# Dunford property for composition
operators on $H^{p}$-spaces

E. A. Gallardo-Gutiérrez  Departamento de Análisis Matemático y Matemática
Aplicada, Facultad de Matemáticas, Universidad Complutense de Madrid, Plaza de
Ciencias 3, 28040 Madrid, Spain and Instituto de Ciencias Matemáticas ICMAT
(CSIC-UAM-UC3M-UCM), Madrid, Spain<EMAIL_ADDRESS>,  F. J.
González-Doña  Departamento de Matemáticas, Escuela Politécnica Superior,
Universidad Carlos III de Madrid, Avda. de la Universidad 30, 28911 Leganés
(Madrid), Spain<EMAIL_ADDRESS>and  M. Monsalve-López  Departamento
de Análisis Matemático y Matemática Aplicada, Facultad de Matemáticas,
Universidad Complutense de Madrid, Plaza de Ciencias 3, 28040 Madrid, Spain.
<EMAIL_ADDRESS>
(Date: July 18th, 2023, Revised: April 1st, 2024)

###### Abstract.

The Dunford property $(C)$ for composition operators on $H^{p}$-spaces
($1<p<\infty$), as well as for their adjoints, is completely characterized
within the class of those induced by linear fractional transformations of the
unit disc. As a consequence, it is shown that the Dunford property is stable
in such a class addressing a particular instance of a question posed by
Laursen and Neumann.

###### Key words and phrases:

Dunford property $(C)$, decomposable operators, local spectral theory,
composition operators

###### 2020 Mathematics Subject Classification:

47A15, 47A11, 47B37, 47B38

Authors are partially supported by Plan Nacional I+D grants no.
PID2019-105979GB-I00 and PID2022-137294NB-I00(Spain), the Spanish Ministry of
Science and Innovation, through the “Severo Ochoa Programme for Centres of
Excellence in R&D” (CEX2019-000904-S) and from the Spanish National Research
Council, through the “Ayuda extraordinaria a Centros de Excelencia Severo
Ochoa” (20205CEX001).

##  1\. Introduction and Preliminaries

In 1959, E. Bishop used a Banach space version of the analytic duality
principle established by Silva, Köthe, Grothendieck and others operator
theorists to study connections between spectral decomposition properties of a
Banach space operator and its adjoint. Of particular interest in this setting
are operators satisfying the _Bishop property_ $(\beta)$ since E. Albrecht and
J. Eschmeier [2] developed a complete duality theory for them. Actually, the
property $(\beta)$ turns out to characterize restrictions of decomposable
operators to closed invariant subspaces, and Albrecht and Eschmeier’s analytic
functional model shows that every Banach space operator is similar to the
quotient of an operator with the Bishop property. Similarly, quotients of
decomposable operators are determined by the decomposition property $(\delta)$
and in fact, both properties $(\beta)$ and $(\delta)$ are completely dual: an
operator has one if and only if its adjoint has the other.

The class of decomposable operators, introduced by Foiaş [16] in the sixties,
constitutes a generalization of spectral operators (in the sense of Dunford)
and many operators in Hilbert spaces as unitary operators, self-adjoint
operators or more generally, normal operators are decomposable. Recall that a
linear bounded operator $T$ acting on a Banach space $X$ is _decomposable_ if
for every open cover $\\{U_{1},U_{2}\\}$ of the complex plane $\mathbb{C}$,
there exist closed $T$-invariant subspaces $X_{1}$ and $X_{2}$ of $X$ so that
$X=X_{1}+X_{2}$ and the spectrum of the restriction $T|_{X_{i}}$ is contained
in $U_{i}$, for $i=1,2$. It turns out that $T$ is decomposable if and only if
$T$ satisfies both properties $(\beta)$ and $(\delta)$.

Although the class of operators which are decomposable is not stable by
restricting them to closed invariant subspaces, it turns out that the class of
those which satisfy the _Dunford property_ $(C)$ is. This property, pioneered
by Dunford in the sixties, has played a major role in the development of the
theory of spectral operators. Being the Dunford property $(C)$ a weaker
property than the Bishop one (see the Millers’ example [22]), one has that an
operator $T$ is decomposable if and only if it satisfies both properties $(C)$
and $(\delta)$. We refer to the monograph by Laursen and Neumann [21] for more
on this subject.

Studying local spectral properties for classes of _concrete operators_ acting
on function spaces leads naturally to interesting questions from the function
theoretic perspective, and this is one of the driving aim of the present work.
At this regard, it is worthy to point out the works of the Millers’ and
coauthors [4] or [23] or the one by Aleman and Persson [3] regarding the
classical Cesàro operator or Cesàro type operators.

One of the most significant classes of operators having the property $(C)$ is
the one consisting of _multipliers_ acting on _functional Banach spaces_ (see
[11] for the definition and [21, Proposition 1.6.9] for the result). In
particular, it turns out that every multiplier of the Hardy spaces
$H^{p}(\mathbb{D})$, $1\leq p<\infty$ has the property $(C)$ but fails
property $(\delta)$. In this work, we characterize the Dunford property $(C)$
for composition operators $C_{\varphi}$, and their adjoints $C_{\varphi}^{*}$,
whenever they are induced by linear fractional transformations $\varphi$ of
the unit disc $\mathbb{D}$ in $H^{p}$-spaces ($1<p<\infty$). They constitute a
first approach to a more comprehensive analysis of the study of composition
operators induced by any holomorphic self map of $\mathbb{D}$. At this regard,
it is worthy to remark that while the spectrum of composition operators
induced by linear fractional transformations of $\mathbb{D}$ in $H^{p}$-spaces
is known (see [11, Chapter 8] and the references therein), determining the
spectrum of general composition operators in $H^{p}(\mathbb{D})$ remains as an
open (and difficult) question (see [17] for recent results in this setting).

As we will show, the Dunford property $(C)$ is strongly dependent on the
fixed-point configuration of the inducing symbol $\varphi$, and fails
drastically in those cases when the operator does not have it, that is, the
operator does not have even the _single-valued extension property_. Moreover,
when both $C_{\varphi}$ and $C_{\varphi}^{*}$ have the Dunford property $(C)$,
it turns out that they are decomposable operators. In Table 1 we summarise the
characterization, and before proceeding further with comments, we recall some
preliminaries.

### The setting

For $1\leq p<\infty$, the _Hardy space_ $H^{p}(\mathbb{D})$ is defined as the
complex Banach space of analytic functions $f:\mathbb{D}\to\mathbb{C}$ with
finite norm given by

$\|f\|_{H^{p}(\mathbb{D})}:=\lim_{r\to
1^{-}}\bigg{(}\frac{1}{2\pi}\int_{0}^{2\pi}|f(re^{i\theta})|^{p}\,d\theta\bigg{)}^{1/p}.$

For $p=\infty$, the space $H^{\infty}(\mathbb{D})$ consists of all bounded
analytic function on $\mathbb{D}$ equipped with the supremum norm.

Fixed $1<p<\infty$, it is a well-known fact that each element of the dual
space $H^{p}(\mathbb{D})^{*}$ may be regarded univocally as a continuous
functional of the form

$\varphi_{h}(f):=\frac{1}{2\pi}\int_{0}^{2\pi}f(e^{i\theta})\,h(e^{i\theta})\,d\theta,$

for some function $h\in
L^{p^{\prime}}(\mathbb{T})/H_{0}^{p^{\prime}}(\mathbb{D})$, where $\mathbb{T}$
denotes the unit circle, $\frac{1}{p}+\frac{1}{p^{\prime}}=1$ and

$H_{0}^{p^{\prime}}(\mathbb{D}):=\bigg{\\{}g\in
H^{p^{\prime}}(\mathbb{D}):\int_{0}^{2\pi}g(e^{i\theta})\,d\theta=0\bigg{\\}}=zH^{p^{\prime}}(\mathbb{D}).$

Equivalently, any functional $\varphi:H^{p}(\mathbb{D})\to\mathbb{C}$ in the
dual space $H^{p}(\mathbb{D})^{*}$ can be identified with a function $g\in
H^{p^{\prime}}(\mathbb{D})$ such that

$\varphi(f):=\frac{1}{2\pi}\int_{0}^{2\pi}f(e^{i\theta})\,\overline{g(e^{i\theta})}\,d\theta.$

In fact, this correspondence establishes a sesquilinear dual pairing between
$H^{p}(\mathbb{D})^{*}$ and $H^{p^{\prime}}(\mathbb{D})$. In particular, the
Hardy space $H^{p}(\mathbb{D})$ is reflexive for every $1<p<\infty$.

### Composition operators.

If $\varphi:\mathbb{D}\to\mathbb{D}$ is a holomorphic map, the _composition
operator_ $C_{\varphi}$ is defined as

$\displaystyle C_{\varphi}:H^{p}(\mathbb{D})$ $\displaystyle\to
H^{p}(\mathbb{D})$ $\displaystyle f$ $\displaystyle\mapsto f\circ\varphi.$

The boundedness of each composition operator $C_{\varphi}:H^{p}(\mathbb{D})\to
H^{p}(\mathbb{D})$ is guaranteed by the _Littlewood Subordination Theorem_.
Clearly, $C_{\varphi}$ is invertible in $H^{p}(\mathbb{D})$ if and only if the
function $\varphi$ is an _automorphism_ of the unit disk $\mathbb{D}$.

In this work, we consider composition operators induced by linear fractional
transformations of $\mathbb{D}$, namely, holomorphic maps of $\mathbb{D}$
given by

$\varphi(z)=\frac{az+b}{cz+d},\quad ad-bc\neq 0$

where

$|b\overline{d}-a\overline{c}|+|ad-bc|\leq|b|^{2}-|d|^{2}$

(see, for instance, [9]). The linear fractional transformations can be
classified according to their fixed points in the Riemann sphere
$\widehat{\mathbb{C}}:=\mathbb{C}\cup\\{\infty\\}$. A linear fractional map
$\varphi$ that takes $\mathbb{D}$ into itself is called parabolic if it has
just one fixed point, which must lie in $\mathbb{T}$. We will distinguish the
cases of parabolic automorphisms (PA) and parabolic non-automorphisms (PNA).

The other possible situation is when $\varphi$ has two fixed points, and the
classification depends on the location of such fixed points. One of them must
lie in $\overline{\mathbb{D}}.$ If it lies on $\mathbb{T}$, then $C_{\varphi}$
is called hyperbolic. If the other fixed point lies in $\mathbb{T}$, then
$\varphi$ is an automorphism and will be called hyperbolic automorphism (HA).
If it lies in $\widehat{\mathbb{C}}\setminus\overline{\mathbb{D}}$, $\varphi$
is a hyperbolic non-automorphism of first kind (HNA I), and if it lies in
$\mathbb{D}$ it will be called a hyperbolic non-automorphism of second kind
(HNA II).

Finally, if $\varphi$ has no fixed points in $\mathbb{T}$, one of them must
lie in $\mathbb{D}$, and the other fixed point must lie in
$\widehat{\mathbb{C}}\setminus\overline{\mathbb{D}}.$ Here, we distinguish two
situations: if $\varphi$ is an automorphism it is called elliptic (EA), and
otherwise, $\varphi$ is said to be loxodromic (LOX).

After conjugations with appropriate linear fractional transformations, every
of these maps can be expressed in a standard form as described in Table 1.
This classification is well-known in the literature and has been used
extensively (see, for instance, [18, 20]).

### Local spectral theory

Local spectral theory arose as an effort to extend some of the most important
properties of normal operators to broader classes of operators in the context
of Banach spaces. In this regard, the notion of _decomposable operators_
emerges: bounded linear operators $T$ on a Banach space $X$ for which each
splitting of the spectrum $\sigma(T)$ corresponds to a sum decomposition of
the space $X$ consisting of $T$-invariant subspaces. The class of decomposable
operators is considerably large: for instance, all operators admitting a
sufficiently rich functional calculus are decomposable, as well as any
operator with totally disconnected spectrum. Among the local spectral
properties, of particular relevance is the single-valued extension property.

######  Definition 1.1.

A linear bounded operator $T$ in $X$ has the single-valued extension property
(abbreviated as SVEP) if for every open subset $U\subset\mathbb{C}$ and every
analytic function $f:U\rightarrow X$ satisfying

$(T-zI)f(z)=0,\qquad\text{for all }z\in U$

it follows that $f\equiv 0.$

It is clear that every operator whose point spectrum has empty interior
automatically possesses the SVEP. This property is closely related to the
concept of local resolvent of an operator. Given a linear bounded operator $T$
in $X$ and a vector $x\in X$, the local resolvent of $T$ at $x$ (denoted by
$\rho_{T}(x)$) is the union of all the open sets $U\subset\mathbb{C}$ for
which there exists an analytic function $f_{x}:U\rightarrow X$ satisfying

(1.1) $(T-zI)f_{x}(z)=x,\qquad z\in U.$

For those operators $T$ enjoying the SVEP, the functional equation (1.1) has a
unique solution for each $x\in X$. Hence, in that case, there exists a unique
local resolvent function $f_{x}:\rho_{T}(x)\to X$ satisfying (1.1).

The local spectrum of $T$ at a vector $x\in X$ is defined as
$\sigma_{T}(x):=\mathbb{C}\setminus\rho_{T}(x)$. With this notion, one may
define the local spectral subspaces (or simply the spectral subspaces) of $T$
associated to a subset $F\subset\mathbb{C}$ as

$X_{T}(F):=\\{x\in X\,:\,\sigma_{T}(x)\subseteq F\\}.$

For each $F\subset\mathbb{C}$ the spectral subspace $X_{T}(F)$ is a (non-
necessarily closed) linear manifold in $X$ which is hyperinvariant by the
operator $T$. With the definition of (local) spectral subspaces at hands, we
recall the Dunford property $(C)$:

######  Definition 1.2.

A linear bounded operator $T$ in $X$ has the Dunford’s property $(C)$ if the
local spectral subspace $X_{T}(F)$ is closed for every closed set
$F\subset\mathbb{C}.$

It turns out that every decomposable operator has property $(C)$, and every
operator with property $(C)$ enjoys the SVEP (see [21, Theorem 1.2.7 and
Proposition 1.2.19]).

As it was pointed out in the introduction, in this work we provide a complete
characterization of the Dunford’s property $(C)$ for composition operators
$C_{\varphi}$ induced by linear fractional maps in $H^{p}(\mathbb{D})$,
$(1<p<\infty)$, as well as for their adjoints $C_{\varphi}^{*}.$ Next table
indicates, in terms of the configuration of the fixed points of the symbol
$\varphi$, whether the operators $C_{\varphi}$ or $C_{\varphi}^{*}$ have the
property $(C)$.

Symbol | Fixed points | Canonical form of $\varphi$ | Property $(C)$
---|---|---|---
HA | $1,\,-1$ | $\varphi(z)=\frac{r+z}{1+rz},\quad 0<r<1$ |  $C_{\varphi}^{*}$ (Thm. 3.1)
EA | $0,\,\infty$ | $\varphi(z)=\omega z,\quad\omega\in\mathbb{T}$ |  $C_{\varphi}$ and $C_{\varphi}^{*}$ ([29])
PA | $1$ | $\varphi(z)=\frac{(2-a)z+a}{-az+2+a},\quad\operatorname{Re}(a)=0$ |  $C_{\varphi}$ and $C_{\varphi}^{*}$ ([29])
HNA I | $1,\,\infty$ | $\varphi(z)=rz+1-r,\quad 0<r<1$ |  $C_{\varphi}^{*}$ (Thm. 4.2)
HNA II | $0,\,1$ | $\varphi(z)=\frac{rz}{1-(1-r)z},\quad 0<r<1$ |  $C_{\varphi}$ (Thm. 4.5)
PNA | $1$ | $\varphi(z)=\frac{(2-a)z+a}{-az+2+a},\quad\operatorname{Re}(a)>0$ |  $C_{\varphi}$ and $C_{\varphi}^{*}$ ([27])
LOX | $c\in\mathbb{D},\,\infty$ | $\varphi(z)=a(z-c)+c,\quad|a|+|1-a||c|\leq 1$ |  $C_{\varphi}$ and $C_{\varphi}^{*}$ ([19])
Table 1. Dunford property $(C)$ for composition operators and their adjoints
acting on $H^{p}(\mathbb{D})$ for $1<p<\infty$

It is worth mentioning that the characterization of the Dunford’s property
$(C)$ is independent of $1<p<\infty$ and relies upon exclusively on the nature
of $\varphi$. Likewise, in the cases in which $\varphi$ is an elliptic
automorphism, a parabolic self-map of $\mathbb{D}$ or a loxodromic one, the
induced composition operators are either known to be decomposable in
$H^{p}(\mathbb{D})$ or the result is straightforward: if $\varphi$ is an
elliptic or parabolic automorphism, $C_{\varphi}$ is generalized scalar [29,
Theorems 1.1 and 1.2], and therefore decomposable. If $\varphi$ is a parabolic
non-automorphism, Shapiro showed that $C_{\varphi}$ is decomposable in [27].
Finally, if $\varphi$ is loxodromic, the spectrum of $C_{\varphi}$ is totally
disconnected [19], so $C_{\varphi}$ is decomposable as well. As a consequence,
the adjoints of such operators are also decomposable [21, Theorem 2.5.3], and
both $C_{\varphi}$ and $C_{\varphi}^{*}$ acting on $H^{p}(\mathbb{D})$ and
$H^{p}(\mathbb{D})^{*}$ respectively for $1<p<\infty$ have the Dunford
property $(C)$.

In order to deal with the remaining cases, in Section 2 we prove a sufficient
condition for linear bounded operators acting on Banach spaces ensuring the
Dunford property $(C)$. In Section 3, we consider invertible composition
operators and show that if $\varphi$ is a hyperbolic automorphism of
$\mathbb{D}$, $C_{\varphi}$ does not have the SVEP for $1\leq p<\infty.$ In
other words, all the invertible composition operators in $H^{p}(\mathbb{D})$
are decomposable except the hyperbolic ones. Nevertheless, in this case, we
prove that $C_{\varphi}^{*}$ has the Dunford property $(C)$ for every
$1<p<\infty.$ As a consequence, we are able to describe all the local spectral
subspaces and characterize the local spectra. In Section 4, we consider the
non-invertible composition operators, dealing with the cases of hyperbolic
non-automorphisms of first and second kind. In particular, as a consequence,
we describe the local spectra of such operators. Finally, as a byproduct of
the results summarized in Table 1, we show that the class of composition
operators induced by linear fractional transformations is stable under the
Dunford property $(C)$.

##  2\. A sufficient condition

In this section, we present a general result for linear bounded operators
acting on Banach spaces which provides a sufficient condition in order to have
the Dunford property $(C)$, along with some more (local) spectral features.
This result, which can be understood as an extension of [21, Proposition
1.6.12], will be applied in the context of composition operators.

We start by recalling the glocal spectral subspaces associated to an operator
$T$ in order to deal with a variant of (local) spectral subspaces which are
better suited for operators which do not have the SVEP. Given a closed set
$F\subseteq\mathbb{C}$, the glocal spectral subspace $\mathcal{X}_{T}(F)$
consists of all $x\in X$ for which there exists an analytic function
$f:\mathbb{C}\setminus F\rightarrow X$ such that

$(T-zI)f(z)=x,\qquad z\in\mathbb{C}\setminus F.$

It is clear that the equality $\mathcal{X}_{T}(F)=X_{T}(F)$ holds for every
closed set $F\subset\mathbb{C}$ whenever the operator $T$ enjoys the SVEP. But
in general, we have the inclusion $\mathcal{X}_{T}(F)\subseteq X_{T}(F)$.

Since we are dealing with Hardy spaces $H^{p}(\mathbb{D})$, for every operator
$T$ and closed set $F\subset\mathbb{C}$, we will denote by $H^{p}_{T}(F)$ and
$\mathcal{H}^{p}_{T}(F)$ the associated local spectral subspaces and glocal
spectral subspaces, respectively.

Glocal spectral subspaces behave well with respect to the adjoint operation.
Indeed, by [21, Proposition 2.5.1], if $T:X\to X$ is a linear bounded operator
and $F$ and $G$ are two disjoint closed subsets of $\mathbb{C}$, then

(2.1)
$\mathcal{X}_{T}(F)\subseteq{{}^{\perp}\mathcal{X}^{*}_{T^{*}}(G)}\qquad\text{and}\qquad\mathcal{X}^{*}_{T^{*}}(F)\subseteq\mathcal{X}_{T}(G)^{\perp},$

where $M^{\perp}$ denotes the annihilator of a linear manifold $M\subseteq X$,
while ${{}^{\perp}N}$ denotes the preannihilator of a linear manifold
$N\subseteq X^{*}$. Actually, since $H^{p}(\mathbb{D})$ is reflexive for
$1<p<\infty$, having in mind the dual pairing between $H^{p}(\mathbb{D})$ and
$H^{p^{\prime}}(\mathbb{D})$, ($\frac{1}{p}+\frac{1}{p^{\prime}}=1$), the
property (2.1) reads as

$\mathcal{H}^{p}_{T}(F)\subseteq\mathcal{H}^{p^{\prime}}_{T^{*}}(G^{*})^{\perp},$

where $A^{*}:=\\{\overline{z}:z\in A\\}$ for every subset
$A\subseteq\mathbb{C}$.

Before stating the main result of the section, let us recall that for each
$x\in X$, the local spectral radius of $T$ at $x$ is the quantity

$r_{T}(x):=\limsup\limits_{n\rightarrow\infty}\left\|T^{n}x\right\|^{1/n}.$

If $T$ has the SVEP, then
$r_{T}(x)=\max\\{|\lambda|:\lambda\in\sigma_{T}(x)\\}$ for every non-zero
$x\in X.$ Besides, we denote by $r(T)$ the spectral radius of $T$ and by
$\eta(\sigma(T))$ the full spectrum of $T$, namely, the union of $\sigma(T)$
and every bounded connected component of $\rho(T)$.

######  Theorem 2.1.

Let $T:X\to X$ be a bounded linear operator in a complex Banach space $X$.
Assume further that for each non-empty relatively open subset
$U\subseteq\sigma(T)$, the glocal spectral subspace
$\mathcal{X}_{T}(\overline{U})$ is dense in $X$. Then,
$\mathcal{X}^{*}_{T^{*}}(F)=\\{0\\}$ for every closed subset
$F\subsetneq\sigma(T^{*})$. As a consequence, the following properties hold:

  1. (i)

If $\sigma(T^{*})$ is not a singleton, then
$\sigma_{\text{{p}}}(T^{*})=\emptyset$.

  2. (ii)

$T^{*}$ has the Dunford’s property $(C)$.

  3. (iii)

$\sigma_{T^{*}}(x)=\sigma(T^{*})$ for every $x\in X^{*}\setminus\\{0\\}.$

  4. (iv)

$r_{T^{*}}(x)=r(T^{*})$ for every $x\in X^{*}\setminus\\{0\\}.$

  5. (v)

If $M$ is any non-trivial closed invariant subspace for $T^{*}$, then

$\sigma(T^{*})\subseteq\sigma(T^{*}|_{M})\subseteq\eta(\sigma(T^{*})).$

###### Proof.

Let $F\subsetneq\sigma(T^{*})$ be a closed subset and consider $U$ a non-empty
relatively open set in $\sigma(T^{*})$ such that
$F\cap\overline{U}=\emptyset.$ Now, by (2.1), it follows that

$\mathcal{X}^{*}_{T^{*}}(F)\subseteq\mathcal{X}_{T}(\overline{U})^{\perp}=\\{0\\},$

where the last equality holds by the density of
$\mathcal{X}_{T}(\overline{U})$.

First, let us prove property (i). Assume that $\sigma(T^{*})$ has at least two
points and take $\lambda\in\sigma(T^{*})$. As a consequence,

$\ker(T^{*}-\lambda
I)\subseteq\mathcal{X}^{*}_{T^{*}}(\\{\lambda\\})=\\{0\\},$

so $\sigma_{\text{{p}}}(T^{*})=\emptyset.$ In particular, $T^{*}$ has the SVEP
and $\mathcal{X}^{*}_{T^{*}}(F)=X^{*}_{T^{*}}(F)$ for every closed set
$F\subseteq\sigma(T^{*})$.

In order to prove properties (ii)-(v), first observe that $T^{*}$ also enjoys
the SVEP when $\sigma(T^{*})$ is reduced to a singleton, because in that case,
its point spectrum has empty interior. To show (ii), let
$F\subseteq\sigma(T^{*})$ be an arbitrary closed set. Then,

$X^{*}_{T^{*}}(F)=\begin{cases}\\{0\\},&\text{ if
}F\subsetneq\sigma(T^{*}),\\\ X^{*},&\text{ if }F=\sigma(T^{*}),\end{cases}$

and so $T^{*}$ has the Dunford property $(C)$.

Now, both properties (iii) and (iv) follow immediately from property (ii).
Finally, (v) is a direct application of [21, Proposition 1.2.16 (e)] and [25,
Theorem 0.8]. ∎

##  3\. Dunford property for invertible composition operators

In this section, we consider the Dunford property $(C)$ for invertible
composition operators $C_{\varphi}$ in $H^{p}(\mathbb{D})$ and their adjoints
$C_{\varphi}^{*}$. As it was pointed out previously, the Dunford property
$(C)$ will be independent of $1<p<\infty$ and rely upon exclusively on the
nature of the automorphism $\varphi$.

In 1996, Smith proved that if $\varphi$ is an elliptic or parabolic
automorphism, $C_{\varphi}$ is generalized scalar (see [29, Theorems 1.1 and
1.2]), and therefore a decomposable operator. Accordingly, we are left with
composition operators induced by hyperbolic automorphisms.

######  Theorem 3.1.

Let $\varphi$ be a hyperbolic automorphism of the unit disc $\mathbb{D}$.
Then, $C_{\varphi}$ does not have the SVEP in $H^{p}(\mathbb{D})$ for any
$1\leq p<\infty$ and $C_{\varphi}^{*}$ has the Dunford’s property $(C)$ for
every $1<p<\infty$.

In order to prove such a result, our first step is to understand the set of
eigenvectors of $C_{\varphi}$ when $\varphi$ is a hyperbolic automorphism. In
this case, the linear fractional model establishes similarity between
$C_{\varphi}$ and a composition operator $C_{\varphi_{r}}$ with hyperbolic
symbol $\varphi_{r}$ of the canonical form:

$\varphi_{r}(z):=\frac{z+r}{1+rz},\quad 0<r<1,$

where $\varphi_{r}$ fixes the points $\pm 1$, being $1$ the Denjoy-Wolff
point. In this light, the spectrum of $C_{\varphi_{r}}$ in
$H^{p}(\mathbb{D})$, was characterized by Nordgren [24]:

$\sigma(C_{\varphi_{r}}|_{H^{p}(\mathbb{D})})=\bigg{\\{}\lambda\in\mathbb{C}\,:\,\bigg{(}\frac{1+r}{1-r}\bigg{)}^{-1/p}\leq|\lambda|\leq\bigg{(}\frac{1+r}{1-r}\bigg{)}^{1/p}\bigg{\\}}$

and its point spectrum is
$\sigma_{\text{{p}}}(C_{\varphi_{r}}|_{H^{p}(\mathbb{D})})=\mathrm{int}\big{(}\sigma(C_{\varphi_{r}}|_{H^{p}(\mathbb{D})})\big{)}$.
In fact, for each eigenvalue $\left(\frac{1+r}{1-r}\right)^{\lambda}$
($-1/p<\operatorname{Re}(\lambda)<1/p$) of $C_{\varphi_{r}}$, the associated
eigenspace is infinite-dimensional, and an example of associated eigenvector
is given by

$w_{\lambda}(z):=\bigg{(}\frac{1+z}{1-z}\bigg{)}^{\lambda},\quad\text{ where
}\;-1/p<\mathrm{Re}(\lambda)<1/p.$

It is clear that the adjoint $C_{\varphi}^{*}$ of any composition operator
with hyperbolic automorphism symbol $\varphi$ is similar to
$C_{\varphi_{r}}^{*}$ for some $0<r<1$ and
$\sigma(C_{\varphi_{r}}^{*}|_{H^{p}(\mathbb{D})^{*}})=\sigma(C_{\varphi_{r}}|_{H^{p}(\mathbb{D})})$.

In order to apply Theorem 2.1 in this context, we prove that, in general, some
subsets of eigenfunctions of $C_{\varphi}$ feature strong spanning properties
in each Hardy space $H^{p}(\mathbb{D})$ with $1<p<\infty$ (see [12, Lemma 1]
in the Hilbert space setting).

######  Proposition 3.2.

Let $\varphi:\mathbb{D}\to\mathbb{D}$ be a hyperbolic automorphism and
consider the composition operator $C_{\varphi}:H^{p}(\mathbb{D})\to
H^{p}(\mathbb{D})$ for some $1<p<\infty$. Then, for every set
$F\subseteq\sigma_{\text{{p}}}(C_{\varphi})$ with a cluster point inside
$\sigma_{\text{{p}}}(C_{\varphi})$, the linear manifold

$\mathrm{span}\big{\\{}\\!\ker(C_{\varphi}-\lambda I):\lambda\in F\big{\\}}$

is dense in $H^{p}(\mathbb{D})$.

###### Proof.

Without loss of generality, suppose that $\varphi=\varphi_{r}$ for some
$0<r<1$. Denote by $B_{1/p}$ the open vertical strip
$\\{\lambda\in\mathbb{C}\,:\,-1/p<\mathrm{Re}(\lambda)<1/p\\}$ and consider
the following $H^{p}(\mathbb{D})$-valued function $\varrho:B_{1/p}\to
H^{p}(\mathbb{D})$, defined by $\varrho(\lambda):=w_{\lambda}$ where, as
above,

$w_{\lambda}(z):=\bigg{(}\frac{1+z}{1-z}\bigg{)}^{\lambda},\quad\text{ for
each }\;-1/p<\mathrm{Re}(\lambda)<1/p.$

Since
$C_{\varphi_{r}}w_{\lambda}=\big{(}\tfrac{1+r}{1-r}\big{)}^{\lambda}w_{\lambda}$,
observe that the mapping
$\lambda\mapsto\big{(}\tfrac{1+r}{1-r}\big{)}^{\lambda}$ sends $B_{1/p}$ onto
the point spectrum $\sigma_{\text{{p}}}(C_{\varphi_{r}})$. Hence, there is a
correspondence between $F$ and a set $\Lambda\subseteq B_{1/p}$ which, by
continuity, has a cluster point inside $B_{1/p}$.

Now, we claim that $\varrho:B_{1/p}\to H^{p}(\mathbb{D})$ is an analytic
function. To do so, consider $p^{\prime}>1$ such that
$\frac{1}{p}+\frac{1}{p^{\prime}}=1$. It is enough to show that, for each
$g\in H^{p^{\prime}}(\mathbb{D}),$ the map

$\lambda\in
B_{1/p}\mapsto\langle\varrho(\lambda),g\rangle=\frac{1}{2\pi}\int_{0}^{2\pi}\left(\frac{1+e^{i\theta}}{1-e^{i\theta}}\right)^{\lambda}\overline{g(e^{i\theta})}d\theta$

is holomorphic, which follows immediately by means of standard methods
involving Morera’s Theorem.

At this point, consider a bounded linear functional $\eta\in
H^{p}(\mathbb{D})^{*}$ such that $\langle w_{\lambda},\eta\rangle=0$ for every
$\lambda\in\Lambda$. The map $\lambda\mapsto\langle w_{\lambda},\eta\rangle$
is a holomorphic function on $B_{1/p}$ which annihilates on the set $\Lambda$.
Taking into account that $\Lambda$ has a cluster point, we conclude that
$\langle w_{\lambda},\eta\rangle=0$ for every $\lambda\in B_{1/p}$.
Accordingly, our quest reduces to prove that
$\mathrm{span}\\{w_{\lambda}:\lambda\in B_{1/p}\\}$ is dense in
$H^{p}(\mathbb{D})$. To do so, we will show that

(3.1)
$\overline{\mathrm{span}\\{w_{it}:t\in\mathbb{R}\\}}^{H^{p}(\mathbb{D})}=H^{p}(\mathbb{D})$

for every $1\leq p<\infty$. In [12, Lemma 1], the authors proved the case
$p=2$ by means of the Paley-Wiener Theorem, which clearly yields $1\leq p\leq
2$.

Fix the principal branch of the logarithm, and consider the band

$\Pi:=\\{z\in\mathbb{C}:\;-\pi/2<\mathrm{Im}(z)<\pi/2\\}.$

Note that proving (3.1) is equivalent to prove that

(3.2) $\overline{\mathrm{span}\big{\\{}e^{\lambda
z}:\,z\in\Pi,\;\mathrm{Re}(\lambda)=0\\}}^{H^{p}(\Pi)}=H^{p}(\Pi),$

where the space $H^{p}(\Pi)$ consists of all holomorphic functions
$f:\Pi\to\mathbb{C}$ for which there exists a harmonic function $u$ such that
$|f(z)|^{p}\leq u(z)$ for all $z\in\Pi$ (we refer to Duren’s book [15, Ch. 10]
for more on these spaces).

The assertion in (3.2) is, indeed, a particular instance of a more general
situation studied in [6] by Bracci, Gallardo-Gutiérrez and Yakubovich, where
the authors give a complete characterization of those non-elliptic semigroups
of holomorphic self-maps of the unit disc for which the linear span of the
eigenfunctions of the generator of the corresponding semigroup of composition
operators is weak-star dense in $H^{\infty}$ of the Koenigs domain $\Omega$
associated to the semigroup. Likewise, the authors give necessary and
sufficient conditions for completeness in $H^{p}(\Omega)$.

We borrow the proof of the instance (3.2) from [6] and include it for the sake
of completeness, referring to the manuscript for more details and further
results.

Write $\lambda=it$ with $t\in\mathbb{R}$ and observe that $e^{itz}\in
H^{\infty}(\Pi)$ for every $t\in\mathbb{R}$. Hence,
$\|e^{itz}\|_{H^{p}(\Pi)}\leq C$ for some absolute constant $C>0$.
Accordingly, the integral

$\int_{0}^{\infty}\|e^{itz}\|_{H^{p}(\Pi)}e^{-t\beta}dt$

converges uniformly on $\Pi$ for every $\beta\in\mathbb{C}$ with
$\mathrm{Re}(\beta)>0$. Given $\varepsilon>0$, take $M>0$ such that

$\int_{M}^{\infty}\|e^{itz}\|_{H^{p}(\Pi)}e^{-t\beta}\,dt<\varepsilon.$

Note that for those $\beta\in\mathbb{C}$ with $\mathrm{Re}(\beta)>-\pi/2$ the
integral

(3.3) $\int_{0}^{\infty}e^{itz}e^{-t\beta}\,dt=\frac{1}{-iz+\beta},$

for $z\in\Pi$. Analogously, for those $\beta\in\mathbb{C}$ with
$\mathrm{Re}(\beta)<\pi/2$ the integral

(3.4) $\int_{-\infty}^{0}e^{itz}e^{-t\beta}\,dt=\frac{1}{iz-\beta},$

for $z\in\Pi$.

Denote by $F_{\beta}$ the function in (3.3), namely
$F_{\beta}(z)=\frac{1}{-iz+\beta}$ for $z\in\Pi$. Clearly, $F_{\beta}\in
H^{\infty}(\Pi)$ and it is not difficult to prove that each
$I_{\beta}^{M}(z)=\int_{0}^{M}e^{itz}e^{-t\beta}\,dt$ converges to $F_{\beta}$
in $H^{\infty}(\Pi)$ as $M\to\infty$. Note that $I_{\beta}^{M}$ can be seen as
the Laplace transform of the complex Borel measure compactly supported in
$[0,+\infty)$ defined by $\mu=\chi_{[0,M]}(t)\,dt$. Approximating $\mu$ (in
the sense of measures) by finite linear combinations of measures $\mu_{n}$
given by

(3.5)
$\mu_{n}=\sum_{j=1}^{n}\mu\bigg{(}\bigg{[}\frac{(j-1)M}{n},\frac{jM}{n}\bigg{)}\bigg{)}\delta_{\frac{jM}{n}},$

it follows that the functions

$P_{n}(z)=\int_{0}^{+\infty}e^{itz}d\mu_{n}(t),\qquad(z\in\Pi)$

belong to $\hbox{span}\\{e^{itz}:t\geq 0\\}$ for all $n\in\mathbb{N}$, and
moreover, there exists $C>0$ such that $\sup_{z\in\Pi}|P_{n}(z)|\leq C$.
Hence, for every $z\in\Pi$ fixed,

$\lim_{n\to\infty}\big{|}P_{n}(z)-I_{\beta}^{M}(z)\big{|}=\lim_{n\to\infty}\left|\int_{0}^{M}e^{itz}\,d\mu_{n}(t)-\int_{0}^{M}e^{itz}e^{-t\beta}\,dt\right|=0,$

where the last limit is $0$ because the latter integral converges to $0$ as
$n\to\infty$ for $z$ fixed since $\\{\mu_{n}\\}$ converges to $\mu$ in the
sense of measures. Accordingly, $\\{P_{n}\\}$ converges weak-star in
$H^{\infty}(\Pi)$ to $I_{\beta}^{M}$.

Arguing similarly with (3.4), it follows that the rational functions with
simple poles outside $\overline{\Pi}$ are contained in the closure of
$\hbox{span}\\{e^{itz}:t\in\mathbb{R}\\}$ in the weak-star topology of
$H^{\infty}(\Pi)$. Now, each rational function in $H^{\infty}(\Pi)$ can be
uniformly approximated in $\Pi$ by rational functions with simple poles. In
addition, it is well-known that rational functions with poles outside
$\overline{\Pi}$ are weak-star dense in $H^{\infty}(\Pi)$ (see, for instance,
[14, Corollary 1]). Accordingly, $\hbox{span}\\{e^{itz}:t\in\mathbb{R}\\}$ is
weak-star dense in $H^{\infty}(\Pi)$ and therefore, weak-star dense in
$H^{p}(\Pi)$. Since $1<p<\infty$, it is dense in $H^{p}(\Pi)$ in the weak
topology and Mazur’s Lemma (see [13, Corollary 3, Chapter 2]) ensures the
density in $H^{p}(\Pi)$. ∎

As previously mentioned, in [12] the authors established that the linear span
of the eigenfunctions of any invertible composition operator of hyperbolic
symbol is dense in $H^{2}(\mathbb{D})$. Using that fact, and reminding that
for each $1<p\leq 2$ the inclusion $H^{2}(\mathbb{D})\hookrightarrow
H^{p}(\mathbb{D})$ is continuous and that $H^{2}(\mathbb{D})$ lies densely
within $H^{p}(\mathbb{D})$, Proposition 3.2 follows directly for $1<p<2$ (and,
by the same token, for $p=1$).

As a consequence of Proposition 3.2, we obtain the density of certain glocal
spectral subspaces associated to $C_{\varphi}$, which will allow us to apply
Theorem 2.1 in order to obtain the desired result.

######  Proposition 3.3.

Let $\varphi$ be a hyperbolic automorphism of $\mathbb{D}$ and $C_{\varphi}$
the induced composition operator in $H^{p}(\mathbb{D})$ for some $1<p<\infty$.
Then, the glocal spectral subspace
$\mathcal{H}^{p}_{C_{\varphi}}(\overline{U})$ is dense in $H^{p}(\mathbb{D})$
for every non-empty relatively open subset $U$ of $\sigma(C_{\varphi})$.

###### Proof.

Let $U$ be a non-empty relatively open subset of $\sigma(C_{\varphi})$. Then,

$\ker(C_{\varphi}-\lambda
I)\subseteq\mathcal{H}^{p}_{C_{\varphi}}(\\{\lambda\\})\subseteq\mathcal{H}^{p}_{C_{\varphi}}(\overline{U}),\quad\text{for
all }\lambda\in\overline{U}.$

Taking into account that $\mathcal{H}^{p}_{C_{\varphi}}(\overline{U})$ is a
linear manifold, we conclude that

$\mathrm{span}\\{\ker(C_{\varphi}-\lambda
I):\lambda\in\overline{U}\\}\subseteq\mathcal{H}^{p}_{C_{\varphi}}(\overline{U}).$

Finally, since the closure of a non-empty open set always has cluster points,
our result follows directly from Proposition 3.2. ∎

With both propositions at hands, the proof of Theorem 3.1 is almost
straightforward.

###### Proof of Theorem 3.1.

As mentioned above, for each $1<p<\infty$, the point spectrum of $C_{\varphi}$
in $H^{p}(\mathbb{D})$ is an annulus and an argument involving the
eigenfunctions yields that $C_{\varphi}$ has not the SVEP (see [29, Theorem
1.4]). On the other hand, Theorem 2.1 and Proposition 3.3 imply that
$C_{\varphi}^{*}$ has the Dunford’s property $(C)$ in $H^{p}(\mathbb{D})^{*}$
for every $1<p<\infty$. ∎

As a consequence, we are able to describe all the local spectral manifolds and
characterize some of the (local) spectral properties of $C_{\varphi}:$

######  Corollary 3.4.

Let $\varphi$ be a hyperbolic automorphism of $\mathbb{D}$ and $C_{\varphi}$
the induced composition operator in $H^{p}(\mathbb{D})$ for some $1<p<\infty$.
Let $C_{\varphi}^{*}$ be its adjoint operator in $H^{p}(\mathbb{D})^{*}$ and
$p^{\prime}>1$ such that $\tfrac{1}{p}+\tfrac{1}{p^{\prime}}=1$. Then:

  1. (i)

The point spectrum of $C_{\varphi}^{*}$ is empty.

  2. (ii)

$H^{p^{\prime}}_{C_{\varphi}^{*}}(F)=\\{0\\}$ for every closed set
$F\subsetneq\sigma(C_{\varphi_{r}}^{*})$.

  3. (iii)

$\sigma_{C_{\varphi}^{*}}(f)=\sigma(C_{\varphi}^{*})$ for every non-zero $f\in
H^{p}(\mathbb{D})^{*}$.

  4. (iv)

$r_{C_{\varphi}^{*}}(f)=r(C_{\varphi}^{*})$ for every non-zero $f\in
H^{p}(\mathbb{D})^{*}$.

  5. (v)

If $M$ is any non-trivial closed invariant subspace for $C_{\varphi}^{*}$,
then

$\sigma(C_{\varphi}^{*}|_{M})=\sigma(C_{\varphi}^{*})\quad\text{ or
}\quad\sigma(C_{\varphi}^{*}|_{M})=\big{\\{}\lambda\in\mathbb{C}:|\lambda|\leq
r(C_{\varphi}^{*})\big{\\}}.$

###### Proof.

Properties (i)-(iv) follow as a byproduct of Theorem 2.1 and Proposition 3.3.
A little extra argument is needed to state (v). Observe that

$\sigma(C_{\varphi}^{*})\subseteq\sigma({C_{\varphi}^{*}}|_{M})\subseteq\eta(\sigma(C_{\varphi}^{*})).$

Now, a result of Scroggs [26] states that if a point $\lambda$ of a hole $K$
of $\sigma(C_{\varphi}^{*})$ belongs to $\sigma({C_{\varphi}^{*}}|_{M})$, then
$K\subset\sigma({C_{\varphi}}|_{M}),$ which yields (v). ∎

##  4\. Dunford property for non-invertible composition operators

In this section, we characterize in $H^{p}(\mathbb{D})$ the Dunford property
$(C)$ for non-invertible composition operators $C_{\varphi}$ induced by linear
fractional maps as well as their adjoints. Since loxodromic maps induce
composition operators with disconnected spectrum in $H^{p}(\mathbb{D})$ (and
hence, they are decomposable) and the parabolic maps induce decomposable
composition operators [27], we are left with the hyperbolic non-automorphisms
cases (of first and second kind).

###  4.1. Hyperbolic non-automorphisms I

Let $\varphi$ be a hyperbolic non-automorphism of first kind of $\mathbb{D}$,
namely, a linear fractional map with a fixed point in $\mathbb{T}$ and the
other one in $\widehat{\mathbb{C}}\setminus\overline{\mathbb{D}}.$

######  Theorem 4.2.

Let $\varphi$ be a hyperbolic non-automorphism of first kind of $\mathbb{D}$.
Then, $C_{\varphi}$ does not have the SVEP in $H^{p}(\mathbb{D})$ for any
$1\leq p<\infty$ and $C_{\varphi}^{*}$ has the Dunford’s property $(C)$.

If $\varphi$ is a hyperbolic non-automorphism of first kind of $\mathbb{D}$,
then $\varphi$ is conjugated to one of its canonical form:

(4.1) $\varphi_{r}(z)=rz+(1-r),\qquad 0<r<1.$

Accordingly, in order to prove Theorem 4.2, it suffices to consider the
transformations described in (4.1). It is straightforward that the functions

$e_{s}(z)=(1-z)^{s},\qquad\operatorname{Re}(s)>-1/p$

belong to $H^{p}(\mathbb{D})$ and satisfy

$C_{\varphi_{r}}e_{s}=r^{s}e_{s}.$

Thus,
$\\{r^{s}:\operatorname{Re}(s)>-1/p\\}=D(0,r^{-1/p})\subseteq\sigma_{\text{{p}}}(C_{\varphi_{r}}).$
Indeed, Kamowitz [19] showed that

$\sigma(C_{\varphi_{r}})=\overline{D(0,r^{-1/p})}.$

Now, we proceed with the proof of Theorem 4.2.

###### Proof of Theorem 4.2.

We may assume without loss of generality that $\varphi=\varphi_{r}$ for some
$0<r<1$ given by (4.1). Note that the map

$s\in\big{\\{}s\in\mathbb{C}:\operatorname{Re}(s)>-1/p\big{\\}}\longmapsto
e_{s}\in H^{p}(\mathbb{D})$

is a vector-valued analytic map and consider $V=\\{z\in
D(0,r^{-1/p}):\operatorname{Re}(z)>0\\}$. The map $f:V\rightarrow
H^{p}(\mathbb{D})$ given by

$f(\lambda)=e_{\frac{\operatorname{Log}(\lambda)}{\operatorname{Log}(r)}},$

where $\operatorname{Log}:\mathbb{C}\setminus(-\infty,0]\rightarrow\mathbb{C}$
is the principal branch of the logarithm, is also a vector-valued analytic map
that satisfies

$(C_{\varphi_{r}}-\lambda)f(\lambda)=0,\qquad\text{for every }\lambda\in V,$

which proves that $C_{\varphi_{r}}$ does not have the SVEP, as claimed.

Now, we will apply Theorem 2.1 to derive the Dunford’s property $(C)$ for the
adjoint $C_{\varphi_{r}}^{*}$. First, let us show that
$\mathcal{H}^{p}_{C_{\varphi_{r}}}(\overline{U})$ is dense in
$H^{p}(\mathbb{D})$ for every non-empty relatively open subset $U$ of
$\sigma(C_{\varphi_{r}})$.

Since $\mathcal{H}^{p}_{C_{\varphi_{r}}}(\overline{U})$ contains
$\ker(C_{\varphi_{r}}-\lambda)$ for every $\lambda\in\overline{U}$, it follows
that

$\textnormal{span}\\{e_{s}:r^{s}\in\overline{U}\\}\subseteq\mathcal{H}^{p}_{C_{\varphi_{r}}}(\overline{U}).$

Now, consider
$\eta\in\textnormal{span}\\{e_{s}:r^{s}\in\overline{U}\\}^{\perp}\subset
H^{p}(\mathbb{D})^{*}.$ Then, the holomorphic map

$s\in\big{\\{}s\in\mathbb{C}:\operatorname{Re}(s)>-1/p\big{\\}}\longmapsto\langle
e_{s},\eta\rangle\in\mathbb{C}$

vanishes in every $s$ such that $r^{s}\in\overline{U}.$ Since $\overline{U}$
has accumulation points, it follows that $\langle e_{s},\eta\rangle=0$ for
every $s\in\mathbb{C}$ with $\operatorname{Re}(s)>-1/p$. Now, observe that

$\textnormal{span}\\{e_{n}:n\in\mathbb{N}\cup\\{0\\}\\}\subseteq\textnormal{span}\big{\\{}e_{s}:\operatorname{Re}(s)>-1/p\big{\\}}.$

But $\textnormal{span}\\{e_{n}:n\in\mathbb{N}\cup\\{0\\}\\}$ contains all the
polynomials, hence it follows that $f\equiv 0$ and so
$\mathcal{H}_{C_{\varphi_{r}}}^{p}(\overline{U})$ is dense as claimed. In
conclusion, Theorem 2.1 yields that $C_{\varphi_{r}}^{*}$ has the Dunford
property in $H^{p}(\mathbb{D})^{*}$. ∎

As a consequence, we deduce the following:

######  Corollary 4.3.

Let $\varphi:\mathbb{D}\rightarrow\mathbb{D}$ be a hyperbolic non-automorphism
map of first kind and $C_{\varphi}$ the induced composition operator in
$H^{p}(\mathbb{D})$ for some $1<p<\infty$. Let $C_{\varphi}^{*}$ be its
adjoint operator in $H^{p}(\mathbb{D})^{*}$ and $p^{\prime}>1$ such that
$\tfrac{1}{p}+\tfrac{1}{p^{\prime}}=1$. Then:

  1. (i)

The point spectrum of $C_{\varphi}^{*}$ is empty.

  2. (ii)

$H^{p^{\prime}}_{C_{\varphi}^{*}}(F)=\\{0\\}$ for every closed set
$F\subsetneq\sigma(C_{\varphi_{r}}^{*})$.

  3. (iii)

$\sigma_{C_{\varphi}^{*}}(f)=\sigma(C_{\varphi}^{*})$ for every non-zero $f\in
H^{p}(\mathbb{D})^{*}$.

  4. (iv)

$r_{C_{\varphi}^{*}}(f)=r(C_{\varphi}^{*})$ for every non-zero $f\in
H^{p}(\mathbb{D})^{*}$.

  5. (v)

If $M$ is a non-trivial closed invariant subspace for $C_{\varphi}^{*}$, then
$\sigma(C_{\varphi}^{*}|_{M})=\sigma(C_{\varphi}^{*}).$

###  4.4. Hyperbolic non-automorphisms II

Now, we consider $\varphi:\mathbb{D}\rightarrow\mathbb{D}$ hyperbolic non-
automorphisms of second kind of $\mathbb{D}$, namely, linear fractional maps
with a fixed point in $\mathbb{D}$ and the other one lying in $\mathbb{T}$.

######  Theorem 4.5.

Let $\varphi:\mathbb{D}\rightarrow\mathbb{D}$ be a hyperbolic non-automorphism
of second kind. Then, $C_{\varphi}$ has the Dunford’s property $(C)$ in
$H^{p}(\mathbb{D})$ for every $1<p<\infty$. Moreover,
$C_{\varphi}^{*}:H^{p}(\mathbb{D})^{*}\to H^{p}(\mathbb{D})^{*}$ does not have
the SVEP for any $1<p<\infty$.

In order to prove this result, we consider the following maps, which have $0$
and $1$ as fixed points:

(4.2) $\varphi_{r}(z)=\frac{rz}{1-(1-r)z},\qquad 0<r<1.$

It turns out that every hyperbolic non-automorphism of second kind is
conjugated to one of the maps of the form (4.2). The spectrum of these
composition operators $C_{\varphi_{r}}$ in $H^{p}(\mathbb{D})$, for each
$1\leq p<\infty$, was characterized by Kamowitz. Specifically,

$\sigma(C_{\varphi_{r}})=\overline{D(0,r^{1/p})}\cup\\{1\\}.$

Our approach is inspired by a construction of Shapiro that appears in [28, p.
864] and traces back to a joint paper with Bourdon [7]. The author shows that
the operator $C_{\varphi_{r}}$ acting on the Hilbert space $H^{2}(\mathbb{D})$
is unitarily equivalent to $I_{1}\oplus(rC_{\psi_{r}})^{*},$ where $I_{1}$ is
the identity acting on the one-dimensional space of constants,
$\psi_{r}(z)=rz+(1-r)$ is a hyperbolic non-automorphism of first kind and
$(rC_{\psi_{r}})^{*}$ acts in $H^{2}(\mathbb{D})$.

######  Lemma 4.6.

Let $\varphi_{r}:\mathbb{D}\to\mathbb{D}$ be a hyperbolic non-automorphism of
second kind of the form (4.2). Then, for every $1<p<\infty,$ the operator
$C_{\varphi_{r}}$ in $H^{p}(\mathbb{D})$ is similar to the operator
$I_{1}\oplus(rC_{\psi_{r}})^{*}$, where $\psi_{r}(z)=rz+(1-r)$ and
$(rC_{\psi_{r}})^{*}$ acts in $H^{p}(\mathbb{D})$.

###### Proof.

Let $1<{p^{\prime}}<\infty$ be such that
$\frac{1}{p}+\frac{1}{{p^{\prime}}}=1$, and consider $C_{\varphi_{r}}^{*}$ in
$H^{p^{\prime}}(\mathbb{D})$. Let
$H_{0}^{p^{\prime}}(\mathbb{D}):=zH^{p^{\prime}}(\mathbb{D}),$ that is, the
closed subspace of $H^{p^{\prime}}(\mathbb{D})$-functions vanishing at $0$. It
is easy to check that each functional in $H_{0}^{p^{\prime}}(\mathbb{D})^{*}$
can be identified with a function of $H_{0}^{p}(\mathbb{D})$, with the same
dual pairing that relates $H^{p^{\prime}}(\mathbb{D})^{*}$ and
$H^{p}(\mathbb{D})$.

Since $1$ is the reproducing kernel at $0$ and $\varphi_{r}(0)=0$, it follows
that the rank-one operator $1\otimes 1$ in $H^{p^{\prime}}(\mathbb{D})$
commutes with $C_{\varphi_{r}}^{*}.$ Observe that $1\otimes 1$ is a projection
onto the one-dimensional space consisting of constants $\langle 1\rangle$, and
so $I-1\otimes 1$ is a projection onto $zH^{p^{\prime}}(\mathbb{D}).$ Hence,
$C_{\varphi_{r}}^{*}$ also commutes with $I-1\otimes 1$ and therefore
$zH^{p^{\prime}}(\mathbb{D})$ is invariant under $C_{\varphi_{r}}^{*}$.

Now, denote by $M_{z}:H^{p^{\prime}}(\mathbb{D})\rightarrow
H_{0}^{p^{\prime}}(\mathbb{D})$ and
$M_{1/z}:H_{0}^{p^{\prime}}(\mathbb{D})\rightarrow H^{p^{\prime}}(\mathbb{D})$
the bounded operators of multiplication by $z$ and $1/z$, respectively.
Observe that these operators are inverses of each other. Similar arguments as
in [7, p. 35-36] yield that

$\langle C_{\varphi_{r}}^{*}f,z^{n}\rangle=\langle
M_{z}(rC_{\psi_{r}})M_{1/z}f,z^{n}\rangle$

for every $f\in H_{0}^{p^{\prime}}(\mathbb{D})$ and $n\in\mathbb{N}$. Since
polynomials are dense in $H_{0}^{p}(\mathbb{D})$, it follows that

$C_{\varphi_{r}}^{*}|_{H_{0}^{p^{\prime}}(\mathbb{D})}=M_{z}\big{(}rC_{\psi_{r}}|_{H^{p^{\prime}}(\mathbb{D})}\big{)}M_{1/z}.$

Finally, since $H^{p^{\prime}}(\mathbb{D})=\langle 1\rangle\oplus
H_{0}^{p^{\prime}}(\mathbb{D}),$ and $C_{\varphi_{r}}^{*}|_{\langle
1\rangle}=I_{1}$, we deduce that $C_{\varphi_{r}}^{*}$ is similar to
$I_{1}\oplus(rC_{\psi_{r}})$, and the result follows by taking adjoints. ∎

With this result at hands, we are in position to prove Theorem 4.5.

###### Proof of Theorem 4.5.

Without loss of generality, we may assume that $\varphi=\varphi_{r}$ for some
$0<r<1$ of the canonical form in (4.2). By Lemma 4.6, using the decomposition
$H^{p}(\mathbb{D})=\langle 1\rangle\oplus H_{0}^{p}(\mathbb{D})$, the operator
$C_{\varphi_{r}}$ is similar to $T:=I_{1}\oplus(rC_{\psi_{r}})^{*}$. Thus, by
Corollary 4.3, it follows that $\sigma_{\text{{p}}}(C_{\varphi_{r}})=\\{1\\}.$
As a consequence, both $C_{\varphi_{r}}$ and $T$ have the SVEP.

Now, let $F\subsetneq\sigma(C_{\varphi_{r}})$ be a closed set. Consider
$f=f_{1}\oplus f_{2}\in H^{p}_{T}(F)$ and
$\Gamma=\Gamma_{1}\oplus\Gamma_{2}:\mathbb{C}\setminus F\rightarrow\langle
1\rangle\oplus H_{0}^{p}(\mathbb{D})$ satisfying

$(T-zI)\Gamma(z)=f\qquad\text{for every }z\in\mathbb{C}\setminus F.$

Observe that $(I_{1}-zI)\Gamma_{1}(z)=f_{1}$ and
$((rC_{\psi_{r}})^{*}-zI)\Gamma_{2}(z)=f_{2}$ for every
$z\in\mathbb{C}\setminus F$, so $H^{p}_{T}(F)=H^{p}_{I_{1}}(F)\oplus
H^{p}_{rC_{\psi_{r}}^{*}}(F).$ Finally, by Theorem 4.2,
$H^{p}_{rC_{\psi_{r}}^{*}}(F)$ is closed, so $H^{p}_{C_{\varphi_{r}}}(F)$ is
closed as well and $C_{\varphi_{r}}$ has the Dunford property $(C)$, as
claimed.

It remains to show that $C_{\varphi_{r}}^{*}$ does not have the SVEP in
$H^{p}(\mathbb{D})^{*}$. Just recall that $C_{\varphi_{r}}^{*}$ is similar to
$I_{1}\oplus(rC_{\psi_{r}})$ acting in $H^{p^{\prime}}(\mathbb{D})$, and it is
straightforward to check that the fact that $C_{\psi_{r}}$ does not enjoy the
SVEP (see Theorem 4.2) implies that $C_{\varphi_{r}}^{*}$ does not either,
which yields the statement. ∎

Indeed, the identity $H^{p}_{T}(F)=H^{p}_{I_{1}}(F)\oplus
H^{p}_{rC_{\psi_{r}}^{*}}(F)$ obtained previously gives us plenty of
information about local spectral features of $C_{\varphi_{r}}$, which we
recollect in our next result:

######  Corollary 4.7.

Let $\varphi_{r}:\mathbb{D}\to\mathbb{D}$ be a hyperbolic non-automorphism of
second kind of the form (4.2) and $1<p<\infty.$ Consider $C_{\varphi_{r}}$
acting on $H^{p}(\mathbb{D})$ and $F\subsetneq\sigma(C_{\varphi_{r}})$ a
closed set. Then:

  1. (i)

$\sigma_{\text{{p}}}(C_{\varphi_{r}})=\\{1\\}$ and its associated eigenspace
is $\langle 1\rangle$.

  2. (ii)

$\displaystyle H^{p}_{C_{\varphi_{r}}}(F)=\begin{cases}\\{0\\},&\text{if
}F\subsetneq\overline{D(0,r^{1/p})},\\\ H_{0}^{p}(\mathbb{D}),&\text{if
}F=\overline{D(0,r^{1/p})},\\\ \langle 1\rangle,&\text{if
}F=\\{1\\}.\end{cases}$

  3. (iii)

If $f$ is a non-zero function in $H^{p}(\mathbb{D})$, then

$\sigma_{C_{\varphi_{r}}}(f)=\begin{cases}\\{1\\},&\text{if }f\in\langle
1\rangle,\\\ \overline{D(0,r^{1/p})},&\text{if }f\in H_{0}^{p}(\mathbb{D}),\\\
\sigma(C_{\varphi_{r}}),&\text{otherwise}.\end{cases}$

  4. (iv)

If $f$ is a non-zero function in $H^{p}(\mathbb{D})$, then

$r_{C_{\varphi_{r}}}(f)=\begin{cases}r^{1/p},&\text{if }f\in
H_{0}^{p}(\mathbb{D}),\\\ 1,&\text{otherwise}.\end{cases}$

  5. (v)

Let $M\subset H^{p}(\mathbb{D})$ be a non-trivial closed invariant subspace
for $C_{\varphi_{r}}$. Then,

$\sigma(C_{\varphi_{r}}|_{M})\in\big{\\{}\\{1\\},\overline{D(0,r^{1/p})},\sigma(C_{\varphi_{r}})\big{\\}}.$

###### Proof.

Note that (i) has been actually proved in the proof of Theorem 4.5. To prove
(ii), recall that $C_{\varphi_{r}}$ is similar to
$T:=I_{1}\oplus(rC_{\psi_{r}})^{*}$ (where $(rC_{\psi_{r}})^{*}$ is acting on
$H^{p}(\mathbb{D})$) and $H^{p}_{T}(F)=H^{p}_{I_{1}}(F)\oplus
H^{p}_{rC_{\psi_{r}}^{*}}(F)$ for every closed set
$F\subsetneq\sigma(C_{\varphi_{r}}).$ By Corollary 4.3, if
$F\subsetneq\overline{D(0,r^{1/p})}$, then
$H^{p}_{rC_{\psi_{r}}^{*}}(F)=\\{0\\}$. Since $H^{p}_{I_{1}}(F)=\\{0\\}$ as
well, we deduce that $H^{p}_{C_{\varphi_{r}}}(F)=\\{0\\}$ in that case.

On the other hand, if $F=\overline{D(0,r^{1/p})}$, then
$H^{p}_{rC_{\psi_{r}}^{*}}(F)=H^{p}(\mathbb{D})$, and bringing it back via the
isomorphism $M_{z}:H^{p}(\mathbb{D})\rightarrow H_{0}^{p}(\mathbb{D})$ we
deduce that $H^{p}_{C_{\varphi_{r}}}(F)=H_{0}^{p}(\mathbb{D}),$ as stated.
Finally, it is straightforward to check that $H^{p}_{T}({{1}})=\langle
1\rangle,$ which completes the proof of (ii). The properties (iii)-(v) follow
easily from (ii) and the basic properties of local spectra and spectral
subspaces. ∎

### A final remark: stability of the Dunford property

Finally, in the class of linear fractional composition operators, we address
the question whether the product of two commuting operators with the Dunford
property $(C)$ inherits this property (see [1] for recent results). It is
well-known that if two linear fractional composition operators $C_{\varphi}$
and $C_{\psi}$ commute, $\varphi$ and $\psi$ has the same fixed points (see
[9], for instance). In addition, $C_{\varphi}C_{\psi}$ is the composition
operator $C_{\psi\circ\varphi}$ and each of the classes in the classification
of the linear transformations of $\mathbb{D}$ according to their fixed points
remains invariant under composition. Hence, keeping in mind the Table 1, we
are left with two cases: if $\varphi$ is a parabolic automorphism and $\psi$ a
parabolic non-automorphism (or viceversa) and if $\varphi$ is an elliptic
automorphism an $\psi$ a loxodromic map with fix points 0 and $\infty$ (or
viceversa). In the first case, $\psi\circ\varphi$ is a parabolic non-
automorphism and in the second one is a loxodromic map. Accordingly, in both
cases $C_{\psi\circ\varphi}$ is decomposable, and therefore satisfies the
Dunford property $(C)$.

## References

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# Searches for Supersymmetry (SUSY) at the Large Hadron Collider

###### Abstract

This paper provides an overview of supersymmetry (SUSY) and the ongoing
efforts to detect SUSY particles at the Large Hadron Collider (LHC). SUSY
proposes corresponding ”sparticles” for each Standard Model particle, with the
potential to resolve the hierarchy problem, identify dark matter, and unify
forces at high energy scales. We explore the key observables central to SUSY
searches at the LHC, such as effective mass ($M_{\text{eff}}$), scalar sum of
transverse momenta ($H_{T}$), and stransverse mass ($M_{T2}$)–each of which is
instrumental in distinguishing SUSY particles by their mass scales and decay
characteristics. The paper also discusses the challenges of differentiating
SUSY signals from Standard Model backgrounds, particularly in scenarios
without significant missing transverse momentum ($E_{T}^{\text{miss}}$).

A critical aspect of these searches is achieving high signal purity while
effectively managing background detection. The LHC employs sophisticated
detection techniques to enhance the signal-to-noise ratio–crucial for
isolating genuine SUSY events from prevalent Standard Model processes such as
multijet productions, gauge bosons with jets, and top quark pairs. Through
this examination, the paper aims to illuminate the challenges and potential
breakthroughs in SUSY searches, assessing their broader implications for
fundamental physics.

University of Pennsylvania, Philadelphia, PA, USA Sophie
<EMAIL_ADDRESS>On behalf of all authors, the corresponding author
states that there is no conflict of interest.

##  1 Introduction

Supersymmetry (SUSY) is a novel theory that expands upon our current Standard
Model of particle physics. While traditional symmetries such as rotational,
Lorentz, and gauge symmetries often relate similar types of particles, SUSY is
unique in that it establishes a relationship between two fundamentally
different classes of particles—bosons and fermions. This theory posits that
for every boson, a corresponding fermion exists, and vice versa. This
relationship suggests that particles should have supersymmetric counterparts,
often referred to as “sparticles,” which share the same mass but differ in
spin by a half-integer value [1][2].

One of the key impacts of SUSY is its potential to unify the fundamental
forces and particles into a single framework, reconciling Standard Model
shortcomings such as the hierarchy problem and the nature of dark matter.
Experimental searches for sparticles, such as those conducted at particle
accelerators, are therefore crucial in verifying this theory. Many such
experiments conducted by the ATLAS group at the Large Hadron Collider (LHC)
are designed to detect the unique signatures that sparticles would leave
behind if they were produced in particle collisions. For instance, certain
supersymmetric models predict the production of sparticles like gluinos [3].

This paper contains six sections. In Section 2 we delve into the theoretical
and experimental underpinnings of supersymmetry, outlining its principles,
implications, and potential role in reconciling Standard Model shortcomings.
In Section 3 we detail the methods and challenges of detecting SUSY,
particularly via high-energy collider experiments conducted at the LHC. In
section 4 we discuss the process of distinguishing between SUSY signal and SM
background, with an emphasis on the detection of key SUSY observables. In
Section 5 we discuss the interpretation of detector results, comparing the
implications of these interpretations in Section 6 after removing assumptions
like R-Parity conservation.

##  2 SUSY Principles and Motivations

###  2.1 The Standard Model

The Standard Model (SM) of particle physics describes fundamental particles
and their interactions with remarkable precision. It describes matter as
fundamentally composed of various generations of quarks and leptons, each with
specific properties such as mass, electric charge, and spin. Likewise, it
sufficiently explains particle interactions through the exchange of gauge
bosons, which serve as carriers for the strong, weak, and electromagnetic
forces. Notably, a recent achievement within this framework has been the
discovery of the Higgs boson: by providing a mechanism for particle mass
generation, the Higgs boson discovery thus completed the SM [4].

Despite its successes, the SM is nonetheless considered an incomplete
representation of the fundamental forces, particularly because it does not
address shortcomings such as the hierarchy problem: this problem highlights
the unnatural fine-tuning required to keep the Higgs boson mass at its
relatively low value despite theoretical predictions that suggest it should be
much higher. Likewise, the SM leaves several crucial questions unanswered,
such as the origins of dark matter, the matter-antimatter asymmetry in the
universe, and the nature of mass. Furthermore, it does not explain the origin
of particle quantum numbers, the mechanisms by which neutrinos acquire mass,
or offer a unified framework for all interactions that incorporates gravity
[3][5].

Supersymmetry (SUSY) stands out as a compelling theoretical extension of the
Standard Model because it proposes solutions to many of these unresolved
issues. SUSY not only provides a framework that could potentially incorporate
gravity with other fundamental interactions but offers an explanation for the
origin and composition of dark matter. In addition to addressing the matter-
antimatter asymmetry, its inclusion of sparticle interactions offers a
compelling resolution to the hierarchy problem. Finally, supersymmetry
suggests a way to explore the origins of particle quantum numbers and neutrino
masses within a unified theoretical framework, possibly pointing towards a
grand unified theory [1][2].

###  2.2 SUSY Principles

Supersymmetry (SUSY) is a theoretical model of particle physics that proposes
a unique type of symmetry between two fundamental classes of particles:
bosons, which are force carriers, and fermions, which constitute matter. In
the SUSY framework, each fermion of the Standard Model, such as quarks and
leptons, is paired with a corresponding bosonic ”sfermion” (e.g.,
($\tilde{f}_{L,R}$) ) with zero spin. These sfermions share the same gauge
quantum numbers111The underlying symmetry of SUSY, wherein its generators
commute with those of internal symmetries, is what preserves these crucial
gauge quantum numbers. This preservation is key to ensuring that the
interactions of the sparticles mirror those of their Standard Model
counterparts. as their fermionic counterparts, maintaining the balance and
symmetry that gauge theories require [5].

SUSY also modifies the role of the Higgs boson by introducing its fermionic
partners, termed Higgsinos, which are integrated within the Higgs chiral
supermultiplets. Further extending its reach, SUSY associates the gauge fields
responsible for mediating the fundamental forces with Majorana fermions known
as gauginos.

The pairings that SUSY offers not only preserves the functional symmetry of
forces but also ensures an equivalence in the number of bosonic and fermionic
degrees of freedom, which is vital for the internal consistency of the model.
By doing so, SUSY provides a structured and unified approach to understanding
particle interactions, bridging the gap between matter particles and force
carriers. Such consistency not only boosts the predictive power of SUSY but
also enhances its potential to be empirically verified in high-energy physics
experiments[4][5].

###  2.3 ”Soft” Breaking

Supersymmetry introduces several distinctive features that set it apart from
the Standard Model, notably its ability to accommodate renormalizable
interactions that can potentially violate baryon and lepton numbers. This is
facilitated by SUSY’s inclusion of scalar sparticles, which could lead to
significant phenomena such as proton decay–especially if the masses of these
sparticles are below the TeV scale [2].

To manage potential issues such as unwanted rapid decay processes, SUSY is
theorized to be a broken symmetry. Known as ”soft” breaking, this phenomena is
crucial because it mitigates SM fine-tuning problems while preventing the
rapid decay phenomena that would otherwise contradict observed physical data.
Such an approach suggests that while sparticles theoretically exist, they
remain undetected due to their significantly higher-than-expected masses [1].

Soft SUSY is supported by the fact that even light sparticles such as the
selectron have not been detected at expected mass scales. These aspects
highlight that evidence of supersymmetry may only become apparent at energy
scales not yet accessible experimentally [6].

###  2.4 Motivations for SUSY

SUSY is appealing because it answers many of the key questions that the
Standard Model cannot address alone. Of these, the hierarchy problem continues
to pose a significant theoretical challenge to current models of the Higgs
boson and its interactions: here, the Higgs boson’s mass requires extreme
fine-tuning to maintain its observed low value against potential high quantum
corrections up to the Planck scale. SUSY offers a compelling solution to this
problem by introducing superpartners for each particle: these interactions
naturally cancel out large quantum corrections at the TeV scale, which
resolves the need for fine-tuning [1][2].

Supersymmetry is also favorable because it provides a potential candidate for
dark matter, particularly when R-parity is conserved. R-parity is a crucial
quantum number derived from the baryon number (B), lepton number (L), and the
spin quantum number (S) of the particles: defined as $P=(-1)^{3(B-L)+2S}$,
this relation posits that Standard Model particles have $P=+1$ while SUSY
particles have $P=-1$. R-parity Conservation (RPC) is particularly favorable
because it implies that SUSY particles can only be produced in pairs. RPC also
theorizes a lightest supersymmetric particle (LSP) that is stable, deeming it
a suitable dark matter candidate. This stability arises because, with
conserved R-parity, the LSP cannot decay into lighter standard model particles
alone222For this exposition, it is assumed that R-parity violating (RPV)
couplings are absent or sufficiently small for the LSP’s lifetime to exceed
the age of the universe. This allows the LSP to still function as a dark
matter candidate. Considerations such as these significantly impact the
potential detection strategies of SUSY–as well as the theoretical implications
of SUSY in providing a comprehensive understanding of dark matter in the
universe. [2][5][6].

Furthermore, supersymmetry inherently incorporates gravitational interactions
by predicting the existence of the graviton and its SUSY partner, the
gravitino–allowing it to potentially unify gravity with other fundamental
forces in a single theoretical framework. Additionally, SUSY predicts that the
coupling constants of the strong, weak, and electromagnetic forces converge at
a grand unification scale when extrapolated to the early universe, theorizing
a unification that is not achieved by the SM alone [4].

These aspects make SUSY a robust theoretical extension of the Standard Model,
promising to address fundamental questions about the universe’s structure and
the underlying principles governing it.

##  3 Searches for Supersymmetry at the LHC

The Large Hadron Collider (LHC) plays a pivotal role in the search for SUSY,
utilizing advanced experimental techniques to explore beyond the Standard
Model. In fact, the detection and analysis strategies it employs are well
designed to reveal the even some of the most subtle signatures of SUSY
particles [7].

###  3.1 The ATLAS Detector

Figure 1: Diagram of the ATLAS Detector, including the components it employs
to accurately measure particle interactions [14].

The ATLAS detector at the Large Hadron Collider (LHC) (Figure 1) is designed
with a cylindrical geometry that provides near-complete coverage in solid
angle, crucial for detecting beyond the Standard Model interactions through
proton collisions. This general-purpose detector utilizes a series of systems
to identify and measure the properties of particles produced in such high-
energy collisions:

  * •

Inner Detector: Central to the detector’s function, the Inner Detector
specializes in reconstructing charged particle tracks in the pseudorapidity
region $|\eta|<2.5$. Located within a 2 T axial magnetic field, it uses a
combination of silicon pixel detectors, silicon microstrip detectors, and a
transition radiation tracker (TRT) to measure charged-particle momenta.
Likewise, it identifies electron trails via ionization in xenon- or argon-
based gases.

  * •

Calorimeters: Above the Inner Detector, the electromagnetic and hadronic
calorimeters span a region up to $|\eta|<4.9$, capturing and measuring
particle energy. These calorimeters function through layers that absorb and
induce particle showers, with interspersed sampling layers quantifying the
energy of these showers. Notably, the hadronic calorimeter plays a key role in
identifying jets initiated by b-quarks.

  * •

Muon Spectrometer: For muons, which are typically too massive to leave
significant energy in the calorimeters, the muon spectrometer (MS) extends up
to $|\eta|<2.7$ and tracks muon momenta using its surrounding trigger and
tracking chambers in a toroidal magnetic field.

  * •

ATLAS Trigger System: Comprised of both hardware and software components, the
ATLAS Trigger System is essential for reconstructing and analyzing real and
simulated data events. It efficiently pinpoints the locations and identities
of particles, facilitating the detection of potential new physics phenomena
like those predicted by supersymmetry (SUSY) [7].

This setup underscores the sophisticated infrastructure required to explore
SUSY and other theoretical models at the frontiers of particle physics.

###  3.2 Theorized Production of SUSY Particles

At the LHC, SUSY particles are theorized to be produced primarily through
high-energy collisions that potentially generate various types of sparticles.
For instance, gluinos and squarks are expected to dominate SUSY production due
to their significant roles in SUSY models. Electroweakinos, sleptons, and
sneutrinos are also sought after but have smaller production cross-sections.
An example where theoretical assumptions play a role in these production
dynamics is the conservation of R-parity in many SUSY models, which predicts
that sparticles must be produced in pairs. This affects both the production
rate and the decay patterns observed.

One of the primary methods of detecting SUSY particles is via the observation
of missing transverse momentum ($\text{E}_{T}^{\text{miss}}$), which signals
the presence of non-interacting particles escaping detection. The measurement
of $\text{E}_{T}^{\text{miss}}$ is crucial, as it often points to the
stability and weakly interacting nature of the lightest SUSY particle (LSP).
The hallmark of many SUSY searches, a stable LSP typically leaves no
detectable signal other than its contribution to missing energy [1][2].

##  4 Signal Purity and Background Measurement

The SUSY phenomenology at the LHC is heavily influenced by the underlying
properties of the LSP and other sparticles. In fact, the expected detector
signatures are determined by how these particles decay and interact with the
LHC’s detectors. However, distinguishing these SUSY signatures from the
background noise created by Standard Model processes poses a far more
significant challenge. This is because Standard Model events often mimic SUSY
signals, necessitating sophisticated analytical techniques to differentiate
between potential discoveries and false positives [8].

###  4.1 SM Background

The difficulty in differentiating SUSY from SM processes is primarily due to
the similarities in their decay products. For instance, the SM can result in
final states that include multiple jets, leptons, or missing transverse
momentum—-all key signatures in SUSY searches. The extent to which a specific
SM process interferes with SUSY detection depends on several factors:

  * •

Production Cross-Section: This refers to the probability of occurrence for a
particular process within the collider. Higher cross-section processes are
more common and thus more likely to contribute to background noise in SUSY
searches.

  * •

Branching Fraction: This metric indicates the likelihood of a process decaying
into the specific final state of interest. A higher branching fraction for a
decay path similar to a SUSY signal increases the background complexity.

  * •

Event Selection: The strategies used to select and analyze events
significantly affect how well SUSY signals can be isolated from SM
backgrounds. Effective event selection criteria are essential to enhance
signal sensitivity and reduce background interference [2][8][9].

Presented by the CMS Collaboration, Figure 2 compares current measurements of
SM cross-sections with their theoretical predictions. This summary is critical
for SUSY search strategies–allowing researchers to understand the landscape of
potential background processes and tailor their detection methods accordingly.
By analyzing these cross-sections, researchers may prioritize which SM
processes might most likely mimic SUSY events and develop more targeted
approaches to filter out these backgrounds [1].

Figure 2: Current measurements of relevant SM cross-sections and their
comparison to theoretical predictions [1].

###  4.2 Background Estimation

To accurately estimate background from Standard Model processes, researchers
at the LHC use simulated samples and actual collision data from control
samples. The control samples that are analyzed are carefully chosen so that
they are representative of specific SM processes but largely free from the
signals being searched for. By using these samples, experimentalists may model
the backgrounds that mimic SUSY signals more accurately.

For backgrounds that result primarily from strong interactions, collision data
is often used due to the significant uncertainties associated with the
theoretical cross-sections of these processes. Conversely, backgrounds from
electroweak processes are usually estimated using simulated samples because
their theoretical predictions are generally more reliable.

Each search develops a tailored method for estimating backgrounds based on the
specific event selection strategies employed. This approach is designed to
maximize the discovery potential by optimizing the discrimination between the
expected signal and the background noise. The ATLAS and CMS collaborations
enhance this process by defining ”signal regions”–categories within the data
characterized by SUSY-like event selection criteria. The classification of
such events based on their content and properties is essential for the
effective identification of genuine SUSY events amidst the vast data collected
at the LHC [1][2][7][8][9].

###  4.3 Key SUSY Observables

SM background is largely dominated by processes like multijet production,
gauge bosons associated with jets, and top quark pairs. Such processes
typically exhibit only mild missing transverse momentum
($E_{T}^{\text{miss}}$), barring significant mismeasurement of jet signal.
Likewise, its decay kinematics are indicative of lower mass particles, with
the heaviest particle at about 175 GeV. To enhance the detection of SUSY
particles from such processes, researchers search for several SUSY observables
that are correlated to the mass scale of the potential parent particle or the
mass differences within the SUSY spectrum:

  * •

Effective Mass ($M_{\text{eff}}$): Defined as the scalar sum of the transverse
momenta ($p_{T}$) of the jets plus $E_{T}^{\text{miss}}$. $M_{\text{eff}}$ is
particularly sensitive to events involving heavier particles like gluinos,
showing a harder spectrum which helps differentiate these events from SM
backgrounds.

  * •

Total Transverse Momentum ($H_{T}$): Similar to $M_{\text{eff}}$, this is the
scalar sum of the transverse momenta of all jets, enhancing sensitivity to
events with high-energy jets.

  * •

Sum of Masses of Large-Radius Jets ($M_{\text{LRJ}}$): Focuses on the combined
mass of large-radius jets, useful for identifying events where these jets may
originate from the decay of heavy particles [9][10].

Another critical observable is the stransverse mass ($M_{T2}$), a
generalization of the transverse mass concept applied to events featuring two
invisible particles. $M_{T2}$ provides a clear endpoint measurement close to
the mass of the parent particle–aiding in the distinction from SM processes
where such endpoints would not typically appear. Optimal sensitivity is
achieved by classifying events based on characteristics such as the presence
of b-jets, whether W/Z/h bosons or top quarks are boosted, and the number,
charge, and flavor of leptons involved [9][10].

###  4.4 Complex Final States

Detecting SUSY is further complicated by additional high cross-section
Standard Model processes. QCD multijet production, a common SM process, does
not typically include energetic leptons or neutrinos. However, jets can be
misidentified as leptons, and missing transverse momentum
($\text{E}_{T}^{\text{miss}}$) may arise from mis-measurement of visible
particles. Such errors can cause multijet events to falsely resemble SUSY
signatures, which often rely on detecting events with leptons and significant
$\text{E}_{T}^{\text{miss}}$. Electroweak (EWK) production of W and Z bosons
and top quark pairs also contribute to the background, particularly when these
particles decay into combinations of jets, leptons, and neutrinos, mimicking
potential SUSY events. Additionally, rare SM processes like triple gauge boson
production or Higgs boson production become relevant due to the LHC’s high
energy and large dataset. These processes can produce complex final states
with multiple jets, leptons, and $\text{E}_{T}^{\text{miss}}$, further
blurring the lines between SM backgrounds and genuine SUSY signals [11].

##  5 Interpreting SUSY Search Results

When observed search data aligns with SM background expectations within
statistical and systematic uncertainties, the findings are used to set an
upper limit on the SUSY production cross-section. This interpretation utilizes
likelihood fits under three hypotheses: a background-only model, a model-
independent signal plus background, and a model-dependent signal plus
background. Each fit incorporates data from both signal and control regions
defined in the analysis–reducing systematic uncertainties by constraining
expected backgrounds to observed yields [12].

###  5.1 Gluino Detection

Historical data from the Tevatron collider and predictions from various SUSY
scenarios suggest that gluinos, a fundamental component in many SUSY models,
likely have masses exceeding several hundred GeV. Despite these high mass
predictions, gluinos are expected to have sufficiently large cross-sections,
making their pair production a classical focus of SUSY searches at high-energy
proton-proton colliders. Assuming R-parity is conserved, gluinos are produced
in pairs and rapidly decay into final states featuring standard model
particles and two neutralinos (denoted as $\tilde{\chi}^{0}_{1}$), which
typically carry away significant missing transverse momentum
($\text{E}_{T}^{\text{miss}}$) due to their high momentum and non-interaction
with the detector [1][2][9][10].

Figure 3: R-parity conserving decays of gluinos under different SUSY spectrum
assumptions: (a) When gluinos are lighter than squarks, they decay via virtual
squarks, but this is suppressed if squarks are significantly heavier. (b) If
gluinos are heavier than squarks, the decay proceeds directly through squarks.
(c, d, e) When electroweakinos and sleptons are lighter than gluinos, these
provide feasible decay paths [1]

The specific decay paths of gluinos depend on the SUSY mass spectrum, which
remains largely unknown. However, many decay scenarios are characterized by
the production of multiple energetic jets—-resulting from the large mass
difference between the gluinos and their decay products—-as well as
substantial $\text{E}_{T}^{\text{miss}}$. This is illustrated in Figure 3,
which depicts several R-parity conserving gluino decay diagrams:

  * •

Gluino Mass Smaller Than Squark Mass (Scenario a): In this configuration, the
gluino can decay through a process involving virtual squarks. However, if the
squark mass is several hundred GeV larger than the gluino mass, this decay
mode becomes suppressed. The large mass difference increases the virtuality of
the squarks involved, thereby reducing the probability of this decay channel
due to the higher energy required to bridge the mass gap.

  * •

Gluinos Heavier Than Squarks (Scenario b): This arrangement allows gluinos to
decay directly via real squarks, which are physically present and lighter than
the gluinos. This mode is typically not suppressed because the real squarks
can more readily mediate the decay, leading to a more straightforward gluino
decay process compared to the virtual squark scenario.

  * •

Electroweakinos and Sleptons Lighter Than Gluinos (Scenarios c, d, e): When
electroweakinos (the supersymmetric partners of the electroweak gauge bosons
and the Higgs boson) and sleptons (the supersymmetric counterparts of leptons)
are lighter than gluinos, these particles provide additional decay channels.
Gluinos can decay into these lighter sparticles, facilitating a range of decay
processes that contribute to the richness of SUSY phenomenology at colliders
[1][2][9][11][12].

Such combinations of high-energy jets and missing energy are therefore a
hallmark feature in the search for gluino sparticles.

###  5.2 Using Jets to Future Search Sensitivity

As part of ongoing efforts to detect signatures of supersymmetry (SUSY) at the
Large Hadron Collider (LHC), the use of jets has become a cornerstone
technique in enhancing the sensitivity of these searches. Jets are algorithms
that help manage and interpret the complex data from collision events. Large-R
jets (R = 1.0), which cover a wide angle (2 radians), effectively consolidate
the showers of particles from these decays into fewer, more manageable
clusters. This capability is particularly beneficial for reconstructing high-
transverse momentum (pT) Higgs bosons, where decay products are tightly
collimated. By using large-R jets, the possible combinations of b-quarks
needed for accurate Higgs reconstruction are simplified, improving the
likelihood of identifying the correct decay chains critical for SUSY detection
[4][13][14].

On the other hand, small-R jets (R = 0.4) play a crucial role in more
precisely mapping the trajectories of individual b-quarks, especially in lower
pT decay scenarios. These jets are adept at capturing more dispersed b-quark
products, which are typical in less energetic Higgs decays, allowing for
better granularity in analysis. However, they face challenges in high pT
conditions where decay products are close together, as they may fail to merge
these closely situated particles effectively [20].

Using jets in this way not only enhances the detection of Higgs bosons in
SUSY-related events but also significantly reduces the interference from
pileup—irrelevant collisions that can obscure the data. By optimizing jet
usage, researchers can improve mass resolution and better distinguish SUSY
signals from Standard Model backgrounds. This technique is integral to
refining the analysis of high-energy collisions at the LHC, aiding in the
broader goal of confirming or refuting the existence of supersymmetry through
precision measurements and detailed reconstructions of potential SUSY events
[13][14].

##  6 Alternate Strategies for SUSY Detection

###  6.1 R-Parity Violating Decays

Despite extensive searches, ATLAS and CMS have reported no significant
deviations from the SM-only hypothesis. Expanding the search, both ATLAS and
CMS have also considered models with gluinos that undergo R-parity violating
(RPV) decays. In these scenarios, SUSY particles decay entirely into SM
particles, resulting in no significant $E_{T}^{\text{miss}}$ and thus,
kinematics that closely mimic SM backgrounds. This absence of missing energy
allows for the full reconstruction of events, leveraging observables like
$M_{\text{LRJ}}$ to examine if large-radius jets could be from the hadronic
decay of heavy SUSY particles. These RPV diagrams are shown in Figure 4.

Figure 4 depicts an example decay that, in scenarios where R-parity is
violated, gluinos can decay directly into three quarks through interactions
governed by the RPV 3-quark couplings, $\lambda^{\prime\prime}$. These decays
bypass typical supersymmetric cascade processes and directly produce Standard
Model particles, which could enhance detectability at colliders. The strength
of the $\lambda^{\prime\prime}$ couplings plays a crucial role in determining
the frequency of these decays, providing a unique pathway to potentially
observe signs of supersymmetry in environments where R-parity conservation is
not assumed [10][11][12].

Figure 4: Example R-parity violating decays of gluinos. Here, the strengths
depicted are proportional to the RPV 3-quark couplings
$\lambda^{\prime\prime}$.

###  6.2 Targetting R-parity Even Neutral Heavy Higgs Bosons

Typically, SUSY searches at colliders face significant challenges due to the
necessity of pair-producing sparticles when R-Parity is conserved. Not only
does the stable LSP often escape detection in these searches, but the need to
produce two sparticles simultaneously demands high-energy collisions. This
further complicates the direct reconstruction of sparticle masses through
traditional resonance techniques–as a particle’s properties are often inferred
from a peak in an energy distribution that is indicative of its mass
[1][2][10].

An alternative to searching for R-parity conserving sparticles focuses on the
R-parity even neutral heavy Higgs bosons, specifically the heavy scalar ($H$)
and the pseudoscalar ($A$). This approach offers significant advantages: for
one, the particles it searches for can be produced individually through
s-channel processes, in which a single particle type is created without the
necessity for a counterpart. This mode of production also allows for the
direct measurement of their invariant mass—a method akin to how the light
scalar Higgs ($h$) was detected. Such direct measurement is possible because
the heavy Higgs bosons do not typically escape detection, resulting in decay
products that can be directly observed and measured [15].

This alternative approach simplifies the detection process at colliders by
circumventing the complexities associated with the pair production of
sparticles and the issues related to the undetected LSPs. It provides a more
straightforward route to probing SUSY manifestations, potentially offering
clearer signals and insights into the structure and implications of SUSY
[1][15].

###  6.3 Co-annihilation and Direct Detection

Another alternative approach to SUSY detection attempts to address the
challenges related to the relic density of dark matter and the observed mass
of the Higgs boson. These factors demand precise adjustments within SUSY
theories, which typically result in the lightest supersymmetric particle (LSP)
being bino-like. However, because bino-like particles do not efficiently
annihilate in the early universe, this potentially leads to a relic density
that exceeds the bounds established by current observations.

To address this issue, SUSY models often employ the mechanism of co-
annihilation. This strategy involves the next-to-lightest supersymmetric
particle (NLSP) having a mass very close to that of the LSP, resulting in a
compressed spectrum of particle masses. Such proximity allows the LSP and NLSP
to effectively co-annihilate, reducing the relic density to within acceptable
limits. However, this scenario also complicates detection efforts because the
resulting particle decays are typically softer and more difficult to detect,
which can obscure signals in experimental data [1][16].

Direct detection experiments are critical in the search for SUSY. Projects
like LUX, Panda, and XENON100 have made significant advancements, achieving
remarkable sensitivities in measuring spin-independent WIMP-nucleon cross-
sections. These experiments aim to detect weakly interacting massive particles
(WIMPs) by looking for the tiny recoils that occur when WIMPs collide with
nuclei within a detector. Projections indicate that future sensitivity could
increase substantially, further enhancing our ability to detect SUSY particles
[16][17].

Another promising line of investigation within SUSY searches involves the
potential for a higgsino-like neutralino. Unlike a bino-like LSP, a higgsino-
like neutralino would likely have higher annihilation rates, which could more
naturally align with the observed relic density of dark matter. However, this
scenario may require the introduction of additional types of dark matter to
fully explain the dark matter composition observed in the universe
[2][16][17][18].

These varied experimental approaches highlight the complex, multi-dimensional
strategies required to explore the landscape of supersymmetry. Each technique
not only contributes to the overarching search for SUSY but also deepens our
understanding of potential new physics beyond the Standard Model.

###  6.4 Additional Efforts

If SUSY exists, it faces two main scenarios: either the supersymmetric
particles are too heavy to be produced by current accelerators, as some
theorists suggest, or they are produced at the LHC but evade detection due to
their subtle signatures.

To address the latter case, researchers are now exploring new models that
produce exotic or experimentally challenging signatures. These signatures may
have been overlooked in previous searches because traditional methods assumed
that new particles would decay immediately at the interaction point. However,
some unconventional SUSY theories propose the existence of long-lived
sparticles that can travel significant distances—from a few micrometers to
hundreds of kilometers—before decaying. Both ATLAS and CMS experiments are
actively searching for such long-lived particles [19].

Moreover, efforts are underway to detect sparticles that decay into low-energy
Standard Model particles. These searches are particularly difficult because
they cannot rely on standard detection techniques. Given the abundance of low-
energy particles in detector backgrounds, researchers must therefore develop
innovative methods to distinguish potential SUSY interactions from ordinary
particle noise–making this area of research both challenging and critical for
advancing our understanding of particle physics [1][19][20].

##  7 Conclusion

While some in the particle physics community have shifted their focus away
from supersymmetry, many experimentalists today remain optimistic about the
capabilities of their detectors in searching for SUSY. Further developments
are also opening new avenues for analysis and experimentation, preparing
researchers for the next phases of data collection. Looking ahead, a High-
Luminosity LHC is planned for the late 2020s–promising further enhancement of
our exploratory capacity [1][2][13].

In the broader scope of future collider projects, many researchers now posit
that the motivation for constructing new, higher-energy colliders is
predominately focused on Higgs boson studies: these perspectives underline a
strategic pivot in high-energy physics towards refining our understanding of
the Higgs boson, while keeping the door open for unexpected discoveries in
areas like supersymmetry.

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