}}:|z|\leq 1/\mu\\}.$

In all cases we have $\sigma_{e}(C_{\varphi})=\sigma(C_{\varphi})$.

The proof of this theorem follows from the two subsections detailed below.

####  3.3.1. The case $x=0$

For $\varphi(s)=\mu s$ with $\mu>0$ we have

$\|C_{\varphi}\|^{2}=\frac{1}{\mu}\sup_{t>0}\frac{w(\mu
t)}{w(t)}=\frac{1}{\mu}\sup_{t>0}\frac{1+1/(\mu
t)}{1+1/t}=\frac{1}{\mu}\sup_{t>0}\frac{\mu t+1}{\mu t+\mu}.$

This is

$\begin{cases}1/\mu^{2}&\hbox{if}\quad 0<\mu<1,\\\
1/\mu&\hbox{if}\quad\mu>1.\end{cases}$

Thus for $\mu>1$ we have

$\|C^{n}_{\varphi}\|^{2}=\frac{1}{\mu^{n}}$

while

$\|C^{-n}_{\varphi}\|^{2}=\mu^{2n}.$

That is, the spectrum of $C_{\varphi}$ lies in the annulus
$\\{z\in{\mathbb{C}}:\frac{1}{\mu}\leq|z|\leq\frac{1}{\sqrt{\mu}}\\}$, but not
in a circle. So we see that Theorem 3.1 (2)(a) does not hold in the general
case of a Zen space.

We now show that the (essential) spectrum equals the whole annulus. To do this
we use Theorem 3.4. The function $t^{\alpha}/(1+1/t)$ lies in the space
$L^{2}(0,\infty,(1+1/t)\,dt)$ if and only if

$\int_{0}^{\infty}\frac{|t^{\alpha}|^{2}}{(1+1/t)^{2}}(1+1/t)\,dt<\infty,$

so that $-1<\mathop{\rm Re}\nolimits\alpha<-1/2$. From this we see that all
points in the annulus
$\\{z\in{\mathbb{C}}:\frac{1}{\mu}<|z|<\frac{1}{\sqrt{\mu}}\\}$ are
eigenvalues of $A^{*}$ so that for $\mu>1$ we have

$\sigma(C_{\varphi})=\left\\{z\in{\mathbb{C}}:\frac{1}{\mu}\leq|z|\leq\frac{1}{\sqrt{\mu}}\right\\}.$

By considering inverses, we see that for $0<\mu<1$

$\sigma(C_{\varphi})=\left\\{z\in{\mathbb{C}}:\frac{1}{\sqrt{\mu}}\leq|z|\leq\frac{1}{\mu}\right\\}.$

####  3.3.2. The case $x>0$

We begin with the spectral radius formula (6). Elementary calculus shows that
the supremum in (6) is at $t=0$ except if $2x_{n}<\mu^{n}-1$. This requires
$\mu>1$. If the supremum is at $t=0$ it is $1/\mu^{n}$.

Otherwise (with $\mu>1$) we can estimate the supremum for each $n$ by
considering the two cases (recall that $x_{n}\to\infty$ as $n\to\infty$):

  1. (1)

$t\geq\dfrac{n\log\mu}{2x_{n}}$, when the expression is clearly at most
$\mu^{-n}$ (the second factor is bounded by 1);

  2. (2)

$0\leq t\leq\dfrac{n\log\mu}{2x_{n}}$, when we may ignore the exponential and
obtain an upper bound

$\frac{1+1/(\mu^{n}t)}{1+1/t}\leq\frac{1+2x_{n}/(n\mu^{n}\log\mu)}{2x_{n}/n\log\mu}\leq\frac{1+C_{1}/n}{C_{2}\mu^{n-1}/n},$

where $C_{1}$ and $C_{2}$ are independent of $n$.

Taking the maximum of these two estimates and then the $2n$-th root gives a
limit $1/\sqrt{\mu}$. That is
$\rho(C_{\varphi})\leq\frac{1}{\sqrt{\mu}}\times\frac{1}{\sqrt{\mu}}=\frac{1}{\mu}$
for all $x>0$.

Now we use Propositions 3.5 and 3.6 to find eigenvectors.

For $\mu>1$ the range of admissible $\alpha$ is $\\{\mathop{\rm
Re}\nolimits\alpha>0\\}$ and the eigenvalue is $1/\mu^{\alpha+1}$, so that
every point of the disc $\\{z\in{\mathbb{C}}:0<|z|<1/\mu\\}$ is an eigenvector
of $A$ and we deduce that $\sigma(C_{\varphi})=\\{z\in{\mathbb{C}}:|z|\leq
1/\mu\\}$.

Similarly, for $0<\mu<1$ the range of admissible $\alpha$ is $\\{\mathop{\rm
Re}\nolimits\alpha>-1\\}$ and the eigenvalue is $\mu^{\alpha}$, so again we
deduce that $\sigma(C_{\varphi})=\\{z\in{\mathbb{C}}:|z|\leq 1/\mu\\}$, and
the same holds for $\sigma_{e}(C_{\varphi})$.

##  4\. Semigroups of composition operators

###  4.1. Norm estimates

Our first task is to generalize the result of Arvanitidis [3] showing that a
necessary and sufficient condition for boundedness of a semigroup
$(C_{\varphi_{t}})$ of composition operators on $H^{2}({\mathbb{C}}_{+})$ with
generator $A:f\mapsto Gf^{\prime}$ is that
$\delta:=\angle\lim_{z\to\infty}G(z)/z$ exists. In this case
$\|C_{\varphi_{t}}\|_{H^{2}({\mathbb{C}}_{+})}=e^{-\delta t/2}$. This relies
on a theorem in [7] (which is stated for the disc), which gives the angular
derivative, and hence the norm, in terms of $G$. In particular,
$L_{t}=e^{-\delta t}$.

By virtue of Theorem 1.2 (Kucik) we know that the condition for boundedness
does not depend on which Zen space we use, and all that changes is the norm of
the operator. We may thus state the following easy theorem.

######  Theorem 4.1.

For a semigroup of analytic self-maps $(\varphi_{t})$ on ${\mathbb{C}}_{+}$,
the following are equivalent:

  1. (1)

the non-tangential limit $\delta:=\angle\lim_{z\to\infty}G(z)/z$ exists;

  2. (2)

the semigroup $(C_{\varphi_{t}})_{t\geq 0}$ consists of bounded operators on
$A^{2}_{\nu}$.

In this case, we have

$L_{t}\inf_{x>0}\frac{w(x)}{w(L_{t}x)}\leq\|C_{\varphi_{t}}\|^{2}\leq
L_{t}\sup_{x>0}\frac{w(x)}{w(L_{t}x)},$

where $L_{t}=e^{-\delta t}$ for $t\geq 0$.

###### Proof.

This follows from [3, Thm. 3.4] together with the estimate (1) from [13]. ∎

For example, with the Hardy space $H^{2}({\mathbb{C}}_{+})$ considered by
Arvanitidis, we have $\|C_{\varphi_{t}}\|=e^{-\delta t/2}$, while for the
standard Bergman space $A^{2}({\mathbb{C}})$ we have the apparently new result
that $\|C_{\varphi_{t}}\|=e^{-\delta t}$.

In [2] there is a result applying to analytic function spaces, i.e., Banach
spaces $X$ of holomorphic functions on a domain $\Omega$ for which point
evaluations are continuous functionals. The result applies to such spaces
satisfying the following supplementary condition:

(E) If $(z_{n})$ is a sequence in $\Omega$ such that $z_{n}\to
z\in\overline{\Omega}\cup\\{\infty\\}$ and $\lim_{n\to\infty}f(z_{n})$ exists
in ${\mathbb{C}}$ for all $f\in X$ then $z\in\Omega$.

######  Theorem 4.2.

Suppose that $(T_{t})_{t\geq 0}$ is a $C_{0}$ semigroup on a function space
$X$ with property (E) such that for some $G\in\mathop{\rm Hol}(\Omega)$ the
generator $A$ is the operator $f\mapsto Gf^{\prime}$ for all
$f\in\operatorname{dom}(A)$. Then $(T_{t})$ is a semigroup of composition
operators.

Similar results are given in [10], for function spaces on the disc
${\mathbb{D}}$. We may now apply Theorem 4.2 to our situation.

######  Theorem 4.3.

Suppose that $(T_{t})_{t\geq 0}$ is a $C_{0}$ semigroup on a Zen space
$A^{2}_{\nu}$ such that for some $G\in\mathop{\rm Hol}({\mathbb{C}}_{+})$ the
generator $A$ is the operator $f\mapsto Gf^{\prime}$ for all
$f\in\operatorname{dom}(A)$. Then $(T_{t})$ is a semigroup of composition
operators.

###### Proof.

For the halfplane we need to find sequences $(z_{n})$ in ${\mathbb{C}}_{+}$
tending to $\infty$ or a point on the imaginary axis for which $\lim f(z_{n})$
does not exist for some $f\in A^{2}_{\nu}$.

Now if $(z_{n})$ is such that $(f(z_{n}))$ converges for all $f\in
A^{2}_{\nu}$ then by the uniform boundedness theorem the reproducing kernels
at $(z_{n})$ are uniformly bounded. But if $\mathop{\rm Re}\nolimits z_{n}\to
0$ monotonically then the integrals in (4) with $\lambda=z_{n}$ tend to
$\infty$ (note that $w$ is a decreasing function). This shows that condition
(E) holds for $A^{2}_{\nu}$ (take sequences tending to either a point on the
imaginary axis or to infinity with real parts tending to 0). The result
follows from Theorem 4.2. ∎

######  Remark 4.4.

We may also consider the question of groups of composition operators on
$A^{2}_{\nu}$. However, since we have same semigroups for all the spaces under
consideration, with different norms, we have the same groups. Proposition 4.4
of [4] asserts that for a $C_{0}$-quasicontractive group of bounded
composition operators on on $H^{2}({\mathbb{C}}_{+})$ we have $G(z)=ps+iq$ for
real $p$ and $q$, and $\varphi_{t}(s)=e^{pt}s+\frac{iq}{p}(e^{pt}-1)$ if
$p\neq 0$, or $\varphi_{t}(s)=s+iqt$ if $p=0$. The same will apply to any Zen
space $A^{2}_{\nu}$.

###  4.2. Semigroups of linear fractional mappings

As in Section 3 we may discuss parabolic or hyperbolic mappings fixing
$\infty$. That is, $\varphi_{t}(s)=\mu_{t}s+s_{t}$, where $\mu_{t}>0$ and
$s_{t}\in\overline{{\mathbb{C}}_{+}}$.

The semigroup relation $\varphi_{t+u}=\varphi_{t}\circ\varphi_{u}$ gives us
$\mu_{t}=e^{pt}$ for some $p\in{\mathbb{R}}$, and

  1. (1)

$s_{t}=\alpha(e^{pt}-1)$ for some $\alpha\in\overline{{\mathbb{C}}_{+}}$ if
$p\neq 0$ (the hyperbolic case), or

  2. (2)

$s_{t}=\alpha t$ for some $\alpha\in\overline{{\mathbb{C}}_{+}}$ if $p=0$ (the
parabolic case).

Using (2) we see that $G(z)=pz+p\alpha$ in the hyperbolic case and
$G(z)=\alpha$ in the parabolic case.

An immediate corollary of Theorem 3.7 is the following description of the
spectrum.

######  Corollary 4.5.

Let $X=H^{2}({\mathbb{C}}_{+})\cap A^{2}({\mathbb{C}}_{+})$ be the Hardy-
Bergman space and $(C_{\varphi_{t}})_{t\geq 0}\subset{\mathcal{L}}(X)$. .

  1. a)

If $\varphi_{t}(s)=e^{pt}s+\alpha(e^{pt}-1)$ with $p\neq 0$, then
$\sigma(C_{\varphi_{t}})$ is the annulus

$\sigma(C_{\varphi_{t}})=\left\\{z\in{\mathbb{C}}:\min\left\\{\frac{1}{e^{pt}},\frac{1}{e^{pt/2}}\right\\}\leq|z|\leq\max\left\\{\frac{1}{e^{pt}},\frac{1}{e^{pt/2}}\right\\}\right\\}.$

  2. b)

If $\varphi_{t}(s)=s+\alpha t$, then $\sigma(C_{\varphi_{t}})$ is the closed
disc

$\sigma(C_{\varphi_{t}})=\\{z\in{\mathbb{C}}:|z|\leq e^{-pt}\\}$

In all cases the spectrum and essential spectrum coincide.

## References

  * [1] A.E. Alajyan and J.R. Partington, Weighted operator-valued function spaces applied to the stability of delay systems. Oper. Matrices, 15 (2021), no. 4, 1257–1266.
  * [2] W. Arendt and I. Chalendar. Generators of semigroups on Banach spaces inducing holomorphic semiflows. Israel J. Math., 229(1) (2019), 165–179.
  * [3] A.G. Arvanitidis, Semigroups of composition operators on Hardy spaces of the half-plane. Acta Sci. Math. (Szeged) 81(1–2) (2015), 293–308.
  * [4] C. Avicou, I. Chalendar, and J. R. Partington, Analyticity and compactness of semigroups of composition operators. J. Math. Anal. Appl., 437(1):545–560, 2016.
  * [5] E. Berkson and H. Porta, Semigroups of analytic functions and composition operators. Michigan Math. J. 25 (1978), no. 1, 101–115.
  * [6] I. Chalendar and J. R. Partington, Norm estimates for weighted composition operators on spaces of holomorphic functions. Complex Anal. Oper. Theory 8 (2014), no. 5, 1087–1095.
  * [7] M.D. Contreras, S. Díaz Madrigal, and Ch. Pommerenke, On boundary critical points for semigroups of analytic functions. Math. Scand. 98 (2006), no. 1, 125–142.
  * [8] S. Elliott and M.T. Jury, Composition operators on Hardy spaces of a half-plane. Bull. Lond. Math. Soc. 44 (2012), no. 3, 489–495.
  * [9] S.J. Elliott and A. Wynn, Composition operators on weighted Bergman spaces of a half-plane. Proc. Edinb. Math. Soc. 54 (2011), 373–379.
  * [10] E.A. Gallardo-Gutiérrez and D.V. Yakubovich, On generators of $C_{0}$-semigroups of composition operators. Israel J. Math. 229 (2019), no. 1, 487–500.
  * [11] B. Jacob, J.R. Partington and S. Pott, On Laplace–Carleson embedding theorems. J. Funct. Anal. 264 (2013), no. 3, 783–814.
  * [12] B. Jacob, J.R. Partington and S. Pott, Applications of Laplace–Carleson embeddings to admissibility and controllability, SIAM J. Control. Optim., 52 (2014), 1299–1313.
  * [13] A.S. Kucik, Weighted composition operators on spaces of analytic functions on the complex half-plane. Complex Anal. Oper. Theory 12 (2018), no. 8, 1817–1833.
  * [14] V. Matache, Weighted composition operators on $H^{2}$ and applications. Complex Anal. Oper. Theory 2 (2008), no. 1, 169–197.
  * [15] R. Schroderus, Spectra of linear fractional composition operators on the Hardy and weighted Bergman spaces of the half-plane. J. Math. Anal. Appl. 447 (2017), no. 2, 817–833.