#  Holomorphic functions on complex Banach lattices

Christopher Boyd, Raymond A. Ryan and Nina Snigireva111Partially supported by
funding from Mathematical Modelling and Intelligent Systems for Health and
Environment, Atlantic Technological University Sligo.

###### Abstract

11footnotetext: Keywords: Holomorphic function; Homogeneous polynomial; Banach
lattice; Regular polynomial; Bohr radius.11footnotetext: MSC(2020): Primary
46G20; 46B42; 32A70; Secondary 46E10; 32A05.

We introduce and study the algebraic, analytic and lattice properties of
regular homogeneous polynomials and holomorphic functions on complex Banach
lattices. We show that the theory of power series with regular terms is closer
to the theory of functions of several complex variables than the theory of
holomorphic functions on Banach spaces. We extend the concept of the Bohr
radius to Banach lattices and show that it provides us with a lower bound for
the ratio between the radius of regular convergence and the radius of
convergence of a regular holomorphic function. This allows us to show that the
radius of regular convergence coincides with the radius of convergence for
holomorphic functions on finite dimensional spaces and orthogonally additive
holomorphic functions but that these radii can be radically different on
$\ell_{p}$ for $p>1$.

##  1 Introduction

The aim of this paper is to build a framework for the study of holomorphic
functions on complex Banach lattices that takes account of the lattice
structure of the domain, a feature that is rarely acknowledged in the
literature in this area at present. The crucial concept is regularity — we
require that the homogeneous polynomials that make up the Taylor expansion of
a holomorphic function are regular, meaning that they have a modulus which is
also a homogeneous polynomial, and that the convergence of the series is
regular, in a sense that we will define. This leads to a theory which is
closer in many respects to the classical theory for several complex variables.
For example, it is possible to define the concept of logarithmic convexity for
sets in a complex Banach lattice and we can show that the domain of
convergence of a power series with regular terms has this property.

To date, regular polynomials on Riesz spaces and Banach lattices have largely
been studied for real spaces. The study of regular holomorphic functions means
that we now have to consider homogeneous polynomial on complex Banach
lattices. A complex Banach lattice is, by definition, the complexification of
a real Banach lattice. Therefore, in Section 2 we will concentrate on the
complexification process and see how regular polynomials on a complex Banach
lattice have real and imaginary parts which are the complexifications of
regular polynomials on the associated ‘real part’. Moreover, the space of
regular $m$-homogeneous polynomials on a complex Banach lattice can be given a
norm, known as the regular norm, with respect to which it becomes a complex
Banach lattice.

In Section 3 we will initiate our study of power series of regular polynomials
and introduce the concept of regular convergence. We will see that the natural
domain of regular convergence of such a power series is a logarithmically
convex solid domain. In Section 4 we introduce regular holomorphic functions
as those holomorphic functions whose derivatives at every point are regular
and whose Taylor series at every point is regularly convergent in some
neighbourhood of the point. We shall see that the space of regular holomorphic
functions forms a holomorphy type in the sense of Nachbin, [36], and that the
theory of regular holomorphic functions is more closely related to the theory
of several complex variables than it is to the theory of holomorphic functions
on Banach spaces.

With each regular holomorphic function $f$ and each point $a$ in the domain of
$f$ we associate two radii of convergence, the radius of convergence of $f$
itself about $a$ and the radius of regular convergence. In general, the radius
of regular convergence is smaller than the radius of convergence. In Section 5
we begin by observing how the idea of the homogeneous Bohr radii introduced by
Defant, García and Maestre, [20], for homogeneous polynomials on
$\mathbb{C}^{k}$ can be extended to homogeneous polynomials on general Banach
lattices. From this viewpoint, the Bohr radius can be seen as a measure of the
equivalence of the regular and supremum norms. We will then see how these Bohr
radii provide us with a lower bound for the ratio between the radius of
regular convergence and the radius of convergence. This will allow us to show
that for finite dimensional spaces the radii of convergence and regular
convergence coincide.

In Section 6 we will show that on $\ell_{p}$, for $1<p<\infty$, it is possible
to construct regular holomorphic functions with radius of convergence equal to
$1$ yet having arbitrary small radius of regular convergence. Finally, in
Section 7 we consider complex orthogonally additive polynomials and
holomorphic functions. We prove that for complex orthogonally additive
polynomials the regular and supremum norms coincide and we use this to show
that the radius of convergence is equal to the radius of regular convergence
on the space of orthogonally additive holomorphic functions.

For further reading on the theory of holomorphic functions on Banach spaces we
refer the reader to [21] and [22] while for the theory of Banach lattices we
refer the reader to [2], [33] and [39].

##  2 Regular polynomials on complex Banach lattices

First we recall the definition of a complex Banach lattice. We denote by
$E_{\mathbb{C}}$ the complexification of a real vector space $E$. Thus,
$E_{\mathbb{C}}$ is a complex vector space in which every element $z$ can be
expressed uniquely in the form $z=x+iy$, where $x,y\in E$. The real vectors
$x$, $y$ are referred to as the real and imaginary parts, respectively, of $z$
and we write $x=\operatorname{Re}z$ and $y=\operatorname{Im}z$. If $E$ is a
Banach lattice, the modulus of $z=x+iy\in E_{\mathbb{C}}$ is the element of
$E$ given by

$|z|=\sqrt{x^{2}+y^{2}}=\sup\\{x\cos\theta+y\sin\theta:0\leq\theta\leq
2\pi\\},$ (1)

where these expressions are defined using the Krivine functional calculus [30,
Section 1.d]. A norm is defined on $E_{\mathbb{C}}$ by

$\|z\|=\|\,|z|\,\|$

and the triple consisting of $E_{\mathbb{C}}$ with the modulus and norm is, by
definition, a _complex Banach lattice_. Alternatively, one may take an
axiomatic approach. This was done by Mittelmeyer and Wolff [34] and they
showed that every complex Banach lattice defined in their way is the
complexification of a (real) Banach lattice as outlined above. We recommend
the paper by Buskes and Schwanke [13] for an informative discussion of the
Mittelmeyer-Wolff axioms and a general treatment of complexification of vector
lattices.

Let $E,F$ be (real) Banach lattices. An $m$-homogeneous polynomial $P\colon
E\to F$ is generated by a unique symmetric $m$-linear mapping $A\colon
E^{m}\to F$, in the sense that $P(x)=A(x,\dots,x)$ for every $x\in E$. We
write $P=\widehat{A}$. The polynomial $P$ is said to be _positive_ if
$A(x_{1},\dots,x_{m})\geq 0$ for all $x_{1},\dots,x_{m}\geq 0$ and _regular_
if it is the difference of two positive polynomials. If $F$ is Dedekind
complete, then the space $\mathcal{P}_{r}(^{m}{E};{F})$ of regular
$m$-homogeneous polynomials is a Banach lattice with the _regular norm_ ,
defined by $\|P\|_{r}=\||P|\|$, where $\|\cdot\|$ is the supremum norm [12].

Let $E_{1},\dots,E_{m},F$ be real vector spaces and let $A\colon
E_{1}\times\dots\times E_{m}\to F$ be an $m$-linear mapping. Then $A$ has a
unique extension to a complex $m$-linear mapping
$A_{\mathbb{C}}\colon(E_{1})_{\mathbb{C}}\times\dots\times(E_{m})_{\mathbb{C}}\to
F_{\mathbb{C}}$ [8, Theorem 3]. For
$z_{j}=x^{0}_{j}+ix^{1}_{j}\in(E_{j})_{\mathbb{C}}$, $1\leq j\leq m$, we have

$A_{\mathbb{C}}(z_{1},\dots,z_{m})=\sum_{\delta_{1},\dots,\delta_{m}=0,1}i^{\sum\delta_{j}}A(x^{\delta_{1}}_{1},\dots,x^{\delta_{m}}_{m})\,.$

We shall say that a complex $m$-linear mapping from
$(E_{1})_{\mathbb{C}}\times\dots(E_{m})_{\mathbb{C}}$ into $F_{\mathbb{C}}$ is
real if it is the complexification of a real $m$ linear mapping from
$E_{1}\times\dots\times E_{m}$ into $F$. Every complex $m$-linear mapping $A$
can be decomposed uniquely in the form $A=A_{0}+iA_{1}$, where $A_{0},A_{1}$
are real $m$-linear mappings. We let
$A_{0}(x_{1},\dots,x_{m})=\operatorname{Re}A(x_{1},\dots,x_{m})$ and
$A_{1}(x_{1},\dots,x_{m})=\operatorname{Im}A(x_{1},\dots,x_{m})$ for real
arguments and then extend $A_{0},A_{1}$ to the complexification as described
above. It follows that the vector space of complex $m$-linear mappings is the
complexification of the space of real $m$-linear mappings:

$L(^{m}(E_{1})_{\mathbb{C}},\dots,(E_{m})_{\mathbb{C}};F_{\mathbb{C}})\cong
L(^{m}E_{1},\dots,E_{m};F)_{\mathbb{C}}\,.$

If the spaces $E_{1},\dots,E_{m}$ are the same then clearly an $m$-linear
mapping $A$ on $E^{m}$ is symmetric if and only if its complexification,
$A_{\mathbb{C}}$, is. Thus, if $E$, $F$ are real vector spaces, then every
$m$-homogeneous polynomial $P=\widehat{A}\colon E\to F$ has a unique extension
to a complex $m$-homogeneous polynomial $P_{\mathbb{C}}\colon
E_{\mathbb{C}}\to F_{\mathbb{C}}$. Moreover, we have

$P_{\mathbb{C}}(z)=\sum_{\ell=0}^{\lfloor
m/2\rfloor}\binom{m}{2\ell}(-1)^{\ell}A(x^{m-2\ell},y^{2\ell})+i\Bigl{(}\sum_{\ell=0}^{\lfloor
m/2\rfloor}\binom{m}{2\ell+1}(-1)^{\ell}A(x^{m-2\ell-1},y^{2\ell+1})\Bigr{)}$

for every $z=x+iy\in E_{\mathbb{C}}$. As these complexifications are
canonical, we usually omit the subscript $\mathbb{C}$, so $P(z)$ will be
understood to mean $P_{\mathbb{C}}(z)$. A complex $m$-homogeneous polynomial
that arises in this way, as the complexification of a polynomial from $E$ into
$F$, will be referred to as a real polynomial. It follows from our remarks
above that the space of complex $m$-homogeneous polynomials is the
complexification of the space of real $m$-homogeneous polynomials:

$P(^{m}E_{\mathbb{C}};F_{\mathbb{C}})\cong P(^{m}E;F)_{\mathbb{C}}\,.$

Every complex $m$-homogeneous polynomial $P$ can be written uniquely in the
form $P=P_{0}+iP_{1}$, where $P_{0},P_{1}$ are real $m$-homogeneous
polynomials, defined, for real arguments by

$P_{0}(x)=\operatorname{Re}P(x)\quad\text{and}\quad
P_{1}(x)=\operatorname{Im}P(x)$ (2)

for $x\in E$. The polynomials $P_{0},P_{1}$ are then extended by
complexification to all of $E_{\mathbb{C}}$. We note that it is not correct so
say that $P_{0}(z)=\operatorname{Re}P(z)$ for complex vectors $z$; the
identity $(\operatorname{Re}P)(z)=\operatorname{Re}(P(z))$ is only valid for
real arguments. To give a simple example, consider the $2$-homogeneous
polynomial $P(z)=z^{2}$ on $\mathbb{C}$. We have $P(z)=(x^{2}-y^{2})+i(2xy)$
for $z=x+iy$. However, $x^{2}-y^{2}$ and $2xy$ are not the real and imaginary
parts of $P$. Indeed, $P$ is a real polynomial, being the complexification of
the real polynomial $P(t)=t^{2}$ and so its imaginary part is zero.

We now consider polynomials on Banach lattices. Let $E$ be a complex Banach
lattice. The real part of $E$, denoted by $E_{\mathbb{R}}$, is defined to be
the real linear span of the set $\\{|z|:z\in E\\}$ and is a (real) Banach
lattice, with the norm induced from $E$. Then $E$, as a vector space, is the
complexification of $E_{\mathbb{R}}$. And we recall that the norms on $E$ and
$E_{\mathbb{R}}$ are connected by the relation
$\|z\|_{E}=\||z|\|_{E_{\mathbb{R}}}$ for $z\in E$. Our definition of
regularity, positivity, respectively, for multilinear and polynomial mappings
is the same as the linear case [33, Section 2.2]. So we say that an $m$-linear
mapping on $E^{m}$, or an $m$-homogeneous polynomial on $E$, is regular,
positive, respectively, if both its real and imaginary parts are regular,
positive on $(E_{\mathbb{R}})^{m}$ or $E_{\mathbb{R}}$, respectively.

If $E,F$ are (real) Banach lattices with $F$ Dedekind complete, then the space
$\mathcal{P}_{r}(^{m}{E};{F})$ of regular $m$-homogeneous polynomials, with
the regular norm $\|P\|_{r}=\||P|\|$, is a Banach lattice [12]. Now let $E,F$
be complex Banach lattices, with $F$ Dedekind complete. Then it follows from
the above that the space $\mathcal{P}_{r}(^{n}{E};{F})$ of regular
$m$-homogeneous polynomials is the complexification of the space
$\mathcal{P}_{r}(^{n}{E_{\mathbb{R}}};{F_{\mathbb{R}}})$ of real, regular
$m$-homogeneous polynomials:

$\mathcal{P}_{r}(^{n}{E};{F})=\bigl{(}\mathcal{P}_{r}(^{n}{E_{\mathbb{R}}};{F_{\mathbb{R}}})\bigr{)}_{\mathbb{C}}\,.$

Therefore, $\mathcal{P}_{r}(^{n}{E};{F})$ can be endowed with a complex Banach
lattice structure, with the norm given by $\|P\|_{r}=\|\,|P|\,\|$. We
summarize these observations:

###### Proposition 1

Let $E$, $F$ be complex Banach lattices, with $F$ Dedekind complete. The space
of regular $m$-homogeneous polynomials from $E$ into $F$ with the regular norm
$\|P\|_{r}=\|\,|P|\,\|$ is a Dedekind complete complex Banach lattice.

Note that this result in particular tells us that the absolute value of a
complex regular $m$-homogeneous polynomial $P=P_{0}+iP_{1}$ satisfies

$|P|=\sqrt{|P_{0}|^{2}+|P_{1}|^{2}}\,,$

where this formula is understood in the sense of the Krivine functional
calculus [30].

To illustrate the difference between the regular and supremum norms, we have
the following useful result by Choi-Kim-Ki [16]:

######  Proposition 2 ([16], Theorem 2.4)

Let $a,b,c\in\mathbb{R}$ with $|a|<1$, $|b|<1$ and $2<|c|\leq 4$. Suppose
$P(x,y)=ax^{2}+by^{2}+cxy\in\mathcal{P}(^{2}{\ell_{1}^{2}})$. Then, over both
the real and complex numbers,

$\|P\|=1\quad\text{ if and only if }\quad 4|c|-c^{2}=4(|a+b|-ab)\,.$

We can use this result to construct examples of polynomials on $\ell_{1}^{2}$
for which the regular norm is greater than the supremum norm.

###### Example 1

On $\ell_{1}^{2}$, let

$P(x,y)=\frac{1}{2}x^{2}-\frac{1}{2}y^{2}+(2+\sqrt{3})xy\,.$

Then

$\|P\|=1\quad\text{ and }\quad\|P\|_{r}=\frac{3+\sqrt{3}}{4}>1\,.$

over both the real and complex numbers.

The fact that $\|P\|=1$ follows from the Choi-Kim-Ki result, but can also be
seen by means of an elementary calculation. We have

$|P|(x,y)=\frac{1}{2}x^{2}+\frac{1}{2}y^{2}+(2+\sqrt{3})xy$

and thus

$\|P\|_{r}\geq|P|\Bigl{(}\frac{1}{2},\frac{1}{2}\Bigr{)}=\frac{3+\sqrt{3}}{4}>1\,.$

A calculation shows that in fact, $\|P\|_{r}=\dfrac{3+\sqrt{3}}{4}$ in the
real case. As the coefficients of $|P|$ are positive, the regular norm over
the complex numbers is the same.

The absolute value of a regular linear mapping $T\colon E\to F$ between
complex Banach lattices satisfies

$|T(z)|\leq|T|(|z|)$

for every $z\in E$ [33]. We will show that there is a corresponding result for
complex regular polynomials. For this, we need a little preparation.

Let $E,F$ be real Banach lattices, with $F$ Dedekind complete. The Banach
lattice of regular $m$-linear mappings from $E$ into $F$ is denoted by
$\mathcal{L}_{r}(^{m}{E};{F})$ and the Banach sublattice of symmetric, regular
$m$-linear mappings is denoted by $\mathcal{L}^{s}_{r}(^{m}{E};{F})$. The
definition of the order relation shows that $\mathcal{P}_{r}(^{m}{E};{F})$ and
$\mathcal{L}^{s}_{r}(^{m}{E};{F})$ are isomorphic as vector lattices. Bu and
Buskes [12] showed that for $A\in\mathcal{L}_{r}(^{m}{E};{F})$, the absolute
value $|A|$ is given by

$|A|(x_{1},\dots,x_{m})=\sup\Bigl{\\{}\,\sum_{i_{1}}\dots\sum_{i_{m}}|A(u^{1}_{i_{1}},\dots,u^{m}_{i_{m}})|:u^{1}\in\Pi(x_{1}),\dots,u^{m}\in\Pi(x_{m})\Bigr{\\}}$

for $x_{1},\dots,x_{m}\geq 0$. Here, $\Pi(x)$ denotes the set of partitions of
a positive element $x$, namely, finite sets of positive vectors whose sum is
$x$. Now $\Pi(x)$ is directed by set inclusion and so the iterated suprema
that appear in the formula above can be interpreted as limits of increasing
nets, with $\Pi(x_{1}),\dots,\Pi(x_{m})$ as the indexing sets. Thus, we can
interchange the operations of supremum with the finite sums.

If we define

$A^{(k)}\colon E^{m-k}\to\mathcal{L}_{r}(^{k}{E};{F})$

by

$A^{(k)}(x_{1},\dots,x_{m-k})(x_{m-k+1},\dots,x_{m})=A(x_{1},\dots,x_{m})\,,$

then the correspondence $A\leftrightarrow A^{(k)}$ is an isometric isomorphism
between $\mathcal{L}_{r}(^{m}{E};{F})$ and
$\mathcal{L}_{r}(^{m-k}{E};{\mathcal{L}_{r}(^{k}{E};{F})})$. Furthermore, it
follows from the remarks above that we have $|A|^{(k)}=|A^{(k)}|$ for every
$A\in\mathcal{L}_{r}(^{m}{E};{F})$ and every $k$. Therefore, this
correspondence is also a lattice homomorphism. Thus, the Banach lattices
$\mathcal{L}_{r}(^{m}{E};{F})$ and
$\mathcal{L}_{r}(^{m-k}{E};{\mathcal{L}_{r}(^{k}{E};{F})})$ are isometrically
lattice isomorphic.

###### Proposition 3

Let $E$, $F$ be real or complex Banach lattices, with $F$ Dedekind complete.
Then the mapping

$A\in\mathcal{L}_{r}(^{m}{E};{F})\mapsto
A^{(k)}\in\mathcal{L}_{r}(^{m-k}{E};{\mathcal{L}_{r}(^{k}{E};{F})})$

is a Banach lattice isometric isomorphism for every $k=1,\dots,m-1$.

Proof: We have already established this result for real Banach lattices. Now
let $E,F$ be complex Banach lattices, with $F$ Dedekind complete. Then,
complexifying the above isomorphism, we have

$\displaystyle\mathcal{L}_{r}(^{m}{E};{F})$
$\displaystyle=\mathcal{L}_{r}(^{m}{E_{\mathbb{R}}};{F_{\mathbb{R}}})_{\mathbb{C}}\cong\mathcal{L}_{r}(^{m-k}{E_{\mathbb{R}}};{\mathcal{L}_{r}(^{k}{E_{\mathbb{R}}};{F_{\mathbb{R}}})})_{\mathbb{C}}$
$\displaystyle\cong\mathcal{L}_{r}(^{m-k}{E};{\mathcal{L}_{r}(^{k}{E_{\mathbb{R}}};{F_{\mathbb{R}}})_{\mathbb{C}}})\cong\mathcal{L}_{r}(^{m-k}{E};{\mathcal{L}_{r}(^{k}{E};{F})})\,.$

Therefore $\mathcal{L}_{r}(^{m}{E};{F})$ and
$\mathcal{L}_{r}(^{m-k}{E};{\mathcal{L}_{r}(^{k}{E};{F})})$ are isometrically
isomorphic as complex Banach lattices.  $\blacksquare\ \ \ \ $

###### Proposition 4

Let $E$, $F$ be complex Banach lattices, with $F$ Dedekind complete and let
$P\colon E\to F$ be a regular $m$-homogeneous polynomial. Then

$|P(z)|\leq|P|(|z|)$

for all $z\in E$.

Proof: Let $A$ be the associated regular $m$-linear mapping. We will show that

$|A(z_{1},\dots,z_{m})|\leq|A|(|z_{1}|,\dots,|z_{m}|)$

for all $z_{1},\dots,z_{m}\in E$. The proof is by induction. The case $m=1$ is
the linear result [33]. Assume the result holds for $(m-1)$-linear mappings.
Then, using Proposition 3,

$\displaystyle|A(z_{1},\dots,z_{m})|$
$\displaystyle=|A^{(m-1)}(z_{1})(z_{2},\dots,z_{m})|\leq|A^{(m-1)}(z_{1})|(|z_{2}|,\dots,|z_{m}|)$
$\displaystyle\leq|A^{(m-1)}|(|z_{1}|)(|z_{2}|,\dots,|z_{m}|))=|A|(|z_{1}|,\dots,|z_{m}|)\,.$

Thus the result follows for every $m$.  $\blacksquare\ \ \ \ $

Our next result shows that the complexification process preserves the absolute
value.

###### Proposition 5

Let $E$, $F$ be real Banach lattices, with $F$ Dedekind complete and let
$P\in\mathcal{P}_{r}(^{m}{E};{F})$. Then $|P_{\mathbb{C}}|=|P|_{\mathbb{C}}$.

Proof: We begin with the case $m=1$. Let $T\colon E\to F$ be a regular linear
operator. For $u\in E_{+}$ we have

$\displaystyle|T_{\mathbb{C}}|(u)$
$\displaystyle=\sup\\{|T_{\mathbb{C}}(x+iy)|:x,y\in E,|x+iy|\leq
u\\}=\sup\\{|Tx+iTy|,|x+iy|\leq u\\}$
$\displaystyle=\sup\sup_{\theta\in\mathbb{R}}\\{|(\cos\theta)Tx+(\sin\theta)Ty|,|x+iy|\leq
u\\}$
$\displaystyle\leq\sup\sup_{\theta\in\mathbb{R}}\\{|T||(\cos\theta)x+(\sin\theta)y|,|x+iy|\leq
u\\}$ $\displaystyle\leq\sup\\{|T||x+iy|,|x+iy|\leq
u\\}=|T|(u)=|T|_{\mathbb{C}}(u)\,.$

On the other hand,

$\displaystyle|T|_{\mathbb{C}}(u)$ $\displaystyle=|T|(u)=\sup\\{|T(v)|:v\in
E,|v|\leq u\\}$ $\displaystyle\leq\sup\\{|T(x+iy)|:x,y\in E,|x+iy|\leq
u\\}=|T_{\mathbb{C}}|(u)$

for every $u\in E_{+}$. Therefore $|T_{\mathbb{C}}|=|T|_{\mathbb{C}}$ on
$E_{+}$ and it follows that this holds on $E_{\mathbb{C}}$.

For the case $m>1$ we have, using Proposition 3, Banach lattice isomorphisms

$\mathcal{L}_{r}(^{m}{E};{F})_{\mathbb{C}}\cong\mathcal{L}_{r}(^{m}{E_{\mathbb{C}}};{F_{\mathbb{C}}})\cong\mathcal{L}_{r}({E_{\mathbb{C}}};{\mathcal{L}_{r}(^{m-1}{E_{\mathbb{C}}};{F_{\mathbb{C}}})})\cong\mathcal{L}_{r}({E};{\mathcal{L}_{r}(^{m-1}{E};{F})})_{\mathbb{C}}\,.$

Thus the result follows by induction on $m$.  $\blacksquare\ \ \ \ $

The supremum norm of a polynomial on real Banach spaces is not, in general,
preserved by complexification. Gustavo, Muñoz, Tonge have shown that if $E$,
$F$ are real Banach spaces and $P\colon E\to F$ is a bounded $m$-homogeneous
polynomial, then the norm of its complexification satisfies

$\|P_{\mathbb{C}}\|_{\nu}\leq 2^{m-1}\|P\|$

where $\nu$ is any natural complexification process [35, Prop. 18], and this
inequality is sharp. In the Banach lattice case, the regular norm is much
better behaved.

###### Theorem 1

Let $E$, $F$ be real Banach lattices, with $F$ Dedekind complete and let
$P\in\mathcal{P}_{r}(^{m}{E};{F})$. Then $\|P_{\mathbb{C}}\|_{r}=\|P\|_{r}$.

Proof: Clearly, we have $\|P\|_{r}\leq\|P_{\mathbb{C}}\|_{r}$. Conversely,
using Proposition 5,

$\displaystyle\|P_{C}\|_{r}$
$\displaystyle=\|\,|P_{\mathbb{C}}|\,\|=\|\,|P|_{\mathbb{C}}\|=\sup\\{\bigl{|}|P|_{\mathbb{C}}(z)\bigr{|}:z\in
E_{\mathbb{C}},\|z\|\leq 1\\}$
$\displaystyle\leq\sup\\{|P|_{\mathbb{C}}(|z|):z\in E_{\mathbb{C}},\|z\|\leq
1\\}=\sup\\{|P|(|z|):z\in E_{\mathbb{C}},\|z\|\leq 1\\}=\|P\|_{r}\,.$

$\blacksquare\ \ \ \ $

Now let $E$, $F$ be complex Banach lattices. A complex linear operator
$T\colon E\to F$ is a homomorphism if $|T(z)|=T(|z|)$ for every $z\in E$. This
is equivalent to $T$ being the complexification of a lattice homomorphism
between the real vector lattices $E_{\mathbb{R}}$ and $F_{\mathbb{R}}$ [39, p.
136].

Recall that the $k$th Fréchet derivative of an $m$-homogeneous polynomial
$P=\widehat{A}\colon E\to F$ between real or complex vector spaces is given by

$\widehat{d}^{k}P(x)(y)=m(m-1)\dots(m-k+1)A(x^{m-k}y^{k})\,.$

Thus $\widehat{d}^{k}P$ is, up to a constant multiple, the $k$-homogeneous
polynomial associated with the $(m-k)$-linear mapping $A^{(k)}$. The following
proposition follows immediately from this fact.

###### Proposition 6

Let $E$, $F$ be real or complex Banach lattices, with $F$ Dedekind complete.
Then

$\bigl{|}\widehat{d^{k}}P\bigr{|}=\widehat{d^{k}}|P|$

for every $P\in\mathcal{P}_{r}(^{m}{E};{F})$ and every $k$, $1\leq k\leq m$.
Therefore, the mapping

$\widehat{d^{k}}\colon\mathcal{P}_{r}(^{m}{E};{F})\to\mathcal{P}_{r}(^{m-k}{E};{\mathcal{P}_{r}(^{k}{E};{F})})$

is a vector lattice isomorphism onto its image.

We point out that the mapping $\widehat{d^{k}}$ is not in general an isometry
in either the real or complex cases. Taking $E=\ell_{1}$ and $F=\mathbb{R}$ or
$\mathbb{C}$, the $2$-homogeneous polynomial $P(x)=x_{1}x_{2}$ has regular
norm $\|P\|_{r}=1/4$, while the regular norm of the linear mapping
$\widehat{d^{1}}P\colon\ell_{1}\to\ell_{\infty}$ is
$\|\widehat{d^{1}}P\|_{r}=1/2$.

The Krivine functional calculus [30, 1.d.1] allows one to prove a number of
Hölder type inequalities. For example, we have

$\||x|^{\theta}|y|^{1-\theta}\|\leq\|x\|^{\theta}\|y\|^{1-\theta}$ (3)

for all elements $x,y$ of a real or complex Banach lattice, with $0<\theta<1$.
Our next result will prove useful when dealing with power series whose terms
are regular homogeneous polynomials. We refer to Kusraev [29] for some similar
inequalities.

###### Proposition 7

Let $E,F$ be real or complex Banach lattices, with $F$ Dedekind complete, and
let $P\colon E\to F$ be a positive $m$-homogeneous polynomial. Then

$P(|x|^{\theta}|y|^{1-\theta})\leq P(|x|)^{\theta}\,P(|y|)^{1-\theta}$

for every $0<\theta<1$ and every $x,y\in E$.

Proof: Let $a=|x|\vee|y|$. The ideal $E_{a}$, which contains $x$ and $y$, is
Banach lattice isomorphic to $C(K)$ for some compact, Hausdorff space $K$ and
the expression $|x|^{\theta}|y|^{1-\theta}$ defined by the Krivine functional
calculus coincides with its value in $C(K)$. The restriction of $P$ to $E_{a}$
is positive and so, by Fremlin’s theorem [24], there exists a regular,
positive Borel measure $\mu$ on $K^{m}$ such that

$P(w)=\int_{K^{m}}w(t_{1})\dots w(t_{m})\,d\mu(t_{1},\dots,t_{m})$

for every $w\in E_{a}$. Let $\theta\in(0,1)$. Applying Hölder’s inequality
with $p=1/\theta$, we have

$\displaystyle P(|x|^{\theta}|y|^{1-\theta})$
$\displaystyle\leq\int_{K^{m}}|x(t_{1})|^{\theta}\dots|x(t_{m})|^{\theta}|y(t_{1})|^{1-\theta}\dots|y(t_{m})|^{1-\theta}\,d\mu(t_{1},\dots,t_{m})$
$\displaystyle\leq\Bigl{(}\int_{K^{m}}|x(t_{1})\dots
x(t_{m})|\,d\mu(t)\Bigr{)}^{\theta}\Bigl{(}\int_{K^{m}}|y(t_{1})\dots
y(t_{m})|\,d\mu(t)\Bigr{)}^{1-\theta}$
$\displaystyle=P(|x|)^{\theta}\,P(|y|)^{1-\theta}\,.$

$\blacksquare\ \ \ \ $

Taking $P=\varphi^{m}$, where $\varphi\in E^{\prime}$, and $F=\mathbb{C}$, we
see that (3) is a consequence of this result.

##  3 Power series with regular terms

Absolute convergence of a power series $\sum_{m}c_{m}z^{m}$ in one complex
variable at a point requires that the numbers $|c_{m}z^{m}|$ are summable. Of
course, this is the same as summability of the numbers $|c_{m}||z|^{m}$.
However, on a Banach lattice, these lead to two different conditions.

We shall say that a power series $\sum_{m}P_{m}$ with regular terms on a
complex Banach lattice $E$ is _regularly convergent_ at a point $z\in E$ if
the series $\sum_{m}|P_{m}|(|z|)$ converges. Note that this is stronger than
absolute convergence, since

$|P_{m}(z)|\leq|P_{m}|(|z|)$

for all $z\in E$. We shall see that regular convergence is in fact a strictly
stronger condition than absolute convergence.

If $E$ has a $1$-unconditional Schauder basis, then, by [25], every regular
$m$-homogeneous polynomial $P_{m}$ on $E$ has a pointwise unconditionally
convergent monomial expansion of the form

$P_{m}(z)=\sum_{|\alpha|=m}c_{\alpha}z^{\alpha}\,.$

The converse is also true: every unconditionally pointwise convergent monomial
expansion of degree $m$ defines a regular $m$-homogeneous polynomial.
Furthermore, by [25], we have

$|P_{m}|(z)=\sum_{|\alpha|=m}|c_{\alpha}|z^{\alpha}\,.$

Thus a power series $\sum_{m}P_{m}(z)$ with regular terms can be written as a
formal monomial expansion

$\sum_{m=0}^{\infty}\sum_{|\alpha|=m}c_{\alpha}z^{\alpha}$

and regular convergence of the power series at a point $z$ is equivalent to
the condition

$\sum_{\alpha\in\mathbb{N}_{0}^{(\mathbb{N})}}|c_{\alpha}||z|^{\alpha}<\infty\,.$

So we see that regular convergence of a power series is a natural abstraction
to the Banach lattice setting of the familiar concept of pointwise absolute
convergence of a monomial expansion.

For a power series $f=\sum_{m}P_{m}$ on a complex Banach space, the radius of
convergence is given by the formula

$r(f,0)=\Bigl{(}\limsup\|P_{m}\|^{1/m}\Bigr{)}^{-1}\,.$

This number is the supremum of the set of nonnegative real numbers $r$ for
which the series is uniformly convergent on the closed ball of radius $r$.
Bearing in mind that the regular norm of a regular $m$-homogeneous polynomial
$P_{m}$ is given by $\|P_{m}\|_{r}=\||P_{m}|\|$, we may define the _radius of
regular convergence_ of a power series $f=\sum_{m}P_{m}$ to be

$|r|(f,0)=\Bigl{(}\limsup\|P_{m}\|_{r}^{1/m}\Bigr{)}^{-1}\,.$

This is the supremum of the set of nonnegative real numbers $\rho$ for which
the series is uniformly regularly convergent in the closed ball of radius
$\rho$. Since the regular norm is at least as big as the uniform norm, we see
that the radius of regular convergence is no bigger than the radius of
convergence, i.e.,

$|r|(f,0)\leq r(f,0)\,.$

We shall see that these radii can be different.

The radius of regular convergence gives some useful information about the
behaviour of a power series. However, this information is somewhat limited as
the natural domain of convergence is not generally a ball. Let us recall some
facts about power series in several complex variables. A subset $D$ of
$\mathbb{C}^{k}$ is a complete Reinhardt domain if $z\in D$ implies that $w\in
D$ whenever $|w_{j}|\leq|z_{j}|$ for $1\leq j\leq k$. A complete Reinhardt
domain $D$ is said to be logarithmically convex if, for every $z,w\in D$ and
every $\theta\in(0,1)$ the point
$(|z_{1}|^{\theta}|w_{1}|^{1-\theta},\dots,|z_{k}|^{\theta}|w_{k}|^{1-\theta})$
belongs to $D$. The domain of convergence of a power series on
$\mathbb{C}^{k}$ is defined to be the interior of the set of points at which
the series is absolutely convergent. In general, the domain of convergence is
either a non-empty open set, or is empty. It is a fundamental result for
several complex variables that the domain of convergence of a power series is
a logarithmically convex Reinhardt domain [27, Section 2.4].

###### Example 2

This result of Matos [31] illustrates the difference in behaviour of the
domain of convergence in finite and infinite dimensions: if $z=(z_{j})$ is a
sequence of complex numbers, then the series

$\sum_{\alpha\in\mathbb{N}_{0}^{(\mathbb{N})}}z^{\alpha}$

is absolutely convergent if and only if $z$ belongs to $\ell_{1}$ and
$|z_{j}|<1$ for every $j$. Therefore, on $\ell_{1}$ this monomial expansion
has a non-empty domain of convergence. However, on $c_{0}$, the set of points
of absolute convergence is a dense subset of the closed unit ball with no
interior points.

We now see how some results from several complex variables can be formulated
for power series on complex Banach lattices. We recall the following
elementary fact: If $\sum c_{\alpha}(z-w)^{\alpha}$ is a monomial expansion on
$C^{k}$ whose terms are bounded at some point $z=w+a$ where $a>0$, then the
expansion is absolutely convergent on the polydisc $D(w,a)=\\{z\in
C^{k}:|z_{j}-w_{j}|\leq a_{j},1\leq j\leq n\\}$ and the convergence is uniform
on smaller polydiscs $D(w,\lambda a)$, $0<\lambda<1$. We extend the definition
of a polydisc to complex Banach lattices in the obvious way. If $a$ is a
positive element in a complex Banach lattice $E$, the polydisc with centre $w$
and polyradius $a$ is $D(w,a)=\\{z\in E:|z-w|\leq a\\}$. We note that $D(0,a)$
is the closed unit ball of the principal ideal $E_{a}$ with the order unit
norm defined by $a$. Note that if a power series centred at the origin is
regularly convergent at a point $z$, then it is regularly convergent at every
point in the polydisc $D(0,|z|)$.

###### Lemma 1

Let $E$ be a complex Banach lattice and let $(P_{m})$ be a sequence of regular
$m$-homogeneous polynomials on $E$. If $\\{P_{m}(a)\\}$ is a bounded subset of
$\mathbb{C}$ for some positive element $a\in E$, then the power series
$\sum_{m}P_{m}$ is regularly convergent on the polydisc $D(0,a)$ and the
convergence is uniform on every smaller polydisc $D(0,\lambda a)$,
$0<\lambda<1$.

Proof: This follows from the fact that if $|z|\leq\lambda a$, then
$||P_{m}|(z)|\leq\lambda^{m}|P_{m}|(a)$.  $\blacksquare\ \ \ \ $

This is stronger than the corresponding result for Banach spaces. In general,
all we can deduce from the boundedness of the values $P_{m}(z)$ at some point
in a Banach space is that the power series $\sum_{m}P_{m}$ is absolutely
convergent in the one dimensional complex disc determined by the point $z$.

We recall that a set $D$ in a real or complex Banach lattice is said to be
solid if $z\in D$ and $|w|\leq|z|$ imply that $w$ belongs to $D$. For subsets
of $\mathbb{C}^{n}$, the solid sets are precisely the complete Reinhardt
domains.

A solid subset $D$ of $E$ is said to be _logarithmically convex_ if, for every
$x,y\in D$, $\theta\in(0,1)$, the point $|x|^{\theta}|y|^{1-\theta}$ also
belongs to $D$. The domain of convergence of a power series in finite
dimensions is logarithmically convex. The same is true for regularly
convergent power series on a Banach lattice:

###### Theorem 2

Let $E$ be a complex Banach lattice and let $\sum_{m}P_{m}$ be a power series
on $E$ with regular terms. Then the set of points at which the series
converges regularly is solid and logarithmically convex.

Proof: Let $z\in D$ and suppose $w\in$ satisfies $|w|\leq|z|$. It follows from
the fact that $|P_{m}|(|w|)\leq|P_{m}|(|z|)$ that $w\in D$. Thus $D$ is a
solid set.

Now let $z,w\in D$. Using Proposition 7 and the inequality between the
weighted geometric and arithmetic means, we get

$|P_{m}|(|z|^{\theta}|w|^{1-\theta})\leq|P_{m}|(|z|)^{\theta}|P_{m}|(|w|)^{1-\theta}\leq\theta|P_{m}|(|z|)+(1-\theta)|P_{m}|(|w|)$

and it follows immediately that $|z|^{\theta}|w|^{1-\theta}\in D$.
$\blacksquare\ \ \ \ $

##  4 Regular holomorphic functions

Let $E,F$ be complex Banach lattices, with $F$ Dedekind complete. Let $U$ be
an open subset of $E$. A function $f\colon U\to F$ is regularly holomorphic if

  1. (a)

$f$ is holomorphic on $U$.

  2. (b)

For every $z\in U$, the derivatives
$\displaystyle\frac{1}{m!}\widehat{d^{m}}{f}(z)$ are regular $m$-homogeneous
polynomials.

  3. (c)

For every $z\in U$, the Taylor series of $f$ at $z$ is regularly convergent in
some neighbourhood of $z$.

Suppose a power series $\sum P_{m}$ is regularly convergent at some point $z$.
This means that the series $\sum|P_{m}|(|z|)$ converges. Then, since
$\bigl{|}|P_{m}|(z)\bigr{|}\leq|P_{m}|(|z|)$, it follows that the series
$\sum|P_{m}|(z)$ is absolutely convergent. So, applying the Cauchy-Hadamard
condition to the power series $\sum|P_{m}|$, we see that condition (c) above
is equivalent to

$\limsup_{m}\,\Bigl{\|}\frac{1}{m!}\widehat{d^{m}}{f}(z)\Bigl{\|}_{r}^{\frac{1}{m}}<\infty$
(4)

for every $z\in U$. We denote the space of regularly holomorphic functions by
$\mathcal{H}_{r}({U};{F})$, or by $\mathcal{H}_{r}({U})$ when $F=\mathbb{C}$.

Let $U$ be an open set and let $z$ be a point in $U$. We shall say that a
function $f\colon U\to F$ is regularly holomorphic at $z$ if the derivatives
$\displaystyle\frac{1}{m!}\widehat{d^{m}}{f}(z)$ of $f$ at $z$ are regular and
(4) holds.

###### Proposition 8

If a holomorphic mapping is regularly holomorphic at a point in its domain,
then it is regularly holomorphic in some neighbourhood of that point.

Proof: Suppose that $f$ is holomorphic on an open set $U$ and that $f$ is
regular at a point $z_{0}$ in $U$. Let $\displaystyle
P_{m}=\widehat{A_{m}}=\frac{1}{m!}\widehat{d^{m}}{f}(z_{0})$. Then there exist
$C,\rho>0$ such that

$\|P_{m}\|_{r}\leq C\,\rho^{m}$

for every $m\in\mathbb{N}$. If $\|z-z_{0}\|<\rho$, then the derivatives of $f$
at $z$ are given by

$\frac{1}{k!}\widehat{d^{k}}{f}(z)=\sum_{m\geq
k}\binom{m}{k}A_{m}(z-z_{0})^{m-k}$

where this series converges in $\mathcal{P}(^{k}{E};{F})$ with the supremum
norm. We will show that this series is absolutely convergent in
$\mathcal{P}_{r}(^{k}{E};{F})$ with the regular norm. Taking
$\|z-z_{0}\|\leq\sigma<(2e\rho)^{-1}$, we have

$\displaystyle\sum_{m\geq k}$
$\displaystyle\Bigl{\|}{\binom{m}{k}A_{m}(z-z_{0})^{m-k}}\Bigr{\|}_{r}=\sum_{m\geq
k}\Bigl{\|}{\binom{m}{k}\bigl{|}A_{m}(z-z_{0})^{m-k}\bigr{|}}\Bigr{\|}$
$\displaystyle\leq\sum_{m\geq
k}\Bigl{\|}{\binom{m}{k}\bigl{|}A_{m}\bigr{|}\,\bigl{|}z-z_{0}|^{m-k}\bigr{|}}\Bigr{\|}\leq\sum_{m\geq
k}\binom{m}{k}\bigl{\|}A_{m}\bigr{\|}_{r}\bigl{\|}(z-z_{0})\bigr{\|}^{m-k}$
$\displaystyle\leq\sum_{m\geq
k}\binom{m}{k}e^{m}\bigl{\|}P_{m}\bigr{\|}_{r}\bigl{\|}(z-z_{0})\bigr{\|}^{m-k}\leq
C\,\sum_{m\geq k}2^{m}e^{m}\rho^{m}\sigma^{m-k}$
$\displaystyle=\frac{C}{1-2e\rho\sigma}\bigl{(}2e\rho\bigr{)}^{k}\,.$

Therefore $\displaystyle\frac{1}{k!}\widehat{d^{k}}{f}(z)$ is a regular
$k$-homogeneous polynomial for every $k$, provided
$\|z-z_{0}\|<(2e\rho)^{-1}$. Furthermore, the above calculation shows that

$\limsup_{k}\,\Bigl{\|}{\frac{1}{k!}\widehat{d^{k}}{f}(z)}\Bigr{\|}^{\frac{1}{k}}_{r}<\infty\,.$

Therefore $f$ is regularly holomorphic in the open ball with centre $z_{0}$
and radius $(2e\rho)^{-1}$.

$\blacksquare\ \ \ \ $

We now look at the special case of Banach lattices in which the lattice
structure is defined by an unconditional Schauder basis. Let $E$ be complex
Banach space with an unconditional Schauder basis $(e_{n})_{n}$. We may
assume, without loss of generality, that every point $z$ in $E$ with
coordinate expansion $z=\sum_{j}z_{j}e_{j}\in E$ satisfies

$\|z\|=\sup\Bigl{\\{}\Bigl{\|}\sum_{j}w_{j}e_{j}\Bigr{\|}:|w_{j}|\leq|z_{j}|\quad\text{
for every }j\Bigr{\\}}\,.$

(The norm on $E$ can always be replaced by an equivalent norm with this
property.) Then $E$ is a complex Banach lattice with the modulus defined
coordinatewise:

$|z|=\sum_{j}|z_{j}|e_{j}.$

Now let $P_{m}=\widehat{A}_{m}$ be a bounded $m$-homogeneous polynomial on
$E$. The value of $P_{m}$ at a point $z$ may be expanded, using the
multilinearity of $\widehat{A}_{m}$ to give an expression such as

$P_{m}(z)=\sum_{j_{m}}\dots\sum_{j_{1}}A_{m}(e_{j_{1}},\dots,e_{j_{m}})\,z_{j_{1}}\dots
z_{j_{m}}\,.$

In general, there is no guarantee that this multiple series will converge
absolutely. Matos and Nachbin [32] isolated the space of $m$-homogeneous
polynomials for which this expansion is absolutely convergent at every point
in $E$. They defined a norm for this space and proved that it is a Banach
space in this norm. Grecu and Ryan [25] showed that the Matos-Nachbin
polynomials coincide with the polynomials that are regular with respect to the
Banach lattice structure of $E$, and furthermore, the Matos–Nachbin norm is
precisely the regular norm. In other words, the space of regular
$m$-homogeneous polynomials on $E$ is exactly the space of $m$-homogeneous
polynomials $P_{m}$ that have pointwise absolutely convergent monomial
expansion of the form

$P_{m}(z)=\sum_{\stackrel{{\scriptstyle\alpha\in\mathbb{N}_{0}^{(\mathbb{N})}}}{{|\alpha|=m}}}c_{\alpha}z^{\alpha}$

and the regular norm is

$\|P_{m}\|_{r}=\sup\Bigl{\\{}\Bigl{\|}\sum_{|\alpha|=m}|c_{\alpha}||z|^{\alpha}\Bigr{\|}:\|z\|\leq
1\Bigr{\\}}\,.$

The following result, and its proof, is based on Theorem 3.10 in [31].

######  Theorem 3 (cf. [31], Prop. 3.10)

Let $E$ be a complex Banach space with an unconditional Schauder basis and let
$U$ be an open subset of $E$. A function $f\colon U\to\mathbb{C}$ is regularly
holomorphic if and only if, for every $z\in U$, the monomial expansion of $f$
around $z$ is absolutely convergent to $f$ in some neighbourhood of $z$.

Proof:

First, suppose that $f$ is regularly holomorphic on $U$. Let $z\in U$. Then

$\limsup_{m}\,\Bigl{\|}\frac{1}{m!}\widehat{d^{m}}{f}(z)\Bigl{\|}_{r}^{\frac{1}{m}}=\rho<\infty$

and so there is a positive constant $C$ satisfying

$\Bigl{\|}\frac{1}{m!}\widehat{d^{m}}{f}(z)\Bigl{\|}_{r}\leq C\,\rho^{m}$

for every $m\in\mathbb{N}$. Now $\frac{1}{m!}\widehat{d^{m}}{f}(z)$, being
regular, has an absolutely pointwise convergent monomial expansion of the form
[31]

$\frac{1}{m!}\widehat{d^{m}}{f}(z)(w)=\sum_{|\alpha|=m}c_{\alpha}w^{\alpha}$

and so we have

$\sum_{|\alpha|=m}|c_{\alpha}|\,|w-z|^{\alpha}\leq\Bigl{\|}\frac{1}{m!}\widehat{d^{m}}{f}(z)\Bigl{\|}_{r}\|w-z\|^{m}\leq
C\,\rho^{m}\,\|w-z\|^{m}\ \,.$

It follows that the monomial expansion

$\sum_{\alpha\in\mathbb{N}_{0}^{(\mathbb{N})}}c_{\alpha}(w-z)^{\alpha}$

is absolutely convergent to $f(w)$ for $\|w-z\|<\rho^{-1}$.

Now suppose that $f$ is locally representable by pointwise absolutely
convergent monomial expansions. So, for each $z\in U$, there exists $\rho>0$
and a monomial expansion of the form

$f(w)=\sum_{\alpha\in\mathbb{N}_{0}^{(\mathbb{N})}}c_{\alpha}(w-z)^{\alpha}$

that converges absolutely at every point in the ball $\|w-z\|<\rho$. It
follows that $f$ is Gateaux-holomorphic in this ball. By Baire’s theorem, $f$
has at least one point of continuity in this ball and hence $f$ is holomorphic
there. The derivatives of $f$ at $z$ are given by

$\frac{1}{m!}\widehat{d^{m}}{f}(z)(w-z)=\sum_{|\alpha|=m}c_{\alpha}(w-z)^{\alpha}\,.$

Choose $0<\sigma<\rho$ so that the holomorphic function
$\sum_{\alpha}|c_{\alpha}|(w-z)^{\alpha}$ is bounded by $C>0$ in the ball
$\|w-x\|\leq\sigma$. For every non-zero $\xi\in E$,

$\sum_{|\alpha|=m}|c_{\alpha}||\xi|^{\alpha}=\Bigl{(}\frac{\|\xi\|}{\sigma}\Bigr{)}^{m}\sum_{|\alpha|=m}|c_{\alpha}|\bigl{(}\sigma\,\xi/\|\xi\|\bigr{)}^{\alpha}\leq
C\Bigl{(}\frac{\|\xi\|}{\sigma}\Bigr{)}^{m}\,.$

Thus $\frac{1}{m!}\widehat{d^{m}}{f}(z)$ is a regular $m$-homogeneous
polynomial for every $m\in\mathbb{N}$ and the inequality above shows that

$\Bigl{\|}\frac{1}{m!}\widehat{d^{m}}{f}(z)\Bigl{\|}_{r}=\sup\bigl{\\{}\sum_{|\alpha|=m}|c_{\alpha}||\xi|^{\alpha}:\|\xi\|\leq
1\bigr{\\}}\leq C\,\Bigl{(}\frac{1}{\sigma}\Bigr{)}^{m}$

for every $m\in\mathbb{N}$. Therefore $f$ is regularly holomorphic on $U$.

$\blacksquare\ \ \ \ $

Let $E,F$ be complex Banach lattices, with $F$ Dedekind complete. Let $U$ be
an open subset of $E$. A function $f\colon U\to F$ is regularly holomorphic of
bounded type if it is regularly holomorphic on $U$ and, for every $z\in U$,

$|r|(f,z)=\left(\limsup_{m}\,\Bigl{\|}\frac{1}{m!}\widehat{d^{m}}{f}(z)\Bigl{\|}_{r}^{\frac{1}{m}}\right)^{-1}\geq
d(z,\partial U)\,.$ (5)

Thus, the Taylor series of $f$ at every point $z$ in $U$ is required to be
uniformly regularly convergent on every ball with centre $z$ that is contained
in $U$. We denote the space of regularly holomorphic functions of bounded type
by $\mathcal{H}_{rb}({U};{F})$, or by $\mathcal{H}_{rb}({U})$ when
$F=\mathbb{C}$.

###### Proposition 9

Let $E,F$ be complex Banach lattices with $F$ Dedekind complete, let $U$ be a
solid open subset of $E$ and let $f\colon U\to F$ be a regularly holomorphic
mapping of bounded type. Then, for every $z\in U$, the Taylor series of $f$ at
$z$ is uniformly regularly convergent on every polydisc with centre $z$
contained in $U$.

Proof: This follows from the fact that

$\Bigl{|}\frac{1}{m!}\widehat{d^{m}}{f}(z)\Bigl{|}(w-z)\leq\Bigl{|}\frac{1}{m!}\widehat{d^{m}}{f}(z)\Bigl{|}(a)$

whenever $a\in E_{+}$ and $w\in D(z,a)$.  $\blacksquare\ \ \ \ $

We recall the definition of a holomorphy type, introduced by Nachbin [36].

###### Definition 1

Let $\mathcal{B}$ be a class of ordered pairs of Banach spaces. A holomorphy
type on $\mathcal{B}$ is an assignment, to each $(E,F)$ in $\mathcal{B}$, of a
sequence of Banach spaces
$(\mathcal{P}_{\theta}(^{m}{E};{F}),\|\cdot\|_{\theta})_{m}$ with the
following properties:

  1. (a)

Each $\mathcal{P}_{\theta}(^{m}{E};{F})$ is a vector subspace of
$\mathcal{P}(^{m}{E};{F})$.

  2. (b)

$\mathcal{P}_{\theta}(^{0}{E};{F})$ coincides with
$\mathcal{P}(^{0}{E};{F})=F$ as a Banach space.

  3. (c)

There is a real number $\sigma\geq 1$ such that, if $k,m$ are natural numbers
with $k\leq m$, $a\in E$ and $P\in\mathcal{P}_{\theta}(^{m}{E};{F})$, then
$\widehat{d}^{k}P\in\mathcal{P}_{\theta}(^{m-k}{E};{\mathcal{P}_{\theta}(^{k}{E};{F})})$
and

$\Bigl{\|}\frac{1}{k!}\widehat{d}^{k}P(a)\Bigr{\|}_{\theta}\leq\sigma^{m}\|P\|_{\theta}\|a\|^{m-k}\,.$

######  Proposition 10 (cf. [31], Prop. 3.8 )

The sequence
$\displaystyle\bigl{(}\mathcal{P}_{r}(^{m}{E,F}),\|\cdot\|_{r}\bigr{)}_{m}$ is
a holomorphy type on the class of pairs $(E,F)$ of complex Banach lattices
with $F$ Dedekind complete.

Proof: Properties (a) and (b) are obvious. To show that (c) is satisfied, let
$P=\widehat{A}\in\mathcal{P}_{r}(^{m}{E};{F})$. Then, using Proposition 6 and
the polarization inequality,

$\displaystyle\Bigl{\|}\frac{1}{k!}\widehat{d^{k}}{P}(a)\Bigl{\|}_{r}$
$\displaystyle=\Bigl{\|}\;\Bigl{|}\frac{1}{k!}\widehat{d^{k}}{P}(a)\Bigr{|}\;\Bigl{\|}\leq\Bigl{\|}\frac{1}{k!}\widehat{d^{k}}{|P|}(a)\Bigl{\|}$
$\displaystyle=\binom{m}{k}\Bigl{\|}\;|A|a^{m-k}\Bigr{|}\;\Bigr{\|}\leq\binom{m}{k}\frac{m^{m}}{m!}\|\;|P|\;\|\|a\|^{m-k}$
$\displaystyle\leq\sigma^{m}\,\|P_{m}\|_{r}\,\|a\|^{m-k}$

where $\sigma=2e$.  $\blacksquare\ \ \ \ $

In some circumstances, it is possible to define a complex lattice structure on
the space of regularly holomorphic functions. We shall use the axiomatization
of complex vector lattices given by Mittelmeyer–Wolff [34]. They define an
_Archimedean modulus_ on a complex vector space $G$ to be a function $m\colon
G\to G$ satisfying

  * (0)

$m(m(x))=m(x)$ for every $x\in G$.

  * (i)

$m(\alpha x)=|\alpha|m(x)$ for every $\alpha\in\mathbb{C}$, $x\in G$.

  * (ii)

$m\bigl{(}m(m(x)+m(y))-m(x+y)\bigr{)}=m(x)+m(y)-m(x+y)$.

  * (iii)

$G$ is the linear span of $m(G)$.

  * (iv)

$m(m(y)-km(x))=m(y)-km(x)$ for every $k\in\mathbb{N}$ implies $x=0$.

A _complex vector lattice_ is a complex vector space $G$ equipped with a
modulus. It is shown in [34] that the real linear span of the subset $m(G)$ of
$G$ is a real vector lattice with $m(G)$ as the positive cone.

###### Proposition 11

Let $E$ be a complex Banach lattice and let $U$ be an open ball in $E$ with
centre $z_{0}$. Then the space $\mathcal{H}_{rb}({U})$ of regularly
holomorphic functions of bounded type on $E$ is a complex Archimedean Riesz
space with the modulus given by

$m(f)=|f|:=\sum_{m=0}^{\infty}\Bigl{|}\frac{1}{m!}\widehat{d}^{m}f(z_{0})\Bigr{|}\,.$

Proof: The proof consists of straightforward calculations. For example, to
show that property (ii) in the definition is satisfied, let $f=\sum_{m}P_{m}$
and $g=\sum_{m}Q_{m}$ be the Taylor expansions at $z_{0}$. Then, since $m$
coincides with the modulus on $\mathcal{P}_{r}(^{m}{E})$,

$m\bigl{(}m(m(f)+m(g))-m(f+g)\bigr{)}\\\
=\biggl{|}\Bigl{|}\sum_{m}|P_{m}|+|Q_{m}|\Bigr{|}-\sum_{m}|P_{m}+Q_{m}|\biggr{|}=\biggl{|}\sum_{m}|P_{m}|+|Q_{m}|-|P_{m}+Q_{m}|\biggr{|}\\\
=\sum_{m}|P_{m}|+|Q_{m}|-|P_{m}+Q_{m}|=m(f)+m(g)-m(f+g)\,.$

$\blacksquare\ \ \ \ $

Is it possible to define a complex lattice structure on
$\mathcal{H}_{rb}({U})$ for domains other than open balls? It would be
reasonable to expect that, if $B(a,r)$ is an open ball contained in $U$, then
the restriction mapping from $\mathcal{H}_{rb}({U})$ to
$\mathcal{H}_{rb}({B(a,r)})$ would be a lattice homomorphism. If
$\mathcal{H}_{rb}({U})$ carried a complex lattice structure satisfying this
condition, then it would follow that

$|f|=\sum_{m=0}^{\infty}\Bigl{|}\frac{1}{m!}\widehat{d}^{m}f(a)\Bigr{|}$

in the ball $B(a,r)$. However, this attempt to define $|f|$ locally fails, due
to a lack of coherence.

To see this, take $E=\mathbb{C}$, and consider a holomorphic function $f$ on
$B(a,r)$. Then for each point $z_{0}$ in $B(a,r)$ we can expand $f$ as a
Taylor series

$f(z)=\sum_{m=0}^{\infty}a_{m}(z-z_{0})^{m}$

for $|z|<r-|z_{0}|$. Note that the coefficients $(a_{m})_{m}$ depend on the
point $z_{0}$ and are given by $\displaystyle
a_{m}=\frac{f^{(m)}(z_{0})}{m!}$.

For each point $z_{0}$ we define the holomorphic function $|f|_{z_{0}}$ by

$|f|_{z_{0}}(z)=\sum_{n=0}^{\infty}|a_{n}|(z-z_{0})^{n}.$

Note that this series converges on $B(z_{0},r-|z_{0}|)$.

However, these expansions lack coherence. To see this take the function
$f\colon B(0,1)\to\mathbb{C}$ given by

$f(z)=\frac{1}{1-z}.$

We first expand $f$ about the origin to get the Taylor series expansion

$f(z)=\sum_{m=0}^{\infty}z^{m}.$

The Taylor series expansion of $f$ about $\frac{i}{2}$ is

$f(z)=\sum_{m=0}^{\infty}\left(\frac{1}{(1-\frac{i}{2})^{m+1}}\right)\left(z-\frac{i}{2}\right)^{m}=\sum_{m=0}^{\infty}\left(\frac{2}{5}(2+i)\right)^{m+1}\left(z-\frac{i}{2}\right)^{m}.$

Then

$|f|_{0}(z)=f(z)=\sum_{m=0}^{\infty}z^{m}$

and

$|f|_{\frac{i}{2}}(z)=\sum_{m=0}^{\infty}\left(\frac{2}{\sqrt{5}}\right)^{m+1}\left(z-\frac{i}{2}\right)^{m}.$

We now observe that $|f|_{0}(\frac{i}{2})=\frac{2}{5}(2+i)$ while
$|f|_{\frac{i}{2}}(\frac{i}{2})=\frac{2}{\sqrt{5}}$.

So it appears that, in general, it is not possible to define a complex lattice
structure on $\mathcal{H}_{rb}({U})$.

##  5 The Bohr radius on Banach lattices

In this section we show how the Bohr radius for complex sequence spaces can be
generalized to the wider setting of complex Banach lattices. In Theorem 4 we
prove that the growth of the Bohr radii controls the radius of regular
convergence and we show that on finite dimensional sequence spaces, the radii
of convergence and regular convergence are equal.

In 1914 H. Bohr [9] showed that if $\sum_{k=0}^{\infty}c_{k}z^{k}$ is a power
series on the unit disc centred at $0$, $D(0,1)$, with
$\left|\sum_{k=0}^{\infty}c_{k}z^{k}\right|\leq 1$ for all $z$ with $|z|<1$
then $\sum_{k=0}^{\infty}|c_{k}z^{k}|\leq 1$ for all $z$ with
$|z|\leq\frac{1}{3}$. Moreover, $\frac{1}{3}$ is the optimal radius for which
this inequality holds.

More generally, given a Reinhardt domain $R$ in $\mathbb{C}^{n}$ the Bohr
radius of $R$, $K(R)$, is defined as the supremum over all $r\geq 0$ such that
if $\sum_{\alpha}c_{\alpha}z^{\alpha}$ is a power series on $R$ with
$\left|\sum_{\alpha}c_{\alpha}z^{\alpha}\right|\leq 1$ for all $z$ in $R$ then
$\sum_{\alpha}|c_{\alpha}z^{\alpha}|\leq 1$ for all $z$ in $rR$. It follows
from results of Aizenberg [1], Boas [6], Boas and Kavinson [7] and Dineen and
Timoney [23], that there is a constant $c$, independent of $n\in\mathbb{N}$
and $1\leq p\leq\infty$, such that

$\frac{1}{c}\left(\frac{1}{n}\right)^{1-\frac{1}{\min(p,2)}}\leq
K(B_{\ell_{p}^{n}})\leq c\left(\frac{\log
n}{n}\right)^{1-\frac{1}{\min(p,2)}}.$

In 2003, Defant, García and Maestre [20] refined the concept of Bohr radius
and introduced the concept of homogeneous Bohr radius. Given a finite
dimensional Banach space $X=(\mathbb{C}^{n},\|\cdot\|)$ with canonical basis
$(e_{k})_{k=1}^{n}$ they define $K_{m}(B_{X})$ as the supremum over all $r$ in
$[0,1]$ such that if $\sum_{|\alpha|=m}c_{\alpha}z^{\alpha}$ is an
$m$-homogeneous polynomial on $X$ with
$\left|\sum_{|\alpha|=m}c_{\alpha}z^{\alpha}\right|\leq 1$ for all $z$ in
$B_{X}$ then $\sum_{|\alpha|=m}|c_{\alpha}z^{\alpha}|\leq 1$ for all $z$ in
$rB_{X}$.

Let $E$ be a Banach space with an unconditional basis, $(x_{n})_{n}$. Then the
unconditional basis constant of $(x_{n})_{n}$, $\chi((x_{n})_{n})$, is defined
by

$\chi((x_{n})_{n})=\inf\left\\{C:\left\|\sum_{k=1}^{\infty}\epsilon_{k}\mu_{k}x_{k}\right\|\leq
C\left\|\sum_{k=1}^{\infty}\mu_{k}x_{k}\right\|:\mu_{j}\in\mathbb{C},|\epsilon_{j}|=1,j\in\mathbb{N}\right\\}.$

In [20, Lemma 2.1] it is shown that if $X=(\mathbb{C}^{n},\|\cdot\|)$ then

$K_{m}(B_{X})=\frac{1}{\root m \of{\chi_{mon}(\mathcal{P}(^{m}{X}))}}$

where $\chi_{mon}(\mathcal{P}(^{m}{X}))$ denotes the unconditional basis
constant of the monomials in $\mathcal{P}(^{m}{X})$.

Given any $m$-homogeneous polynomial
$P=\sum_{|\alpha|=m}c_{\alpha}z^{\alpha}$, when we regard
$\mathcal{P}(^{m}{X})$ as a Banach lattice, the absolute value of $P$ is given
by $|P|=\sum_{|\alpha|=m}|c_{\alpha}|z^{\alpha}$. Motivated by this
observation we now introduce the $m$-th Bohr radius of a general Banach
lattice.

###### Definition 2

Let $E$ be a complex Banach lattice and $m$ be a positive integer. We define
the $m$-th Bohr radius of $E$, $K_{m}(B_{E})$ by

$\displaystyle K_{m}(B_{E})$ $\displaystyle:=\sup\\{\rho:\sup_{z\in\rho
B_{E}}||P|(z)|\leq\|P\|\hbox{ for all }P\in\mathcal{P}_{r}(^{m}{E})\\}$
$\displaystyle=\sup\\{\rho:\rho^{m}\|P\|_{r}\leq\|P\|\text{ for all
$P\in\mathcal{P}_{r}(^{m}{E})$}\\}.$

Note that $0\leq K_{m}(B_{E})\leq 1$ and that $K_{m}(B_{E})$ may be $0$.

Indeed, we observe that $K_{m}(B_{E})>0$ if and only if the regular and
supremum norms are equivalent when restricted to $\mathcal{P}_{r}(^{m}{E})$.
In this case, ${K_{m}(B_{E})}^{-m}$ is the norm of the identity mapping from
$(\mathcal{P}_{r}(^{m}{E}),\|\cdot\|)$ onto
$(\mathcal{P}_{r}(^{m}{E}),\|\cdot\|_{r})$.

Let $E$ be a complex Banach lattice, $U$ be an open subset of $E$ and $f\colon
U\to\mathbb{C}$ be a homomorphic function such that
$f(a+z)=\sum_{m=0}^{\infty}P_{m}(z)$ is the Taylor series of $f$ about $a$
with each $P_{m}$ a regular $m$-homogeneous polynomial. Our next theorem
provides a lower bound for the radius of regular convergence, $|r|(f,a)$, in
terms of the homogeneous Bohr radii.

###### Theorem 4

Let $E$ be a complex Banach lattice and $U$ an open subset of $E$. Let
$f\colon U\to\mathbb{C}$ be a regular holomorphic function and $a\in U$. Then,

$\liminf_{m\to\infty}(K_{m}(B_{E}))r(f,a)\leq|r|(f,a)\leq r(f,a).$

Moreover, for each $a\in U$, both the upper and lower bounds are sharp.

Proof: If $\liminf_{m\to\infty}K_{m}(B_{E})=0$ then there is nothing to prove.
Let us now suppose that there is $m_{0}\in\mathbb{N}$ so that $K_{m}(B_{E})>0$
for each $m\geq m_{0}$. Write the Taylor series of $f$ about $a$ as

$f(a+z)=\sum_{m=0}^{\infty}P_{m}(z).$

Then, by the definition of the Bohr radius, we have that

$(K_{m}(B_{E}))^{m}\|P_{m}\|_{r}\leq\|P_{m}\|$

for each $m$ which we will rewrite as

$\|P_{m}\|_{r}\leq\frac{1}{(K_{m}(B_{E}))^{m}}\|P_{m}\|$

for all $m\geq m_{0}$. Taking $m$-th roots we get that

$\|P_{m}\|_{r}^{1/m}\leq\frac{1}{(K_{m}(B_{E}))}\|P_{m}\|^{1/m}.$

We now let $m$ tend to infinity to get

$\limsup_{m\to\infty}\|P_{m}\|_{r}^{1/m}\leq\limsup_{m\to\infty}\frac{1}{(K_{m}(B_{E}))}\limsup_{m\to\infty}\|P_{m}\|^{1/m}.$

Noting that
$\displaystyle\frac{1}{\limsup_{m\to\infty}a_{m}}=\liminf_{m\to\infty}a_{m}$
we get that

$\limsup_{m\to\infty}\|P_{m}\|_{r}^{1/m}\leq\frac{1}{\liminf_{m\to\infty}(K_{m}(B_{E}))}\limsup_{m\to\infty}\|P_{m}\|^{1/m}$

and inverting we obtain

$\liminf_{m\to\infty}(K_{m}(B_{E}))r(f,a)\leq|r|(f,a).$

Therefore, we have

$\liminf_{m\to\infty}(K_{m}(B_{E}))r(f,a)\leq|r|(f,a)\leq r(f,a).$

Let us now see that both sides of this inequality are sharp. If we consider a
holomorphic function $f=\sum_{m=0}^{\infty}P_{m}$ with $P_{m}\geq 0$ for each
$m\in\mathbb{N}$ then we have that $|r|(f,0)=r(f,0)$.

To show that the lower bound is optimal we consider two cases. We first assume
that $K_{m}(B_{E})>0$ all but finitely many $m$. Then, discarding finitely
many terms, we can assume that $K_{m}(B_{E})>0$ for all $m$. For each
$m\in\mathbb{N}$ we choose $P_{m}\not=0$ so that

$\frac{1}{2}\|P_{m}\|\leq K_{m}(B_{E})^{m}\|P_{m}\|_{r}\leq\|P_{m}\|.$ (6)

Dividing $P_{m}$ by $\|P_{m}\|$ we may assume that $\|P_{m}\|=1$ for each
$m\in\mathbb{N}$. If we let $f=\sum_{m=1}^{\infty}P_{m}$ then $r(f,0)=1$.

From (6) we obtain

$\left(\|P_{m}\|^{1/m}\right)^{-1}\leq
K_{m}(B_{E})^{-1}(\|P_{m}\|_{r}^{1/m})^{-1}\leq
2^{1/m}(\|P_{m}\|^{1/m})^{-1}\,,$

or

$K_{m}(B_{E})\left(\|P_{m}\|^{1/m}\right)^{-1}\leq(\|P_{m}\|_{r}^{1/m})^{-1}\leq
2^{1/m}K_{m}(B_{E})(\|P_{m}\|^{1/m})^{-1}.$

As $\|P_{m}\|=1$ for all $m$ we get

$K_{m}(B_{E})\leq(\|P_{m}\|_{r}^{1/m})^{-1}\leq 2^{1/m}K_{m}(B_{E}).$

Letting $m$ tend to infinity we get that

$\left(\limsup_{m\to\infty}\|P_{m}\|_{r}^{1/m}\right)^{-1}=\liminf_{m\to\infty}K_{m}(B_{E})$

and therefore we see that the lower bound is attained.

Let us now suppose that we have infinitely many $m$ with $K_{m}(B_{E})=0$.
Then we can choose a subsequence $(m_{k})_{k}$ so that $K_{m_{k}}(B_{E})=0$
for all $k$. Fix $j\in\mathbb{N}$ and for each $k$ in $\mathbb{N}$ choose
$P_{k}\in\mathcal{P}_{r}(^{m_{k}}{E})$ so that

$\sup_{z\in\frac{1}{j}B_{E}}||P_{k}|(z)|>\|P_{k}\|$

or that

$\|P_{k}\|_{r}>j^{m_{k}}\|P_{k}\|.$

Then repeating the above argument with $K_{m}(B_{E})$ replaced with
$\frac{1}{j}$ we get

$\limsup_{k\to\infty}\|P_{k}\|_{r}^{1/m_{k}}\geq\liminf_{m_{k}\to\infty}{j}$

and therefore

$|r|(f,0)\leq\frac{1}{j}.$

As this holds for all $j\in\mathbb{N}$ we see that we can find regular
holomorphic functions with radius of convergence $1$ but radius of absolute
convergence arbitrarily small.  $\blacksquare\ \ \ \ $

As we have mentioned above, it follows from [20, Lemma 2.1] that if $X$ is
$(\mathbb{C}^{n},\|\cdot\|)$ then $K_{m}(B_{X})=\frac{1}{\root m \of{\chi_{\rm
mon}(\mathcal{P}(^{m}{X}))}}$. In this case Theorem 4 gives us that

$\frac{1}{\limsup_{m\to\infty}\root m \of{\chi_{\rm
mon}(\mathcal{P}(^{m}{X}))}}r(f,a)\leq|r|(f,a)\leq r(f,a).$

If $X$ is the finite dimensional Banach lattice $(\mathbb{C}^{n},\|\cdot\|)$,
with any norm for which the unit vector basis is a $1$-unconditional Schauder
basis, then it follows from [19, Lemma 11..2.2] that
$\limsup_{m\to\infty}\sqrt[m]{\chi_{\rm mon}(\mathcal{P}(^{m}{X}))}=1$. Hence,
we have the following result.

###### Theorem 5

Let $f$ be a holomorphic function on $(\mathbb{C}^{n},\|\cdot\|)$, with any
norm for which the unit vector basis is a $1$-unconditional Schauder basis.
Then $r(f,a)=|r|(f,a)$ for every $a$.

Theorem 5 is not true for real analytic functions on subsets of
$\mathbb{R}^{n}$. Indeed, Hayman, [26], shows that if
$f(x)=\sum_{k=0}^{\infty}P_{k}(x)$, where each $P_{k}$ is a harmonic
$k$-homogeneous polynomial, converges on the polydisc
$\\{(x_{i})_{i=1}^{n}:|x_{i}|<r\\}$ in $\ell_{\infty}^{n}$ then
$\sum_{k=0}^{\infty}|P_{k}|$ converges on the polydisc
$\\{(x_{i})_{i=1}^{n}:|x_{i}|<r/\sqrt{2}\\}$. Moreover, an example is provided
in [26] to show that the factor of $r/\sqrt{2}$ is sharp.

##  6 Regular Holomorphic Functions on $\ell_{p}$

In this section we will look at regular holomorphic functions on $\ell_{p}$
for $1<p<\infty$. We will show that it possible to construct holomorphic
functions with radius of convergence $1$ yet having arbitrarily small radius
of regular convergence.

In our constructions we need to consider the Bohr radius,
$K_{m}(B_{\ell_{p}})$, of $\mathcal{P}(^{m}{\ell_{p}})$. We begin with the
observation that for any positive integer $k$ we have $K_{m}(B_{\ell_{p}})\leq
K_{m}(B_{\ell_{p}^{k}})$. In addition, by [20, Lemma 2.1], for each $m$ and
$k$ we have

$K_{m}(B_{\ell_{p}^{k}})=\frac{1}{\root m \of{\chi_{\rm
mon}(\mathcal{P}(^{m}{\ell_{p}^{k}}))}}.$

Let $\pi_{k}$ denote the canonical projection from $\ell_{p}$ onto
$\ell_{p}^{k}$. Our plan is to define a holomorphic function $f$ on $\ell_{p}$
of the form $f(z)=\sum_{m=0}^{\infty}P_{m}\circ\pi_{n_{m}}(z)$ where for each
$m$, $P_{m}$ is an $m$-homogeneous polynomial on $\ell_{p}^{n_{m}}$ with
$n_{m}\to\infty$ as $m\to\infty$.

We need some notation. Let $(A_{mn})_{m,n}$ and $(B_{mn})_{m,n}$ be two doubly
indexed sequences of positive real numbers. Following [18], we will write
$A_{mn}\sim B_{mn}$ if there is $C\geq 1$ so that for all $m$ and $n$ we have

$(1/C^{m})A_{mn}\leq B_{mn}\leq C^{m}A_{mn}.$

It follows from [18, Page 133] that for $1\leq p\leq\infty$ we have

$\chi_{\rm
mon}(\mathcal{P}(^{m}{\ell_{p}^{n}}))\sim\left(1+\frac{n}{m}\right)^{(m-1)\left(1-\frac{1}{\min(p,2)}\right)}.$

Let $\alpha_{m}>0$. Then for each fixed positive integer $m$, let us choose
$n_{m}$ so that

$1+\frac{n_{m}}{m}>\left((1+\alpha_{m})^{\frac{m}{m-1}}\left(C^{\frac{m}{m-1}}\right)\right)^{\frac{1}{1-\frac{1}{\min(p,2)}}}.$

Then we have

$\displaystyle\root m \of{\chi_{\rm
mon}(\mathcal{P}(^{m}{\ell_{p}^{n_{m}}}))}$
$\displaystyle\geq\frac{1}{C}\left(\left(1+\frac{n_{m}}{m}\right)^{1-\frac{1}{\min(p,2)}}\right)^{\frac{m-1}{m}}$
$\displaystyle>1+\alpha_{m}.$

Therefore, for each positive integer $m$ we have that
$K_{m}(B_{\ell_{p}^{n_{m}}})<\frac{1}{1+\alpha_{m}}$.

Thus, for each $m$ we can choose an $m$-homogeneous polynomial $P_{m}$ in
$\mathcal{P}(^{m}{\ell_{p}^{n_{m}}})$ so that $\|P_{m}\|=1$ and
$\|P_{m}\|_{r}^{\frac{1}{m}}>{1+\alpha_{m}}$. If we define $f$ on the unit
ball of $\ell_{p}$ by $f(z)=\sum_{n=0}^{\infty}P_{m}(\pi_{n_{m}}(z))$ then
$r(f,0)=1$ while
$|r|(f,0)\leq\left(\limsup_{m\to\infty}\frac{1}{1+\alpha_{m}}\right)^{-1}$.

Let us show that if $p>1$ then for each $m$ in $\mathbb{N}$ and each
$1\leq\eta<\infty$ we can find $n_{m}\in\mathbb{N}$ and an $m$-homogeneous
polynomial $P_{\eta}$ in $\mathcal{P}(^{m}{\ell_{p}^{n_{m}}})$ with
$\|P_{\eta}\|=1$ and $\|P_{\eta}\|_{r}^{\frac{1}{m}}=\eta$.

We begin by choosing $\alpha_{m}$ so that $1+\alpha_{m}>\eta$. From the above
construction we know that we can find $n_{m}\in\mathbb{N}$ and $P_{m}$ in
$\mathcal{P}(^{m}{\ell_{p}^{n_{m}}})$ with $\|P_{m}\|=1$ and
$\|P_{m}\|_{r}^{1/m}>1+\alpha_{m}$. We now choose a positive $m$-homogeneous
polynomial $Q_{m}$ on $\mathcal{P}(^{m}{\ell_{p}^{n_{m}}})$ with
$\|Q_{m}\|=1$. As $S_{\mathcal{P}(^{m}{\ell_{p}^{n_{m}}})}$ is path connected,
we can find a path $\gamma\colon[0,1]\to
S_{\mathcal{P}(^{m}{\ell_{p}^{n_{m}}})}$ with $\gamma(0)=Q_{m}$ and
$\gamma(1)=P_{m}$.

Let us now consider the function $\lambda\colon[0,1]\to\mathbb{R}$ given by
$\lambda(t)=\|\gamma(t)\|_{r}^{\frac{1}{m}}$. Since the supremum and regular
norms are equivalent, we know that $\lambda$ is continuous. As $\lambda(0)=1$
and $\lambda(1)\geq 1+\alpha_{m}>\eta>1$ the Intermediate Value Theorem tells
us that we can find $t_{o}\in(0,1)$ so that $\lambda(t_{o})=\eta$. If we set
$\gamma(t_{0})=P_{m,\eta}$ then we have an $m$-homogeneous polynomial
$P_{m,\eta}$ on $\ell_{p}^{n_{m}}$ with $\|P_{m,\eta}\|=1$ and
$\|P_{m,\eta}\|_{r}^{\frac{1}{m}}=\eta$. We now define $f$ on $B_{\ell_{p}}$
by $f(z)=\sum_{m=0}^{\infty}P_{m,\eta}(\pi_{n_{m}}(z))$ to obtain the
following result.

###### Proposition 12

Let $p>1$. Then for each $\tau\in(0,1)$ there is a holomorphic function $f$ on
$B_{\ell_{p}}$ with $r(f,0)=1$ and $|r|(f,0)=\tau$.

For $\ell_{1}$, Matos [31, Prop. 3.7] showed that every bounded
$m$-homogeneous polynomial is regular and that the regular and supremum norms
on $\mathcal{P}(^{m}{\ell_{1}})$ are equivalent. More precisely,

$\|P\|\leq\|P\|_{r}\leq e^{m}\|P\|$

for every $P\in\mathcal{P}(^{m}{\ell_{1}})$. Thus, on $\ell_{1}$, we have
$e^{-1}r(f,0)\leq|r|(f,0)\leq r(f,0)$ for every holomorphic function $f$ on
$\ell_{1}$ or its unit ball. We do not know if this inequality is sharp.

##  7 Orthogonally Additive Holomorphic Functions

In this section we study orthogonally additive holomorphic functions on Banach
lattices and show that for this class of functions the radii of convergence
and regular convergence coincide.

We begin by looking at what is known for finite dimensional Banach lattices.
Let us suppose that $X$ is $\mathbb{C}^{n}$ endowed with a norm under which
$(e_{j})_{j=1}^{n}$ is a Schauder basis with unconditional basis constant $1$.
Given a subset $J$ of $\\{1,\ldots,n\\}^{m}$ for some natural number $m$,
Bayart, Defant and Schlüters, [5], use $\mathcal{P}(^{J}{X})$ to denote the
closed subspace of all holomorphic function $f$ in
$\mathcal{H}^{\infty}(B_{X})$ for which $c_{\alpha}(f)=0$ if $\alpha\not\in
J$. In other words, ${\cal P}(^{J}X)$ is the span of $\\{z^{j};j\in J\\}$.

In the case where we take $J=\\{(k,\ldots,k):k\in\\{1,\dots,n\\}\\}$ the space
$\mathcal{P}(^{J}{X})$ is the space of $m$-homogeneous orthogonally additive
polynomials on $X$. Using [4, Theorem 1.3] Bayart, Defant and Schlüters,
deduce on [5, Page 113] that $\chi_{\rm mon}(\mathcal{P}(^{J}{X}))=1$. In
particular, this means that $\|P\|=\|P\|_{r}$ for every $m$-homogeneous
orthogonally additive polynomial on $X$. We now extend this result to
orthogonally additive polynomials on any complex Banach lattice.

Let $E$ be a complex Banach lattice. Then $z,w\in E$ are said to be _disjoint_
, denoted by $z\perp x$, if $|z|\wedge|w|=0$ in $E_{\mathbb{R}}$.

###### Lemma 2

Let $E$ be a complex Banach lattice and let $z=x+iy,w=u+iv$ be elements of
$E$. Then $z\perp w$ if and only if $x,y\perp u,v$.

Proof: We have

$|z|\wedge|w|=\\\
\sup\\{|x|\cos\theta+|y|\sin\theta:\theta\in[0,2\pi]\\}\wedge\sup\\{|u|\cos\phi+|v|\sin\phi:\phi\in[0,2\pi]\\}\\\
=\sup_{\theta}\sup_{\phi}\;\bigl{(}|x|\cos\theta+|y|\sin\theta\bigr{)}\wedge\bigl{(}|u|\cos\phi+|v|\sin\phi\bigr{)}\,,$
(7)

using the infinite distributive property of the lattice operations [2, Theorem
1.8].

Suppose that $z\perp w$. Then, by (7),

$\bigl{(}|x|\cos\theta+|y|\sin\theta\bigr{)}\wedge\bigl{(}|u|\cos\phi+|v|\sin\phi\bigr{)}\leq
0\,$

for all $\theta,\phi$. Taking $\theta,\phi=0,\pi/2$, we get

$|x|\wedge|u|=|x|\wedge|v|=|y|\wedge|u|=|y|\wedge|v|=0\,.$

The converse follows easily from (7).  $\blacksquare\ \ \ \ $

We will now make use of the following property of real orthogonally additive
polynomials. Let $Q$ be an $m$-homogeneous polynomial on $E$, generated by the
symmetric $m$-linear form $B$. Then $B$ is said to be _orthosymmetric_ if
$B(x_{1},\dots,x_{m})=0$ whenever any two of $x_{1},\dots,x_{m}$ are disjoint.
This property of $B$ is equivalent to $Q$ being orthogonally additive [12].

###### Proposition 13

Let $E$ be a complex Banach lattice and $P$ be a regular $m$-homogeneous
polynomial on $E$. Then the following are equivalent:

  * (a)

$P$ is orthogonally additive on $E$.

  * (b)

$P$ is orthogonally additive on $E_{\mathbb{R}}$.

  * (c)

The real and imaginary parts of $P$ are orthogonally additive on $E$.

Proof: The implications (a) implies (b) and (c) implies (a) are trivial.

To show that (b) implies (c), suppose that $P=P_{0}+iP_{1}$ is orthogonally
additive on $E_{\mathbb{R}}$. Then for $x,y\in E_{\mathbb{R}}$ with $x\perp
y$, we have $P(x+y)=P(x)+P(y)$. Taking real and imaginary parts of $P$ and
using (2), we get

$P_{0}(x+y)=P_{0}(x)+P_{0}(y)\quad\mbox{and}\quad
P_{1}(x+y)=P_{1}(x)+P_{1}(y).$

That is, $P_{0}$ and $P_{1}$ are orthogonally additive on $E_{\mathbb{R}}$.

To complete the proof, we must show that complexifications of $P_{0}$ and
$P_{1}$ are orthogonally additive on $E$. Let $z=x+iy$, $w=u+iv$ be disjoint
elements of $E$. We have

$P_{0}(z+w)=\sum_{k=0}^{m}i^{k}\binom{m}{k}A_{0}(x+u)^{m-k}(y+v)^{k}\,,$

where $A_{0}$ is the symmetric $m$-linear form that generates $P_{0}$.
Expanding, we get

$P_{0}(z+w)=\sum_{k=0}^{m}i^{k}\binom{m}{k}\sum_{r=0}^{m-k}\sum_{s=0}^{k}\binom{m-k}{r}\binom{k}{s}A_{0}(x^{m-k-r}u^{r}y^{k-s}v^{s})\,.$

Since $P_{0}$ is orthogonally additive on $E_{\mathbb{R}}$, the $m$-linear
form $A_{0}$ is orthosymmetric on $E_{\mathbb{R}}^{m}$. Thus, we have that
$A_{0}(x^{m-k-r}u^{r}y^{k-s}v^{s})$ vanishes if either of $x,y$ appears along
with either of $u,v$ in the argument. Therefore, for each $k$, there are only
two non-zero terms, $A_{0}(x^{m-k}y^{k})$ and $A_{0}(u^{m-k}v^{k})$ and hence

$P_{0}(z+w)=P_{0}(z)+P_{0}(w)\,.$

A similar argument shows that $P_{1}$ is orthogonally additive on $E$.
$\blacksquare\ \ \ \ $

It follows from the above proposition that the Banach lattice of orthogonally
additive $m$-homogeneous polynomials on a complex Banach lattice $E$ is the
complexification of the lattice of orthogonally additive $m$-homogeneous
polynomials on $E_{\mathbb{R}}$.

A regular holomorphic function $f\colon E\to\mathbb{C}$ is said to be
orthogonally additive if $f(x+y)=f(x)+f(y)$ whenever $x$ and $y$ are disjoint.
Orthogonally additive holomorphic functions have previously been defined on
$C(K)$ spaces by Carando, Lassalle and Zalduendo, [15] and on $C^{*}$-algebras
by Jaramillo, Prieto and Zalduendo, [28] and Peralta and Puglisi, [37]. The
proof of [15, Lemma 1.1] for $C(K)$ spaces trivially extends to arbitrary
complex Banach lattices to show that if $f\colon E\to\mathbb{C}$ is a regular
holomorphic function which has a Taylor series expansion
$f(z)=\sum_{k=0}P_{k}(z)$ about $0$ then $f$ is orthogonally additive if and
only if $P_{k}$ orthogonally additive for each $k$ in $\mathbb{N}$.

In [11] the authors proved that if $P$ is an $m$-homogeneous orthogonally
additive polynomial on a real Banach lattice then $\|P\|=\|P\|_{r}$ when $m$
is odd and $\|P\|\leq\|P\|_{r}\leq 2\|P\|$ when $m$ is even. Moreover, the
polynomial $P(x)=x_{1}^{m}-x_{2}^{m}$, with $m$ even, on $\ell_{\infty}^{2}$
shows that this bound is sharp.

Let us now consider what happens for orthogonally additive polynomials on
complex Banach lattices.

###### Proposition 14

Let $P$ be an orthogonally additive $m$-homogeneous polynomial on the complex
Banach lattice $C(K)$. Then $\|P\|_{r}=\|P\|$.

Proof:

Using [3] we know that every orthogonally additive $m$-homogeneous polynomial
$P$ on $C(K)$ has an extension, $\tilde{P}$, to $C(K)^{\prime\prime}$, the
bidual of $C(K)$. Moreover, it follows from [17] that $\|\tilde{P}\|=\|P\|$
and from [14, Corollary 2.1] that $\tilde{P}$ is orthogonally additive. We
observe that $\mathcal{B}(K)$, the space of all bounded Borel measurable
functions on $K$ is a closed subspace of $C(K)^{\prime\prime}$. This means
that each $m$-homogeneous orthogonally additive polynomial $P$ on $C(K)$ has
an extension $P_{B}$ to $\mathcal{B}(K)$ as an orthogonally additive
polynomial with $\|P_{B}\|=\|P\|$.

Since $P$ is orthogonally additive on $C(K)$, there is a complex measure $\mu$
on $K$ such that

$P(x)=\int_{K}x^{m}(t)\,d\mu(t)$

for all $x$ in $C(K)$. This implies that $|P|$ is represented by the measure
$|\mu|$. It follows from [14, Corollary 2.1] that

$P_{B}(y)={\tilde{P}}(y)=\int_{K}y^{m}(t)\,d\mu(t)$

for all bounded Borel measurable functions $y$ on $K$.

By [38, Theorem 6.12], there exists a Borel measurable function $\rho\colon
K\to\mathbb{C}$ with $|\rho(t)|=1$ for all $t$ in $K$ such that $d|\mu|=\rho
d\mu$.

Let us choose a branch $\rho^{1/m}$ of the $m$-th root of $\rho$. For each $x$
in $C(K)$, we note that $x\rho^{1/m}$ is a bounded Borel function on $K$.
Therefore, given an orthogonally additive $m$-homogeneous polynomial $P$ on
$C(K)$ we have

$\displaystyle|P|(x)$ $\displaystyle=\int_{K}x(t)^{m}\,d|\mu|(t)$
$\displaystyle=\int_{K}x(t)^{m}\rho(t)\,d\mu(t)$
$\displaystyle=\int_{K}(x\rho^{1/m}(t))^{m}\,d\mu(t)$
$\displaystyle=P_{B}(x\rho^{1/m}).$

Thus, we have

$\displaystyle\|P\|_{r}$ $\displaystyle=\sup_{\|x\|\leq 1,x\in C(K)}||P|(x)|$
$\displaystyle=\sup_{\|x\|\leq 1,x\in C(K)}|{P_{B}}(x\rho^{1/m})|$
$\displaystyle\leq\sup_{\|y\|\leq 1,y\in B(K)}|{P_{B}}(y)|$
$\displaystyle=\|P\|.$

As $\|P\|\leq\|P\|_{r}$ in general, we have $\|P\|=\|P\|_{r}$.

$\blacksquare\ \ \ \ $

We can extend this result to all Banach lattices by localising to principal
ideals. If $E$ is a real or complex Banach lattice, then for $a\in E_{+}$, the
principal ideal $E_{a}$ is the ideal in $E$ consisting of all $x$ satisfying
$|x|\leq c\,a$ for some $c>0$. With the norm

$\|x\|_{a}=\inf\\{c:|x|\leq c\,a\\}\,,$

it becomes a Banach lattice. As it is an AM-space with unit, the Kakutani
Representation Theorem says that $E_{a}$ is Banach lattice isometrically
isomorphic to $C(K_{a})$ for some compact Hausdorff space $K_{a}$. It is easy
to see that, in the complex case, $E_{a}$ is the Banach lattice
complexification of the real Banach lattice $(E_{\mathbb{R}})_{a}$.

The norm of a real or complex Banach lattice $E$ is determined by the norms of
its principal ideals. For every $z\in E$, we have

$\|z\|=\inf\\{\|a\|\,\|z\|_{a}:a\in E,a>0\\}$

and this infimum is attained when $a=|z|$. It follows that the closed unit
ball of $E$ is

$B_{E}=\bigcup_{a>0}\|a\|^{-1}\,B_{E_{a}}$ (8)

and so the norm of a regular $m$-homogeneous polynomial $P$ is given by

$\|P\|=\sup\\{\|a\|^{-m}\|P_{a}\|:a\in E,a>0\\}\,,$

where $P_{a}$ denotes the restriction of $P$ to the Banach lattice $E_{a}$,
equipped with the principal ideal norm. Applying this to the regular norm, we
have

$\|P\|_{r}=\sup\\{\|a\|^{-m}\||P|_{a}\|:a\in E,a>0\\}\,.$

In the real case, the restriction mapping $P\in\mathcal{P}_{r}(^{m}{E})\mapsto
P_{a}\in\mathcal{P}_{r}(^{m}{E_{a}})$ is a lattice homomorphism [10]. It is
easy to see that, in the complex case, the restriction mapping is the
complexification of the real restriction mapping. Therefore the restriction
mapping is a complex lattice homomorphism [39, p. 136]. So we have
$|P|_{a}=|P_{a}|$ and it follows that

$\displaystyle\|P\|_{r}$ $\displaystyle=\sup\\{\|a\|^{-m}\||P|_{a}\|:a\in
E,a>0\\}=\sup\\{\|a\|^{-m}\||P_{a}|\|:a\in E,a>0\\}$
$\displaystyle=\sup\\{\|a\|^{-m}\|P_{a}\|_{r}:a\in E,a>0\\}\,.$

Thus, from Proposition 14 we obtain the following result.

###### Theorem 6

Let $P$ be an orthogonally additive $m$-homogeneous polynomial on a complex
Banach lattice $E$. Then $\|P\|=\|P\|_{r}$.

In a similar way we can show that if $m$ is an odd integer and $P$ is an
$m$-homogeneous polynomial on a (real) Banach lattice $E$ then
$\|P\|=\|P\|_{r}$, recovering the first part of [11, Corollary 1].

From Theorem 6 we immediately obtain the following result.

###### Theorem 7

Let $f$ be an orthogonally additive holomorphic function on a complex Banach
lattice $E$. Then for each $a\in E$ we have

$|r|(f,a)=r(f,a).$

This contrasts with the results of Section 6, where we have seen that the
radii of convergence can be different in general.

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Christopher Boyd, School of Mathematics & Statistics, University College
Dublin, Belfield, Dublin 4, Ireland.
e-mail<EMAIL_ADDRESS>
Raymond A. Ryan, School of Mathematical and Statistical Sciences, University
of Galway, Ireland.
e-mail<EMAIL_ADDRESS>
Nina Snigireva, School of Mathematical and Statistical Sciences, University of
Galway, Ireland.
e-mail<EMAIL_ADDRESS>