* [53] M. Gutperle, N. Klein, and D. Rathore, Holographic 6d co-dimension 2 defect solutions in M-theory, arXiv:2304.12899.
  * [54] D. Gaiotto and J. Maldacena, The Gravity duals of N=2 superconformal field theories, JHEP 10 (2012) 189, [arXiv:0904.4466].
  * [55] L. J. Romans, Gauged $N=4$ Supergravities in Five-dimensions and Their Magnetovac Backgrounds, Nucl. Phys. B 267 (1986) 433–447.
  * [56] J. P. Gauntlett and O. Varela, D=5 SU(2) x U(1) Gauged Supergravity from D=11 Supergravity, JHEP 02 (2008) 083, [arXiv:0712.3560].
  * [57] D. Cassani, G. Josse, M. Petrini, and D. Waldram, $\mathcal{N}$ = 2 consistent truncations from wrapped M5-branes, JHEP 02 (2021) 232, [arXiv:2011.04775].
  * [58] N. Bobev, V. Dimitrov, and A. Vekemans, Wrapped M5-branes and AdS5 black holes, JHEP 05 (2023) 012, [arXiv:2212.10360].
  * [59] K. C. Matthew Cheung, J. P. Gauntlett, and C. Rosen, Consistent KK truncations for M5-branes wrapped on Riemann surfaces, Class. Quant. Grav. 36 (2019), no. 22 225003, [arXiv:1906.08900].
  * [60] A. Zaffaroni, AdS black holes, holography and localization, Living Rev. Rel. 23 (2020), no. 1 2, [arXiv:1902.07176].
  * [61] M. Pernici, K. Pilch, and P. van Nieuwenhuizen, Gauged Maximally Extended Supergravity in Seven-dimensions, Phys. Lett. 143B (1984) 103–107.
  * [62] H. Nastase, D. Vaman, and P. van Nieuwenhuizen, Consistent nonlinear K K reduction of 11-d supergravity on AdS(7) x S(4) and selfduality in odd dimensions, Phys. Lett. B469 (1999) 96–102, [hep-th/9905075].
  * [63] H. Nastase, D. Vaman, and P. van Nieuwenhuizen, Consistency of the AdS(7) x S(4) reduction and the origin of selfduality in odd dimensions, Nucl. Phys. B581 (2000) 179–239, [hep-th/9911238].
  * [64] J. P. Gauntlett, D. Martelli, J. Sparks, and D. Waldram, Supersymmetric AdS(5) solutions of M theory, Class. Quant. Grav. 21 (2004) 4335–4366, [hep-th/0402153].
  * [65] P. Bomans and C. Couzens, Equivariant localisation in the LLM background, To appear (2023).
  * [66] P. Bomans and C. Couzens, On the class $\mathcal{S}$ origin of spindle solutions, To appear (2023).
  * [67] M. Gutperle and M. Vicino, Holographic Surface Defects in $D=5$, $N=4$ Gauged Supergravity, Phys. Rev. D 101 (2020), no. 6 066016, [arXiv:1911.02185].
  * [68] M. Gutperle and C. F. Uhlemann, Surface defects in holographic 5d SCFTs, JHEP 04 (2021) 134, [arXiv:2012.14547].
  * [69] S. Gukov and E. Witten, Gauge Theory, Ramification, And The Geometric Langlands Program, hep-th/0612073.
  * [70] P. Benetti Genolini, J. P. Gauntlett, and J. Sparks, Equivariant localization in supergravity, arXiv:2306.03868.
  * [71] D. Martelli and A. Zaffaroni, Equivariant localization and holography, arXiv:2306.03891.
  * [72] I. Bah and E. Nardoni, Structure of Anomalies of 4d SCFTs from M5-branes, and Anomaly Inflow, arXiv:1803.00136.
  * [73] I. Bah, F. Bonetti, R. Minasian, and E. Nardoni, Class $\mathcal{S}$ Anomalies from M-theory Inflow, Phys. Rev. D99 (2019), no. 8 086020, [arXiv:1812.04016].
  * [74] I. Bah, F. Bonetti, R. Minasian, and E. Nardoni, Anomaly Inflow for M5-branes on Punctured Riemann Surfaces, arXiv:1904.07250.
  * [75] I. Bah, F. Bonetti, R. Minasian, and E. Nardoni, Anomalies of QFTs from M-theory and Holography, arXiv:1910.04166.
  * [76] J. T. Liu and R. Minasian, Black holes and membranes in AdS(7), Phys. Lett. B457 (1999) 39–46, [hep-th/9903269].
  * [77] J. P. Gauntlett, E. O Colgain, and O. Varela, Properties of some conformal field theories with M-theory duals, JHEP 02 (2007) 049, [hep-th/0611219].


# Derivative bounded functional calculus of power bounded operators on Banach
spaces

LORIS ARNOLD  LABORATOIRE DE MATHÉMATIQUES DE BESANÇON, UMR 6623, CNRS
UNIVERSITÉ DE FRANCHE-COMTÉ
25030 BESANÇON CEDEX
FRANCE<EMAIL_ADDRESS>
###### Abstract.

In this article we study bounded operators $T$ on Banach space $X$ which
satisfy the discrete Gomilko Shi-Feng condition

$\int_{0}^{2\pi}|\langle
R(re^{it},T)^{2}x,x^{*}\rangle|dt\leq\frac{C}{(r^{2}-1)}\left\|x\right\|\left\|x^{*}\right\|,\quad
r>1,x\in X,x^{*}\in X^{*}.$

We show that it is equivalent to a certain derivative bounded functional
calculus and also to a bounded functional calculus relative to Besov space.
Also on Hilbert space discrete Gomilko Shi-Feng condition is equivalent to
power-boundedness. Finally we discuss the last equivalence on general Banach
space involving the concept of $\gamma$-boundedness.

###### Key words and phrases:

$\gamma$-boundedness, power bounded operators, functional calculus, Besov
spaces

###### 2010 Mathematics Subject Classification:

Primary REQUIRED; Secondary OPTIONAL

###### 2010 Mathematics Subject Classification:

47A60, 46B28, 42B35

This work is supported by the French “Investissements d’Avenir" program,
project ISITE-BFC (contract ANR-15-IDEX-03).

##  1\. Introduction

Let $X$ be a Banach space, a bounded operator $T:X\rightarrow X$ is called
power-bounded when
$\underset{n\in\mathbb{N}}{\sup}\left\|T^{n}\right\|<\infty$. In this case the
spectrum of $T$ is contained in the closed unit ball $\overline{\mathbb{D}}$.
We say that $T$ is polynomially bounded when it satisfies an estimate

(1.1) $\left\|P(T)\right\|\leq C\sup\\{|P(z)|:z\in\mathbb{D}\\},\quad P\text{
polynomial. }$

When $X=H$ is an Hilbert space and $T$ is a contraction, that is
$\left\|T\right\|\leq 1$ (which is a power-bounded operator) it is well-known
that $T$ satisfies (1.1) with $C=1$. This is the so-called von Neumann’s
inequality. However even on Hilbert space we cannot expect that any power-
bounded operator satisfies (1.1) (see [12]). A natural question is whether one
can obtain similar estimates, for power-bounded operators, replacing the norm
uniform in the right-hand side of (1.1) by an another function norm. An answer
is given by Peller in [13] for power-bounded operators on Hilbert space. If
$T$ is such operator, then it satisfies

(1.2) $\left\|P(T)\right\|\leq
C\left\|P\right\|_{\mathcal{B}(\mathbb{D})},\quad P\text{ polynomial. }$

where $\left\|\cdot\right\|_{\mathcal{B}(\mathbb{D})}$ is an appropriate Besov
norm (see section 3).

In the first part of this paper we deal with operators which satisfy the
discrete Gomilko Shi-Feng condition: an operator $T:X\rightarrow X$ satisfies
this condition if the spectrum of $T$ is included in $\overline{\mathbb{D}}$
and

(1.3) $\int_{0}^{2\pi}|\langle
R(re^{it},T)^{2}x,x^{*}\rangle|dt\leq\frac{C}{(r^{2}-1)}\left\|x\right\|\left\|x^{*}\right\|,\quad
r>1,x\in X,x^{*}\in X^{*}.$

This condition was first introduced in [7] and [15].The continuous case had
been introduced before in [6] and studied extensively in [4], [1] and [3]. We
will show that this condition implies power boundedness and that the converse
is true when $X=H$ is an Hilbert space. We will be able to show (see section
3) that a bounded operator $T$ on a general Banach space $X$ satisfies an
estimate (1.2) if and only if $T$ satisfies the discrete Gomilko Shi-Feng
condition (1.3). Furthermore one of the main results of this paper is to show
that the discrete Gomilko Shi-Feng condition is equivalent to the boundedness
of the set

(1.4) $\Big{\\{}(r-1)\phi^{\prime}(T):r>1\text{ and }\phi\in
H^{\infty}(r\mathbb{D})\text{ with }\sup\\{|\phi(z)|:z\in r\mathbb{D}\\}\leq
1\Big{\\}}.$

This is weaker than (1.1) in the sense that if an operator $T:X\rightarrow X$
satisfies (1.1) then (1.4) is bounded. The converse is false even on Hilbert
space.

The boundedness of $\eqref{primefuncalc}$ is a way to characterize power
bounded operators on Hilbert space or operators with discrete Gomilko Shi-Feng
condition on general Banach space. However (1.4) is not a usual bounded
functional calculus because we have an estimate of
$\left\|\phi^{\prime}(T)\right\|$ and not $\left\|\phi(T)\right\|$.

The second part of this article (section 4) is devoted to ‘$\gamma$-versions’
of the previous results. By a ‘$\gamma$-version’, we mean replacing operator
norm boundedness by the so-called stronger notion of $\gamma$-boundedness (see
[8] and references therein). This section starts with some basics about
$\gamma$-boundedness, $\gamma$-operators and $\gamma^{\prime}$-operators (see
[9]). We will introduce a $\gamma$-analogue of the discrete Gomilko Shi-Feng
condition for a bounded operator $T:X\rightarrow X$ with spectrum included in
$\overline{\mathbb{D}}$ as follows : there exists $C>0$ such that for any
$N\in\mathbb{N}$, for any $r_{1},\ldots,r_{N}>1$, and for any
$x_{1},\ldots,x_{N}\in X$ and $x_{1}^{*},\ldots,x_{N}^{*}\in X^{*}$, we have

(1.5)
$\displaystyle\sum_{k=1}^{N}\int_{\mathbb{R}}|\langle(r_{k}+1)(r_{k}-1)R(r_{k}e^{it},T)^{2}x_{k},x_{k}^{*}\rangle|dt$
$\displaystyle\leq
C\left\|(x_{k})_{k\in\mathbb{N}_{N}}\right\|_{\gamma(\mathbb{N}_{N};X)}\left\|(x^{*}_{k})_{k\in\mathbb{N}_{N}}\right\|_{\gamma^{\prime}(\mathbb{N}_{N};X^{*})}.$

We will show that $T$ satisfies this condition if and only if the set in (1.4)
is $\gamma$-bounded. Moreover, we will show that $T$ is power
$\gamma$-bounded, that is the set $\\{T^{n}:n\in\mathbb{N}\\}$ is
$\gamma$-bounded, if and only if $T$ satisfies (1.5). It is important to
notice that these results are stated without any assumption on $X$. We will
conclude this paper with a $\gamma$-version of [13, Corollary 3.7].

Finally we give a few notation to be used along this paper. We write
$\mathbb{N}=\\{1,2,\ldots\\}$ for the natural numbers and for $N\in\mathbb{N}$
we write $\mathbb{N}_{N}=\\{1,2,\ldots,N\\}$ the first N natural numbers. For
any Banach spaces $X$ and $Y$, we denote by $B(X,Y)$ the algebra of all
bounded operators from $X$ into $Y$ equipped with the operator norm, and we
set $B(X):=B(X,X)$. For $T\in B(X)$ we denote by $\sigma(T)$ the spectrum of
$T$. For $\lambda\in\mathbb{C}\backslash\sigma(T)$, we put
$R(\lambda,T)=(\lambda I_{X}-T)^{-1}$ the resolvent operator. We let
$\mathbb{D}$ and $\overline{\mathbb{D}}$ respectively the open and closed unit
disk. Also the open disk of radius $r$ centered at $0$ and will be denote by
$r\mathbb{D}$ while the closed disk of radius $r$ centered at $0$ and will be
denoted by $r\overline{\mathbb{D}}$. Also. For any $r\in\mathbb{R}$ we denote
by $H^{\infty}(r\mathbb{D})$ the space of all bounded analytic functions
$\phi:r\mathbb{D}\rightarrow\mathbb{C}$. This is a Banach space for the norm

$\left\|\phi\right\|_{H^{\infty}(r\mathbb{D})}:=\sup\\{|\phi(z)|:z\in
r\mathbb{D}\\}.$

##  2\. Discrete Gomilko Shi-Feng condition and derivative functional
calculus.

### 2.1. Discrete Gomilko Shi-Feng condition

Let $T\in B(X)$ with $\sigma(T)\subset\overline{\mathbb{D}}$. For $r>1$,
$\phi\in H^{\infty}(r\mathbb{D})$ and $m\in\mathbb{N}$, we denote by
$\phi^{(m)}(T)$ the operator obtained with Riesz-Dunford calculus. Since
$\phi\in H^{\infty}(r\mathbb{D})$ then for each $1<\rho<r$, $\phi^{(m)}\in
H^{\infty}(\rho\mathbb{D})$ and therefore the Riesz-Dunford calculus can be
applied to the function $\phi^{(m)}$ and the operator $T$. We obtain

(2.1) $\phi^{(m)}(T)=\frac{1}{2\pi}\int_{0}^{2\pi}\rho e^{it}\phi^{(m)}(\rho
e^{it})R(\rho e^{it},T)dt\in B(X),\quad 1<\rho<r.$

######  Lemma 2.1.

Let $T\in B(X)$ with $\sigma(T)\subset\overline{\mathbb{D}}$ and let
$m\in\mathbb{N}$. Then for each $r>1$ and $\phi\in H^{\infty}(r\mathbb{D})$,
one has

(2.2) $\phi^{(m)}(T)=\frac{m!}{2\pi}\int_{0}^{2\pi}\rho e^{it}\phi(\rho
e^{it})R(\rho e^{it},T)^{m+1}dt,\quad 1<\rho<r.$

###### Proof.

Let $r>1$, $\phi\in H^{\infty}(r\mathbb{D})$ and $1<\rho<r$. For each
$n\in\mathbb{N}$,

$\frac{d}{dt}R(\rho e^{it},T)^{n}=-in\rho e^{it}R(\rho e^{it},T)^{n+1}.$

Using (2.1) it follows by $m$ integrations by parts,

$\displaystyle\phi^{(m)}(T)$ $\displaystyle=\frac{1}{2\pi}\int_{0}^{2\pi}\rho
e^{it}\phi^{(m-1)}(\rho e^{it})R(\rho e^{it},T)^{2}dt$
$\displaystyle=\ldots=\frac{m!}{2\pi}\int_{0}^{2\pi}\rho e^{it}\phi(\rho
e^{it})R(\rho e^{it},T)^{m+1}dt.$

∎

We now investigate the discrete Gomilko Shi-Feng condition which appears in
[7] in the case $m=1$. The case $m>1$ is inspired by [4, Proposition 6.3.].

######  Definition 2.2.

Let $m\geq 1$ be an integer. We say that $T\in B(X)$ has the property
$(GFS)_{m}$ if $\sigma(T)\subset\overline{\mathbb{D}}$ and there exists $C>0$
such that

(2.3) $\int_{0}^{2\pi}|\langle
R(re^{it},T)^{m+1}x,x^{*}\rangle|dt\leq\frac{C}{(r+1)(r-1)^{m}}\left\|x\right\|\left\|x^{*}\right\|,\quad
r>1,x\in X,x^{*}\in X^{*}.$

######  Proposition 2.3.

Let $T$ with $(GFS)_{m}$. Then $T$ is power bounded.

###### Proof.

Let $\phi(z)=\frac{z^{n+m}}{(n+1)\ldots(n+m)}$, then $\phi\in
H^{\infty}(r\mathbb{D})$ for any $r>1$ with

$\left\|\phi\right\|_{H^{\infty}(r\mathbb{D})}=\frac{r^{n+m}}{(n+1)\ldots(n+m)}$

and furthermore $\phi^{(m)}(z)=z^{n}$. By (2.2), one has for $r>1$,

$T^{n}=\frac{m!}{2\pi}\int_{0}^{2\pi}re^{it}\phi(re^{it})R(re^{it},T)^{m+1}dt.$

It follows, for $x\in X$ and $x^{*}\in X^{*}$,

$\displaystyle|\langle T^{n}x,x^{*}\rangle|$
$\displaystyle\leq\frac{rm!}{2\pi}\left\|\phi\right\|_{H^{\infty}(r\mathbb{D})}\int_{0}^{2\pi}|\langle
R(re^{it},T)^{m+1}x,x^{*}\rangle|dt$ (2.4)
$\displaystyle\leq\frac{Cr^{n+m+1}m!}{2\pi(n+1)\ldots(n+m)(r+1)(r-1)^{m}}\left\|x\right\|\left\|x^{*}\right\|.$

Therefore,

$\left\|T^{n}\right\|\leq\frac{m!Cr^{n+m+1}}{2\pi(n+1)\ldots(n+m)(r-1)^{m}}.$

Letting $r=1+\frac{1}{n}$ one obtains

$\frac{m!Cr^{n+m+1}}{2\pi(n+1)\ldots(n+m)(r-1)^{m}}=\frac{m!Cn^{m}(1+\frac{1}{n})^{n}(1+\frac{1}{n})^{m+1}}{2\pi(n+1)\ldots(n+m)}.$

But for each $n\in\mathbb{N}$ one has

$\frac{n^{m}}{(n+1)\ldots(n+m)}\leq
1\quad\text{and}\quad\bigg{(}1+\frac{1}{n}\bigg{)}^{m+1}\leq 2^{m+1}$

and it is well-known that

$\sup_{n\in\mathbb{N}}\Big{(}1+\frac{1}{n}\Big{)}^{n}=e.$

Finally, for each $n\in\mathbb{N}$,

$\left\|T^{n}\right\|\leq\frac{2^{m}m!Ce}{\pi}.$

∎

######  Remark 2.4.

We recall that $T\in B(X)$ is a Ritt operator if
$\sigma(T)\subset\overline{\mathbb{D}}$ and there exists $C\geq 0$ such that

$\forall\lambda\in\mathbb{C}\textbackslash{\overline{\mathbb{D}}},\quad\left\|R(\lambda,T)\right\|\leq\displaystyle\frac{C}{|\lambda-1|}.$

Ritt operators play a prominent role in the theory of functional calculus.
Indeed they have a specific $H^{\infty}$ functional calculus (see [11] and
references therein).

It is easy to check that a Ritt operator has $(GFS)_{1}$. Indeed let $T$ a
Ritt operator and $r>1$, let $x\in X$ and $x^{*}\in X^{*}$, one has

$\displaystyle\int_{0}^{2\pi}|\langle R(re^{it},T)^{2}x,x^{*}\rangle|dt$
$\displaystyle\leq\int_{0}^{2\pi}\frac{C}{|re^{it}-1|^{2}}\left\|x\right\|\left\|x^{*}\right\|dt$
$\displaystyle=\int_{0}^{2\pi}\frac{C}{r^{2}+1-2rcos(t)}\left\|x\right\|\left\|x^{*}\right\|dt.$

Now by the residue method (see for example [5, p.99])

(2.5) $\int_{0}^{2\pi}\frac{1}{r^{2}+1-2rcos(t)}dt=\frac{2\pi}{(r+1)(r-1)},$

therefore one has

$\int_{0}^{2\pi}|\langle R(re^{it},T)^{2}x,x^{*}\rangle|dt\leq\frac{2\pi
C}{(r+1)(r-1)}\left\|x\right\|\left\|x^{*}\right\|.$

We will need the following stability property. We skip the easy proof.

######  Proposition 2.5.

Let $S\in B(X)$ which has $(GFS)_{m}$. If $T$ is similar to $S$, that is there
exists $U\in B(X)$ invertible such that $T=USU^{-1}$, then $T$ has
$(GFS)_{m}$.

We will see that $T$ has $(GFS)_{m}$ if and only if $T$ has $(GFS)_{1}$. Let
us begin by one implication. For the reverse implication we will use a
derivative functional calculus that we introduce in the next sub-section.

######  Proposition 2.6.

Let $m\geq 1$ an integer and $T\in B(X)$. If $T$ has $(GFS)_{m}$ then $T$ has
$(GFS)_{k}$ for each $1\leq k\leq m$.

###### Proof.

Let $r>1$ and assume (2.3). We show by downward induction that for $1\leq
k\leq m$,

(2.6) $\int_{0}^{2\pi}|\langle
R(re^{it},T)^{k+1}x,x^{*}\rangle|dt\leq\frac{Cm}{k(r+1)(r-1)^{k}}\left\|x\right\|\left\|x^{*}\right\|,\quad
x\in X,x^{*}\in X^{*}.$

Let $x\in X$, $x^{*}\in X^{*}$. Suppose that (2.6) is satisfied for $k$ with
$1<k+1\leq m$. Using the fact that
$\frac{d}{dr}R(re^{it},T)^{n}=-ne^{it}R(re^{it},T)^{n+1}$ for each
$n\in\mathbb{N}$ and Fubini’s Theorem, one has

$\displaystyle\int_{0}^{2\pi}\langle R(re^{it},T)^{k}x,x^{*}\rangle dt$
$\displaystyle=\int_{0}^{2\pi}\int_{r}^{+\infty}\langle
ke^{it}R(ue^{it},T)^{k+1}x,x^{*}\rangle dudt$ (2.7)
$\displaystyle=k\int_{r}^{+\infty}\int_{0}^{2\pi}e^{it}\langle
R(ue^{it},T)^{k+1}x,x^{*}\rangle dtdu.$

It follows using (2.6),

$\displaystyle\int_{0}^{2\pi}|\langle R(re^{it},T)^{k}x,x^{*}\rangle|dt$
$\displaystyle\leq k\int_{r}^{+\infty}\int_{0}^{2\pi}|\langle
R(ue^{it},T)^{k+1}x,x^{*}\rangle|dtdu$ $\displaystyle\leq
k\int_{r}^{+\infty}\frac{Cm}{k(u+1)(u-1)^{k}}\left\|x\right\|\left\|x^{*}\right\|du$
$\displaystyle\leq\int_{r}^{+\infty}\frac{Cm}{(r+1)(u-1)^{k}}\left\|x\right\|\left\|x^{*}\right\|du$
$\displaystyle=\frac{Cm}{(k-1)(r+1)(r-1)^{k-1}}\left\|x\right\|\left\|x^{*}\right\|.$

∎

### 2.2. ${m}$-derivative bounded functional calculus

We begin by giving a basic result which is a straighforward consequence of
Cauchy inequalities.

######  Lemma 2.7.

Let $r>0$ and $\phi\in H^{\infty}(r\mathbb{D})$. Then for each $\rho<r$,
$\phi^{(m)}\in H^{\infty}(\rho\mathbb{D})$ with

(2.8)
$\left\|\phi^{(m)}\right\|_{H^{\infty}(\rho\mathbb{D})}\leq\frac{m!}{(r-\rho)^{m}}\left\|\phi\right\|_{H^{\infty}(r\mathbb{D})}.$

This Lemma justifies the following definition.

######  Definition 2.8.

Let $T\in B(X)$ with $\sigma(T)\subset\overline{\mathbb{D}}$ and
$m\in\mathbb{N}$. Then $T$ is said to have  $m$-derivative bounded functional
calculus if there exists $C>0$ such that

(2.9)
$\left\|\phi^{(m)}(T)\right\|\leq\frac{C}{(r-1)^{m}}\left\|\phi\right\|_{H^{\infty}(r\mathbb{D})}\;\text{
for each $\phi\in H^{\infty}(r\mathbb{D})$ \text{ and } $r>1$}.$

######  Remark 2.9.

It is easy to check that (2.9) is equivalent to:

$\left\|\phi^{(m)}\big{(}rT\big{)}\right\|\leq\frac{C}{(1-r)^{m}}\left\|\phi\right\|_{H^{\infty}(\mathbb{D})}\;\text{
for each $\phi\in H^{\infty}(\mathbb{D})$ \text{ and } $0<r<1$}.$

######  Theorem 2.10.

Let $T\in B(X)$ with $\sigma(T)\subset\overline{\mathbb{D}}$. The following
assertions are equivalent for $m\in\mathbb{N}$,

  1. (i)

T has $(GFS)_{m}$;

  2. (ii)

T has ${1}$-derivative bounded functional calculus;

  3. (iii)

T has ${m}$-derivative bounded functional calculus.

###### Proof.

$(i)\Rightarrow(ii)$: According to Proposition 2.6 $T$ has $(GFS)_{1}$. Let
$r>\rho>1$ and $\phi\in H^{\infty}(r\mathbb{D})$. Then using (2.2),

$\displaystyle\left\|\phi^{\prime}(T)\right\|$
$\displaystyle=\frac{1}{2\pi}\underset{\left\|x\right\|=1}{\sup}\underset{\left\|x^{*}\right\|=1}{\sup}\Big{|}\int_{0}^{2\pi}\langle\rho
e^{it}\phi(\rho e^{it})R(\rho e^{it},T)^{2}x,x^{*}\rangle dt\Big{|}$
$\displaystyle\leq\frac{\rho\left\|\phi\right\|_{H^{\infty}(\rho\mathbb{D})}}{2\pi}\underset{\left\|x\right\|=1}{\sup}\underset{\left\|x^{*}\right\|=1}{\sup}\int_{0}^{2\pi}|\langle
R(\rho e^{it},T)^{2}x,x^{*}\rangle|dt$
$\displaystyle\leq\frac{C\rho}{2\pi(\rho-1)(\rho+1)}\left\|\phi\right\|_{H^{\infty}(\rho\mathbb{D})}\leq\frac{C}{2\pi(\rho-1)}\left\|\phi\right\|_{H^{\infty}(r\mathbb{D})}.$

The second to last inequality comes from (2.3) with $m=1$. Letting
$\rho\rightarrow r$ yields

$\left\|\phi^{\prime}(T)\right\|\leq\frac{C}{2\pi(r-1)}\left\|\phi\right\|_{H^{\infty}(r\mathbb{D})}.$

$(ii)\Rightarrow(iii)$: Let $1<\rho<r$ and $\phi\in H^{\infty}(r\mathbb{D})$.
According to Lemma 2.7 $\phi^{(m-1)}\in H^{\infty}(\rho\mathbb{D})$ so
applying the ${1}$-derivative bounded functional calculus to $\phi^{(m-1)}$
and again Lemma 2.7 one has

$\displaystyle\left\|\phi^{(m)}(T)\right\|$
$\displaystyle=\left\|(\phi^{(m-1)})^{\prime}(T)\right\|\leq\frac{C}{\rho-1}\left\|\phi^{(m-1)}\right\|_{H^{\infty}(\rho\mathbb{D})}$
$\displaystyle\leq\frac{(m-1)!C}{(\rho-1)(r-\rho)^{m-1}}\left\|\phi\right\|_{H^{\infty}(r\mathbb{D})}.$

Taking $\rho=\frac{r+1}{2}$ yields

$\left\|\phi^{(m)}(T)\right\|\leq\frac{2^{m}(m-1)!C}{(r-1)^{m}}\left\|\phi\right\|_{H^{\infty}(r\mathbb{D})}.$

$(iii)\Rightarrow(i)$: Let $r>1$, $1<\rho<r$, $x\in X$ and $x^{*}\in X^{*}$.
There exists a measurable function
$\varepsilon:[0,2\pi]\rightarrow\\{z\in\mathbb{C},\;|z|=1\\}$ such that for
each $t\in[0,2\pi)$

$|\langle R(re^{it},T)^{m+2}x,x^{*}\rangle|=\varepsilon(t)\langle
R(re^{it},T)^{m+2}x,x^{*}\rangle.$

Let define
$\phi(z):=\displaystyle\frac{1}{(m+1)!}\int_{0}^{2\pi}\frac{\varepsilon(t)}{(re^{it}-z)^{2}}dt$
for $z\in\rho\mathbb{D}$. Then $\phi\in H^{\infty}(\rho\mathbb{D})$, with

(2.10)
$\left\|\phi\right\|_{H^{\infty}(\rho\mathbb{D})}\leq\frac{2\pi}{(m+1)!(r+\rho)(r-\rho)}.$

Indeed for each $\theta\in[0,2\pi)$, using (2.5),

$\displaystyle\int_{0}^{2\pi}\frac{1}{|re^{it}-\rho e^{i\theta}|^{2}}dt$
$\displaystyle=\frac{1}{\rho^{2}}\int_{0}^{2\pi}\frac{1}{|\frac{r}{\rho}e^{it}-1|^{2}}dt$
$\displaystyle=\frac{2\pi}{\rho^{2}(\frac{r}{\rho}+1)(\frac{r}{\rho}-1)}$
$\displaystyle=\frac{2\pi}{(r+\rho)(r-\rho)}.$

Moreover

$\phi^{(m)}(z)=\int_{0}^{2\pi}\frac{\varepsilon(t)}{(re^{it}-z)^{m+2}}dt,\;z\in\rho\mathbb{D}.$

By Fubini’s Theorem one has

$\phi^{(m)}(T)=\int_{0}^{2\pi}\varepsilon(t)R(re^{it},T)^{m+2}dt,$

and finally using (2.9) and (2.10),

$\displaystyle\int_{0}^{2\pi}|\langle R(re^{it},T)^{m+2}x,x^{*}\rangle|$
$\displaystyle=\int_{0}^{2\pi}\varepsilon(t)\langle
R(re^{it},T)^{m+2}x,x^{*}\rangle$
$\displaystyle=|\langle\phi^{(m)}(T)x,x^{*}\rangle|\leq\left\|\phi^{(m)}(T)\right\|\left\|x\right\|\left\|x^{*}\right\|$
$\displaystyle\leq\frac{C}{(\rho-1)^{m}}\left\|\phi\right\|_{H^{\infty}(\rho\mathbb{D})}\left\|x\right\|\left\|x^{*}\right\|$
$\displaystyle\leq\frac{2\pi
C}{(m+1)!(\rho-1)^{m}(r-\rho)(r+\rho)}\left\|x\right\|\left\|x^{*}\right\|$
$\displaystyle\leq\frac{2\pi
C}{(m+1)!(r+1)(\rho-1)^{m}(r-\rho)}\left\|x\right\|\left\|x^{*}\right\|.$

Taking $\rho=\frac{r+1}{2}$ one obtains

$\int_{0}^{2\pi}|\langle R(re^{it},T)^{m+2}x,x^{*}\rangle|\leq\frac{2^{m+2}\pi
C}{(m+1)!(r+1)(r-1)^{m+1}}\left\|x\right\|\left\|x^{*}\right\|.$

Thus $T$ has $(GFS)_{m+1}$ and hence by Proposition 2.6, $T$ has $(GFS)_{m}$.
∎

######  Corollary 2.11.

The condition $(GFS)_{m}$ does not depend of $m\in\mathbb{N}$. In other words,
if $T\in B(X)$ satisfies condition $(GFS)_{m}$ for one $m\in\mathbb{N}$ then
it has $(GFS)_{k}$ for all $k\in\mathbb{N}$.

######  Definition 2.12.

Let $T\in B(X)$. If $T$ satisfies one of the three conditions of Theorem 2.10
then we will say that $T$ is a Gomilko Shi-Feng operator or GFS operator.

When $X$ is an Hilbert space one obtains the following characterisation.

######  Corollary 2.13.

Let $H$ be a Hilbert space and let $T\in B(H)$ with
$\sigma(T)\subset\overline{\mathbb{D}}$. The following assertions are
equivalent,

  1. (i)

$T$ is power-bounded;

  2. (ii)

there is a constant $C>0$ such that for all $r>1$,

$\displaystyle(r^{2}-1)\int_{0}^{2\pi}\left\|R(re^{it},T)x\right\|^{2}dt\leq
C\left\|x\right\|^{2}\quad(x\in H)$
$\displaystyle(r^{2}-1)\int_{0}^{2\pi}\left\|R(re^{it},T)^{*}y\right\|^{2}dt\leq
C\left\|y\right\|^{2}\quad(y\in H);$

  3. (iii)

T is a GFS operator.

###### Proof.

The implication $(iii)\implies(i)$ is given by Proposition 2.3.
$(i)\implies(ii)$: Let $r>1$, one has

$\int_{0}^{2\pi}\left\|R(re^{it},T)x\right\|^{2}dt=\int_{0}^{2\pi}\left\|\sum_{n=0}^{\infty}\frac{T^{n}x}{(re^{it})^{n+1}}\right\|^{2}dt.$

The Fourier-Plancherel Theorem gives

$\int_{0}^{2\pi}\left\|\sum_{n=0}^{\infty}\frac{T^{n}x}{(re^{it})^{n+1}}\right\|^{2}dt=2\pi\sum_{n=0}^{\infty}\bigg{(}\frac{\left\|T^{n}x\right\|}{r^{n+1}}\bigg{)}^{2}.$

Then taking $M=\underset{n}{\sup}\\{\left\|T^{n}\right\|\\}$, we have

$2\pi\sum_{n=0}^{\infty}\bigg{(}\frac{\left\|T^{n}\right\|}{r^{n+1}}\bigg{)}^{2}\leq
2\pi M^{2}\sum_{n=0}^{\infty}\frac{1}{r^{2(n+1)}}=\frac{2\pi M^{2}}{r^{2}-1},$

whence the first inequality in $(ii)$. Since $T^{*}$ is also power bounded,
the second inequality in $(ii)$ holds as well.
$(ii)\implies(iii)$: by Cauchy-Schwarz inequality one has

$\displaystyle\int_{0}^{2\pi}|\langle R(re^{it},T)^{2}x,x^{*}\rangle|$
$\displaystyle=\int_{0}^{2\pi}|\langle
R(re^{it},T)x,R(re^{it},T)^{*}x^{*}\rangle|$
$\displaystyle\leq\bigg{(}\int_{0}^{2\pi}\left\|R(re^{it},T)x\right\|^{2}dt\bigg{)}^{\frac{1}{2}}\bigg{(}\int_{0}^{2\pi}\left\|R(re^{it},T)^{*}x^{*}\right\|^{2}dt\bigg{)}^{\frac{1}{2}}$
$\displaystyle\leq\frac{C}{r^{2}-1}\left\|x\right\|\left\|x^{*}\right\|.$

If follows that $T$ has $(GFS)_{1}$, and so $T$ is a GFS operator. ∎

######  Remarks 2.14.

  1. (1)

We say that $T\in B(X)$ is polynomially bounded when there exists $C>0$ such
that for each polynomial $P$ one has

$\left\|P(T)\right\|\leq C\left\|P\right\|_{H^{\infty}(\mathbb{D})}.$

It is easy to show that if $T$ is polynomially bounded, then $T$ is a GFS
operator. Indeed let $P$ be a polynomial and $r>1$. Then according to Lemma
2.7 and the assumption that $T$ is polynomially bounded:

$\left\|P^{(m)}(T)\right\|\leq
C\left\|P^{(m)}\right\|_{H^{\infty}(\mathbb{D})}\leq\frac{Cm!}{(r-1)^{m}}\left\|P^{(m)}\right\|_{H^{\infty}(r\mathbb{D})}.$

The converse implication is false. In fact, on any infinite-dimensional
Hilbert space there exists a power bounded operator which is not polynomially
bounded (see [12]). But, according to Corollary 2.13, in a Hilbert space an
operator is power bounded if and only if it is a GFS operator.

  2. (2)

When $X$ is not an Hilbert space, the implication $(i)\implies(iii)$ in
Corollary (2.13) is false. Indeed according to [7, Theorem 2.2], if $X$ is a
reflexive Banach space and $T\in B(X)$ with $\sigma(T)\subset\mathbb{T}$ then
$T$ is a scalar type spectral operator if and only if $T$ and $T^{-1}$ have
$(GFS)_{1}$. Let $U$ denote the shift operator defined on $l_{p}(\mathbb{Z})$
$(1<p<\infty)$ by

(2.11) $U((x_{n})_{n\in\mathbb{Z}})=(x_{n+1})_{n\in\mathbb{Z}}.$

It is well-known that for $1<p\neq 2<\infty$, $U$ is not a scalar type
spectral operator. It follows, that either $U$ or $U^{-1}$ does not have
$(GFS)_{1}$. Further $U$ and $U^{-1}$ are similar. Hence by Proposition 2.5,
$U$ does not have $(GFS)_{1}$. Thus $U$ is a power bounded operator which is
not a GFS operator.

##  3\. Polynomial Besov calculus on general Banach spaces

We denote by $\mathcal{B}(\mathbb{D})$ the Banach algebra of all holormorphic
functions $f$ on $\mathbb{D}$ such that

(3.1)
$\int_{0}^{1}\underset{t\in[0,2\pi)}{\sup}|f^{\prime}(ue^{it})|du<\infty,$

endowed with the norm

$\left\|f\right\|_{\mathcal{B}}=\int_{0}^{1}\underset{t\in[0,2\pi)}{\sup}|f^{\prime}(ue^{it})|du+\left\|f\right\|_{H^{\infty}(\mathbb{D})}.$

We remark that by a change of variables $u=\frac{1}{r}$ one has

(3.2)
$\int_{0}^{1}\underset{t\in[0,2\pi)}{\sup}|f^{\prime}(ue^{it})|du=\int_{1}^{\infty}\frac{1}{r^{2}}\underset{t\in[0,2\pi)}{\sup}\Big{|}f^{\prime}\Big{(}\frac{e^{it}}{r}\Big{)}\Big{|}dr.$

Let $H$ be a Hilbert space and let $T\in B(H)$ be a power-bounded operator.
According to [13, Theorem 3.8.] there exists $C>0$ such that for each
polynomial $P$ one has

$\left\|P(T)\right\|\leq C\left\|P\right\|_{\mathcal{B}}.$

Now since the set of polynomials is dense in $\mathcal{B}(\mathbb{D})$ the
bounded algebra homomorphism $P\rightarrow P(T)$ extends to a bounded algebra
homomorphism on $\mathcal{B}(\mathbb{D})$.

We now consider the issue of obtaining a similar bounded Besov calculus on an
arbitrary Banach space $X$. Ideas are based on [3].

######  Proposition 3.1.

Let $T\in B(X)$ be a $GFS$ operator. Then there exists $C>0$ such that for
each polynomial $P$,

$\left\|P(T)\right\|\leq C\left\|P\right\|_{\mathcal{B}}.$

###### Proof.

Let $n\in\mathbb{N}$ and consider $P_{n}(z)=z^{n}$ Applying (2.2) with $m=1$
and $\phi(z)=\frac{z^{n+1}}{n+1}$, one has the following representation,

$T^{n}=\frac{1}{2\pi(n+1)}\int_{0}^{2\pi}r^{n+2}e^{i(n+2)t}R^{2}(re^{it},T)dt,$

for any $r>1$. We deduce that for any $x\in X,x^{*}\in X^{*}$,

$\displaystyle\langle T^{n}x,x^{*}\rangle$
$\displaystyle=2n(n+1)\int_{1}^{\infty}\frac{r^{2}-1}{r^{2n+3}}dr\langle
T^{n}x,x^{*}\rangle$
$\displaystyle=2n(n+1)\int_{1}^{\infty}\frac{r^{2}-1}{r^{2n+3}}\frac{1}{2\pi(n+1)}\int_{0}^{2\pi}r^{n+2}e^{i(n+2)t}\langle
R^{2}(re^{it},T)x,x^{*}\rangle dtdr$
$\displaystyle=\frac{n}{\pi}\int_{1}^{\infty}\frac{r^{2}-1}{r^{n+1}}\int_{0}^{2\pi}e^{i(n+2)t}\langle
R^{2}(re^{it},T)x,x^{*}\rangle dtdr$
$\displaystyle=\frac{n}{\pi}\int_{1}^{\infty}(r^{2}-1)\int_{0}^{2\pi}\frac{e^{3it}}{r^{2}}\frac{e^{i(n-1)t}}{r^{n-1}}\langle
R^{2}(re^{it},T)x,x^{*}\rangle dtdr$ (3.3)
$\displaystyle=\frac{1}{\pi}\int_{1}^{\infty}(r^{2}-1)\int_{0}^{2\pi}\frac{e^{3it}}{r^{2}}P_{n}^{\prime}\Big{(}\frac{e^{it}}{r}\Big{)}\langle
R^{2}(re^{it},T)x,x^{*}\rangle dtdr,$

where, in the first equality, we have used the following identity

$\int_{1}^{\infty}\frac{r^{2}-1}{r^{2n+3}}dr=\frac{1}{2n(n+1)}.$

Now let $P$ be a polynomial. Using the linearity of derivative and
integration, and previous calculations one obtains

(3.4) $\langle
P(T)x,x^{*}\rangle=\frac{1}{\pi}\int_{1}^{\infty}(r^{2}-1)\int_{0}^{2\pi}\frac{e^{3it}}{r^{2}}P^{\prime}\Big{(}\frac{e^{it}}{r}\Big{)}\langle
R^{2}(re^{it},T)x,x^{*}\rangle dtdr+P(0)\langle x,x^{*}\rangle.$

Since $T$ has $(GFS)_{1}$ there exists $C>0$ such that

$(r^{2}-1)\int_{0}^{2\pi}|\langle R(re^{it},T)^{2}x,x^{*}\rangle|dt\leq
C\left\|x\right\|\left\|x^{*}\right\|,\quad r>1,x\in X,x^{*}\in X^{*}.$

Therefore combining this with (3.2),

(3.5) $|\langle P(T)x,x^{*}\rangle|\leq
C^{\prime}\Big{(}\int_{1}^{\infty}\frac{1}{r^{2}}\big{|}P^{\prime}\Big{(}\frac{e^{it}}{r}\Big{)}\big{|}dr+|P(0)|\Big{)}\left\|x\right\|\left\|x^{*}\right\|\leq
C^{\prime}\left\|P\right\|_{\mathcal{B}}\left\|x\right\|\left\|x^{*}\right\|,$

which proves the result.

∎

The above proof leads to the following result:

######  Theorem 3.2.

Let $T\in B(X)$ a $GFS$ operator. Then the mapping $P\rightarrow P(T)$ on the
set of polynomials can be extended uniquely to a bounded homomorphism from
Besov algebra $\mathcal{B}(\mathbb{D})$ to the algebra of bounded operators
$B(X)$. When $f\in\mathcal{B}(\mathbb{D})$ let $f(T)$ denote the operator
obtained by this extension.

Then for each $f\in\mathcal{B}(\mathbb{D})$ one has the following
representation: for $x\in X$ and $x^{*}\in X^{*}$:

(3.6) $\langle
f(T)x,x^{*}\rangle=\frac{1}{\pi}\int_{1}^{\infty}(r^{2}-1)\int_{0}^{2\pi}\frac{e^{3it}}{r^{2}}f^{\prime}\Big{(}\frac{e^{it}}{r}\Big{)}\langle
R^{2}(re^{it},T)x,x^{*}\rangle dtdr+\langle f(0)x,x^{*}\rangle.$

###### Proof.

The first assertion is straighforward since the set of polynomials is dense in
Besov algebra $\mathcal{B}(\mathbb{D})$. Let $f\in\mathcal{B}(\mathbb{D})$ and
denote by $f_{\mathcal{B}}(T)$ the operator defined for $x\in X$ and $x^{*}\in
X^{*}$ by

$\langle
f_{\mathcal{B}}(T)x,x^{*}\rangle=\frac{1}{\pi}\int_{1}^{\infty}(r^{2}-1)\int_{0}^{2\pi}\frac{e^{3it}}{r^{2}}f^{\prime}\Big{(}\frac{e^{it}}{r}\Big{)}\langle
R^{2}(re^{it},T)x,x^{*}\rangle dtdr+\langle f(0)x,x^{*}\rangle.$

The calculation at the end of the proof of Proposition 3.1 shows that this is
a well-defined element of $B(X,X^{**})$. Now let $(P_{n})_{n\in\mathbb{N}}$ a
sequence of polynomials such that

$\left\|f-P_{n}\right\|_{\mathcal{B}}\underset{n\rightarrow\infty}{\longrightarrow}0.$

By definition of $f(T)$, one has for $x\in X$ and $x^{*}\in X^{*}$

$\langle
P_{n}(T)x,x^{*}\rangle\underset{n\rightarrow\infty}{\longrightarrow}\langle
f(T)x,x^{*}\rangle.$

Furthermore by the same arguments leading to the estimate (3.5) one has for
$x\in X$ and $x^{*}\in X^{*}$,

$|\langle(f_{\mathcal{B}}(T)-P_{n}(T))x,x^{*}\rangle|\leq
C\left\|f-P_{n}\right\|_{\mathcal{B}}\left\|x\right\|\left\|x^{*}\right\|.$

We conclude that $f(T)=f_{\mathcal{B}}(T)$ and therefore we have shown
equality (3.6). ∎

In general when we extend a functional calculus to a larger class, we do not
know what the operators obtained on this larger class look like. The above
Theorem has more value because we do not only extend polynomial calculus to
the Besov algebra but also we are able to give a good representation of
operators obtained by extension.

The next result is a converse to Proposition 3.1.

######  Proposition 3.3.

Let $T\in B(X)$ with $\sigma(T)\subset\overline{\mathbb{D}}$. If there exists
$C>0$ such that for each polynomial $P$

$\left\|P(T)\right\|\leq C\left\|P\right\|_{\mathcal{B}},$

is a GFS operator.

###### Proof.

Let $r>1$ and $\phi\in H^{\infty}(r\mathbb{D})$. Then
$\phi^{\prime}\in\mathcal{B}(\mathbb{D})$, indeed by (2.8)

$\displaystyle\int_{0}^{1}\underset{t\in[0,2\pi)}{\sup}|\phi^{\prime\prime}(se^{it})|ds$
$\displaystyle=\int_{0}^{1}\left\|\phi^{\prime\prime}\right\|_{H^{\infty}(s\mathbb{D})}ds$
$\displaystyle\leq\int_{0}^{1}\frac{2}{(r-s)^{2}}\left\|\phi\right\|_{H^{\infty}(r\mathbb{D})}ds$
$\displaystyle=\frac{2}{r(r-1)}\left\|\phi\right\|_{H^{\infty}(r\mathbb{D})}.$

It follows

$\left\|\phi^{\prime}(T)\right\|\leq
C\left\|\phi^{\prime}\right\|_{\mathcal{B}}\leq\frac{4C}{(r^{2}-1)}\left\|\phi\right\|_{H^{\infty}(r\mathbb{D})}.$

Therefore $T$ has ${1}$-derivative bounded functional calculus, hence $T$ is a
GFS operator. ∎

######  Remark 3.4.

It is known (see [16, Lemma 2.3.7] for details) that there exists $C>0$ such
that for each polynomial of degree $N$ one has

$\left\|P\right\|_{\mathcal{B}}\leq
Clog(N+2)\left\|P\right\|_{H^{\infty}(\mathbb{D})}.$

Therefore if $T$ has $(GFS)_{1}$ then there exists $C>0$ such that for each
polynomial of degree $N$,

$\left\|P(T)\right\|\leq Clog(N+2)\left\|P\right\|_{H^{\infty}(\mathbb{D})}.$

##  4\. Generalizations involving $\gamma$-boundedness

This main goal of this section is to obtain an extension of Corollary 2.13 to
Banach spaces. We have noticed that it is not possible to replace $H$ by a
Banach space $X$ without additional assumptions. We need assumptions on the
operator $T$. We prove an analogue of Corallary 2.13 if
$\\{T^{n}:n\in\mathbb{N}\\}$ is $\gamma$-bounded. We start with some
background on the so-called $\gamma$-spaces introduced by Kalton and Weis in
[9].

Let $X,Y$ be Banach spaces and we let $(\gamma_{n})_{n\geq 1}$ be a sequence
of independent complex valued standard Gaussian variables on some probability
space $\Sigma$. We denote by $G(X)$ the closure of

$\big{\\{}\sum_{k=1}^{n}\gamma_{k}\otimes x_{k}\,:\,x_{k}\in
X,\,n\in\mathbb{N}\big{\\}}$

in $L^{2}(\Sigma,X)$. For $x_{1},\ldots,x_{n}\in X$, we let

$\left\|\sum_{k=1}^{n}\gamma_{k}\otimes
x_{k}\right\|_{G(X)}:=\bigg{(}\underset{\Sigma}{\bigint}{\left\|\sum_{k=1}^{n}\gamma_{k}(\lambda)x_{k}\right\|^{2}d\lambda}\bigg{)}^{\frac{1}{2}}$

denote the induced norm.

######  Definition 4.1.

Let $\mathcal{T}\subset B(X,Y)$ be a set of operators. We say that
$\mathcal{T}$ is $\gamma$-bounded if there exists a constant $C\geq 0$ such
that for all finite sequences $(T_{n})_{n=1}^{N}\subset\mathcal{T}$ and
$(x_{n})_{n=1}^{N}\subset X$, the following inequality holds:

(4.1) $\left\|\sum_{n=1}^{N}\gamma_{n}\otimes T_{n}x_{n}\right\|_{G(Y)}\leq
C\left\|\sum_{n=1}^{N}\gamma_{n}\otimes x_{n}\right\|_{G(X)}.$

The least admissible constant in the above inequality is called the
$\gamma$-bound of $\mathcal{T}$ and we denote this quantity by
$\gamma(\mathcal{T})$. If $\mathcal{T}$ fails to be $\gamma$-bounded, we set
$\gamma(\mathcal{T})=\infty$.

The notion of $\gamma$-boundedness is stronger than uniform boundedness,
indeed using definition of $\gamma$-boundedness with $N=1$ it is easy to see
that $\gamma$-boundedness implies uniform boundedness. We will use also two
important facts which we sum up in the following proposition.

######  Proposition 4.2.

Let $\mathcal{T}\subset B(X,Y)$ be a $\gamma$-bounded set. Then

  1. (1)

the closure $\overline{\mathcal{T}}^{so}$ of $\mathcal{T}$ in the strong
operator topology is $\gamma$-bounded with
$\gamma(\overline{\mathcal{T}}^{so})=\gamma(\mathcal{T})$.

  2. (2)

The absolute convex hull of $\mathcal{T}$, $absconv(\mathcal{T})$ is
$\gamma$-bounded with $\gamma(absconv(\mathcal{T}))=\gamma(\mathcal{T})$.

We now turn to the definition of $\gamma$-spaces.

Let $H$ be a Hilbert space. A linear operator $T:H\rightarrow X$ is called
$\gamma$-summing if

$\left\|T\right\|_{\gamma}:=\sup\left\|\sum_{n=1}^{N}\gamma_{n}\otimes
Th_{n}\right\|_{G(X)}<\infty,$

where the supremum is taken over all finite orthonormal systems
$\\{h_{1},...,h_{n}\\}$ in $H$. We let $\gamma_{\infty}(H;X)$ denote the space
of all $\gamma$-summing operators and we endow it with the norm
$\left\|\cdot\right\|_{\gamma}$. Then $\gamma_{\infty}(H;X)$ is a Banach
space. Clearly any finite rank (bounded) operator is a $\gamma$-summing
operator. We let $\gamma(H;X)$ be the closure in $\gamma_{\infty}(H;X)$ of the
space of finite rank operators from $H$ into $X$.

We let $\gamma^{\prime}_{\infty}(H;X)$ be the space of bounded operators
$T:H\mapsto X$ such that

$\left\|T\right\|_{\gamma^{\prime}(H;X)}=\sup\\{trace(T^{*}\circ
S)\;|\;S:H\mapsto X^{*},\,\left\|S\right\|_{\gamma(H;X)\leq
1},\,dim\,S(H)<\infty\\}$

and we denote by $\gamma^{\prime}(H;X)$ the closure of the finite dimensional
operators in $\gamma^{\prime}_{\infty}(H;X)$. See [9, section 5] for details
about spaces $\gamma^{\prime}(H;X)$ and $\gamma_{\infty}^{\prime}(H;X)$. When
$X$ is $K$-convex (see [8, Section 7.4.] for details about $K$-convexity) then
one has

$\gamma^{\prime}(H;X^{*})=\gamma(H;X^{*}).$

Let $(S,\mu)$ be a measure space. We say that a function $f:S\rightarrow X$ is
weakly $L^{2}$ if for each $x^{*}\in X^{*}$, the function $s\mapsto\langle
f(s),x^{*}\rangle$ is measurable and belongs to $L^{2}(S)$. If $f:S\rightarrow
X$ is measurable and weakly $L^{2}$, one can define an operator
$\mathbb{I}_{f}:L^{2}(S)\to X$, given by

$\mathbb{I}_{f}(g):=\int_{S}g(s)f(s)d\mu,\quad g\in L^{2}(S),$

where this integral is defined in the Pettis sense. We let $\gamma(S;X)$
(resp. $\gamma^{\prime}(S,X)$) be the space of all measurable and weakly
$L^{2}$ functions $f:S\rightarrow X$ such that $\mathbb{I}_{f}$ belongs to
$\gamma(L^{2}(S);X)$ (resp. $\gamma^{\prime}(L^{2}(S);X)$). We endow it with
$\left\|f\right\|_{\gamma(S;X)}:=\left\|\mathbb{I}_{f}\right\|_{\gamma(L^{2}(S);X)}$
(resp.
$\left\|f\right\|_{\gamma^{\prime}(S;X)}:=\left\|\mathbb{I}_{f}\right\|_{\gamma^{\prime}(L^{2}(S);X)}$).

The next result is an inequality of Hölder type [9, Corollary 5.5.].

######  Theorem 4.3 ($\gamma$-Hölder inequality).

If $f:S\rightarrow X$ and $g:S\rightarrow X^{*}$ belongs to $\gamma(S;X)$ and
$\gamma^{\prime}(S;X^{*})$, respectively, then $\langle f,g\rangle$ belong to
$L^{1}(S)$ and we have

$\left\|\langle
f,g\rangle\right\|_{L^{1}(S)}\leq\left\|f\right\|_{\gamma(S;X)}\left\|g\right\|_{\gamma^{\prime}(S;X^{*})}.$

######  Remark 4.4.

Let $N\in\mathbb{N}$, $x_{1},\ldots,x_{N}\in X$ and
$x_{1}^{*},\ldots,x_{N}^{*}\in X^{*}$. By construction,

(4.2) $\left\|\sum_{k=1}^{N}\gamma_{k}\otimes
x_{k}\right\|_{G(X)}=\left\|(x_{k})_{k\in\mathbb{N}_{N}}\right\|_{\gamma(\mathbb{N}_{N};X)}.$

As a particular case of Theorem 4.3 one has

$\big{|}\sum_{k=1}^{N}\langle
x_{k},x^{*}_{k}\rangle\big{|}\leq\left\|(x_{k})_{k\in\mathbb{N}_{N}}\right\|_{\gamma(\mathbb{N}_{N};X)}\left\|(x^{*}_{k})_{k\in\mathbb{N}_{N}}\right\|_{\gamma^{\prime}(\mathbb{N}_{N};X^{*})}.$

Further this inequality is optimal, namely

(4.3)
$\left\|(x_{k})_{k\in\mathbb{N}_{N}}\right\|_{\gamma(\mathbb{N}_{N};X)}=\sup\Big{\\{}\big{|}\sum_{k=1}^{N}\langle
x_{k},x^{*}_{k}\rangle\big{|}:(x^{*}_{k})_{k\in\mathbb{N}_{N}}\subset
X^{*},\,\left\|(x^{*}_{k})_{k\in\mathbb{N}_{N}}\right\|_{\gamma^{\prime}(\mathbb{N}_{N};X^{*})}\leq
1\Big{\\}}$

and

(4.4)
$\left\|(x^{*}_{k})_{k\in\mathbb{N}_{N}}\right\|_{\gamma^{\prime}(\mathbb{N}_{N};X^{*})}=\sup\Big{\\{}\big{|}\sum_{k=1}^{N}\langle
z_{k},x_{k}^{*}\rangle\big{|}:(x_{k})_{k\in\mathbb{N}_{N}}\subset
X,\,\left\|(x_{k})_{k\in\mathbb{N}_{N}}\right\|_{\gamma(\mathbb{N}_{N};X)}\leq
1\Big{\\}}.$

Furthermore for each sequence $(x^{*}_{k})_{k\in\mathbb{N}}\subset X^{*}$, one
has

(4.5)
$\left\|(x^{*}_{k})_{k\in\mathbb{N}}\right\|_{\gamma^{\prime}(\mathbb{N};X^{*})}=\lim\limits_{N\rightarrow+\infty}\left\|(x^{*}_{k})_{k\in\mathbb{N}_{N}}\right\|_{\gamma^{\prime}(\mathbb{N}_{N};X^{*})}.$

We finally state an extension result ([8, Theorem 9.6.1.] and [9, Remark
5.4.]).

######  Lemma 4.5.

Let $H$ and $K$ be Hibert spaces and $U\in B(H^{*},K^{*})$. Then $U\otimes
I_{X}$ extends uniquely to a bounded linear operator $\widetilde{U}\in
B(\gamma(H;X),\gamma(K;X))$ of the same norm.
Moreover $U\otimes I_{X^{*}}$ extends uniquely to a bounded linear operator
$\widehat{U}\in B(\gamma^{\prime}(H;X^{*}),\gamma^{\prime}(K;X^{*}))$ of the
same norm.

### 4.1. Power $\gamma$-bounded operators

######  Definition 4.6.

Let $T\in B(X)$ be power bounded. We say that $T$ is power $\gamma$-bounded if
the set $\\{T^{n}:n\in\mathbb{N}\\}$ is $\gamma$-bounded. When $T$ is power
$\gamma$-bounded we will denote by $C_{\gamma}$ the $\gamma$-bound
$\gamma(\\{T^{n}:n\in\mathbb{N}\cup\\{0\\}\\})$.

######  Example 4.7.

The shift operator $U$ defined by (2.11) is obviously power bounded but for
$1<p\neq 2<\infty$, $U$ is not power $\gamma$-bounded. Assume that $1<p<2$ and
let $(\delta_{0})_{n\in\mathbb{Z}}$ the sequence defined by $\delta_{0}(0)=1$
and $\delta(k)=0$ for each $k\neq 0$. Then one has

$\left\|\bigg{(}\sum_{i=1}^{n}|U^{(i-1)}\delta_{0}|^{2}\bigg{)}^{\frac{1}{2}}\right\|_{l_{p}}=n^{\frac{1}{p}}$

whereas

$\left\|\bigg{(}\sum_{i=1}^{n}|\delta_{0}|^{2}\bigg{)}^{\frac{1}{2}}\right\|_{l_{p}}=n^{\frac{1}{2}}.$

The proof in the case $2<p<\infty$ is similar.

######  Example 4.8.

A Ritt operator $T$ is called $\gamma$-Ritt if the set

$\\{(\lambda-1)R(\lambda,T):\lambda\in\mathbb{C}\backslash\overline{\mathbb{D}}\\}$

is $\gamma$-bounded. It turns out $T$ is $\gamma$-Ritt if and only if the two
sets

$\\{T^{n}:n\in\mathbb{N}\\}\quad\text{and}\quad\\{n(I-T)T^{n}:n\in\mathbb{N}\\}$

are $\gamma$-bounded. It follows that any $\gamma$-Ritt operator is power
$\gamma$-bounded. In [2] we construct a Ritt operator such that the set
$\\{T^{n}:n\in\mathbb{N}\\}$ is not $\gamma$-bounded (in particular $T$ is not
$\gamma$-Ritt). An interesting problem which is still open is to construct a
Ritt operator which is not $\gamma$-Ritt but which is power $\gamma$-bounded.

######  Example 4.9.

Let $(\Omega,\mu)$ be a measure space,
$m:\Omega\rightarrow\overline{\mathbb{D}}$ an element of $L^{\infty}(\Omega)$
and $1\leq p<\infty$. We define the bounded multiplier $T_{m}\in
B(L^{p}(\Omega))$ by

$(T_{m}f)(s):=m(s)f(s),\quad s\in\Omega.$

For each $n\in\mathbb{N}$ one has

$(T_{m})^{n}=T_{m^{n}}.$

Since for each $n\in\mathbb{N}$, $\left\|m^{n}\right\|_{\infty}\leq 1$, it
follows (see [8, Example 8.1.9]) that $T_{m}$ is power $\gamma$-bounded.

Polynomial boundedness does not imply power $\gamma$-boundedness. Indeed
according to [10, Proposition 6.6] there exist a Banach space $X$ and an
invertible operator $T\in B(X)$ such that $T$ is not power $\gamma$-bounded
but there exists a bounded unital homomorphism
$\omega:C(\mathbb{T})\rightarrow B(X)$ such that $\omega(z\mapsto z)=T$.
Obviously that operator $T$ is polynomially bounded.

######  Theorem 4.10.

Suppose $X$ has property $(\alpha)$. Let $T\in B(X)$ be invertible and let
$m\in\mathbb{N}$. Then $T$ and $T^{-1}$ are GFS operators if and only if
$\\{T^{n},n\in\mathbb{Z}\\}$ is $\gamma$-bounded.

###### Proof.

Since $X$ has property $(\alpha)$ then according to [10, Proposition 6.3.
(2)], the set $\\{T^{n},n\in\mathbb{Z}\\}$ is $\gamma$-bounded if and only if
there is a bounded unital homomorphism $\omega:C(\mathbb{T})\rightarrow B(X)$
such that $\omega(z\mapsto z)=T$. Furthermore, according to the proof of [7,
Theorem 2.2.], there is a bounded unital homomorphism
$\omega:C(\mathbb{T})\rightarrow B(X)$ such that $\omega(z\mapsto z)=T$ if and
only if $T$ and $T^{-1}$ have $(GFS)_{1}$. The result is now straighforward. ∎

### 4.2. $\gamma$-discrete Gomilko Shi-Feng condition

######  Definition 4.11.

Let $m\geq 1$ be an integer and $T\in B(X)$ with
$\sigma(T)\subset\overline{\mathbb{D}}$. We say that $T$ has property
$(\gamma$-$GFS)_{m}$ if there exists a constant $C>0$ such that for any
$N\in\mathbb{N}$, for any $r_{1},\ldots,r_{N}>1$, and for any
$x_{1},\ldots,x_{N}\in X$ and $x_{1}^{*},\ldots,x_{N}^{*}\in X^{*}$, we have

(4.6)
$\displaystyle\sum_{k=1}^{N}\int_{\mathbb{R}}|\langle(r_{k}+1)(r_{k}-1)^{m}R(r_{k}e^{it},T)^{m+1}x_{k},x_{k}^{*}\rangle|dt$
$\displaystyle\leq
C\left\|(x_{k})_{k\in\mathbb{N}_{N}}\right\|_{\gamma(\mathbb{N}_{N};X)}\left\|(x^{*}_{k})_{k\in\mathbb{N}_{N}}\right\|_{\gamma^{\prime}(\mathbb{N}_{N};X^{*})}.$

Obviously if $T$ has $(\gamma$-$GFS)_{m}$ then $T$ has $(GFS)_{m}$. Moreover
on Hilbert spaces, properties $(\gamma$-$GFS)_{m}$ and $(GFS)_{m}$ are
equivalent.

######  Proposition 4.12.

If $T$ has $(\gamma$-$GFS)_{m}$ then $T$ is power $\gamma$-bounded, that is,
the set $\\{T^{n}:n\in\mathbb{N}\\}$ is $\gamma$-bounded.

###### Proof.

The proof is similar to the proof of Proposition 2.3. Let $N\in\mathbb{N}$,
$n_{1},\cdots,n_{N}\in\mathbb{N}$, $r_{1},\ldots,r_{N}>1$,
$x_{1},\ldots,x_{N}\in X$ and $x_{1}^{*},\ldots,x_{N}^{*}\in X^{*}$. For
$x_{1}^{*},\ldots,x_{N}^{*}\in X^{*}$ such that
$\left\|(x^{*}_{k})_{k\in\mathbb{N}_{N}}\right\|_{\gamma^{\prime}(\mathbb{N}_{N};X^{*})}\leq
1$ the proof of (2.4) and the assumption yield

$\displaystyle\sum_{k=1}^{N}\frac{2\pi(n_{k}+1)\ldots(n_{k}+m)(r_{k}+1)(r_{k}-1)^{m}}{r^{n_{k}+m+1}m!}|<T^{n_{k}}x_{k},x_{k}^{*}>|$
$\displaystyle\leq\sum_{k=1}^{N}\int_{0}^{2\pi}(r_{k}+1)(r_{k}-1)^{m}|\langle
R(r_{k}e^{it},T)^{m+1}x_{k},x_{k}^{*}\rangle|dt$ $\displaystyle\leq
C\left\|(x_{k})_{k\in\mathbb{N}_{N}}\right\|_{\gamma(\mathbb{N}_{N};X)}\left\|(x^{*}_{k})_{k\in\mathbb{N}_{N}}\right\|_{\gamma^{\prime}(\mathbb{N}_{N};X^{*})}$
$\displaystyle\leq
C\left\|(x_{k})_{k\in\mathbb{N}_{N}}\right\|_{\gamma(\mathbb{N}_{N};X)}.$

Taking $r_{k}=1+\frac{1}{n_{k}}$ one obtains

$\sum_{k=1}^{N}|<T^{n_{k}}x_{k},x_{k}^{*}>|\leq\frac{2^{m}m!Ce}{\pi}\left\|(x_{k})_{k\in\mathbb{N}_{N}}\right\|_{\gamma(\mathbb{N}_{N};X)}.$

Taking the supremum over $x_{1}^{*},\ldots,x_{N}^{*}$, such that
$\left\|(x^{*}_{k})_{k\in\mathbb{N}_{N}}\right\|_{\gamma^{\prime}(\mathbb{N}_{N};X^{*})}\leq
1$, one obtains according to (4.3)

$\left\|\big{(}T^{n_{k}}x_{k}\big{)}_{k\in\mathbb{N}_{N}}\right\|_{\gamma(\mathbb{N}_{N};X)}\leq\frac{2^{m}m!Ce}{\pi}\left\|(x_{k})_{k\in\mathbb{N}_{N}}\right\|_{\gamma(\mathbb{N}_{N};X)}.$

Hence the set $\\{T^{n}:n\in\mathbb{N}\\}$ is $\gamma$-bounded. ∎

######  Lemma 4.13.

Let $m\in\mathbb{N}$ and $T\in B(X)$ with
$\sigma(T)\subset\overline{\mathbb{D}}$. Then $T$ has property
$(\gamma\text{-}GFS)_{m}$ if and only if the set

(4.7)
$\Big{\\{}(r+1)(r-1)^{m}\displaystyle\int_{0}^{2\pi}\varepsilon(t)R(re^{it},T)^{m+1}dt\,\big{|}\,\varepsilon:[0,2\pi)\rightarrow\overline{\mathbb{D}}\text{
measurable},\,r>1\Big{\\}}$

is $\gamma$-bounded.

###### Proof.

Let $T\in B(X)$ with $(\gamma$-$GFS)_{m}$. Let $N\in\mathbb{N}$,
$r_{1},\ldots,r_{N}>1$, $x_{1},\ldots,x_{N}\in X$ and
$\varepsilon_{k}:[0,2\pi)\rightarrow\overline{\mathbb{D}}$ measurable. For
$x_{1}^{*},\ldots,x_{N}^{*}\in X^{*}$ with
$\left\|(x_{k}^{*})_{k\in\mathbb{N}_{N}}\right\|_{\gamma^{\prime}(\mathbb{N}_{N};X^{*})}\leq
1$ one has, using (4.6),

$\displaystyle\Big{|}\sum_{k=1}^{N}\big{\langle}\Big{(}$
$\displaystyle(r_{k}+1)(r_{k}-1)^{m}\displaystyle\int_{0}^{2\pi}\varepsilon_{k}(t)R(r_{k}e^{it},T)^{m+1}dt\Big{)}x_{k},x^{*}_{k}\big{\rangle}\Big{|}$
$\displaystyle\leq
C\left\|(x_{k})_{k\in\mathbb{N}_{N}}\right\|_{\gamma(\mathbb{N}_{N};X)}.$

Now take the supremum over $x_{1}^{*},\ldots,x_{N}^{*}\in X^{*}$ one obtains
by (4.3)

$\left\|\Big{(}(r_{k}+1)(r_{k}-1)^{m}\displaystyle\int_{0}^{2\pi}\varepsilon_{k}(t)R(r_{k}e^{it},T)^{m+1}dtx_{k}\Big{)}_{k\in\mathbb{N}_{N}}\right\|_{\gamma(\mathbb{N}_{N};X)}\leq
C\left\|(x_{k})_{k\in\mathbb{N}_{N}}\right\|_{\gamma(\mathbb{N}_{N};X)}.$

This means that the set in (4.7) is $\gamma$-bounded.

Suppose that the set in (4.7) is $\gamma$-bounded. Let $N\in\mathbb{N}$,
$r_{1},\ldots,r_{N}>1$, $x_{1},\ldots,x_{N}\in X$ and
$x_{1}^{*},\ldots,x_{N}^{*}\in X^{*}$. For $k\in\mathbb{N}_{N}$, let
$\varepsilon_{k}:[0,2\pi)\rightarrow\overline{\mathbb{D}}$ measurable such
that

$\displaystyle\int_{\mathbb{R}}|$
$\displaystyle\langle(r_{k}+1)(r_{k}-1)^{m}R(r_{k}e^{it},T)^{m+1}x_{k},x_{k}^{*}\rangle|dt$
$\displaystyle=\big{\langle}\big{(}(r_{k}+1)(r_{k}-1)^{m}\int_{\mathbb{R}}\varepsilon_{k}(t)R(r_{k}e^{it},T)^{m+1}dt\big{)}x_{k},x_{k}^{*}\big{\rangle}.$

Then by Theorem 4.3 and the assumption,

$\displaystyle\big{\langle}$
$\displaystyle\Big{(}(r_{k}+1)(r_{k}-1)^{m}\int_{\mathbb{R}}\varepsilon_{k}(t)R(r_{k}e^{it},T)^{m+1}dt\big{)}x_{k},x_{k}^{*}\big{\rangle}$
$\displaystyle\leq\left\|\Big{(}\sum_{k=1}^{N}(r_{k}+1)(r_{k}-1)^{m}\displaystyle\int_{0}^{2\pi}\varepsilon_{k}(t)R(r_{k}e^{it},T)^{m+1}dtx_{k}\Big{)}_{k\in\mathbb{N}_{N}}\right\|_{\gamma(\mathbb{N}_{N};X)}\left\|(x^{*}_{k})_{k\in\mathbb{N}_{N}}\right\|_{\gamma^{\prime}(\mathbb{N}_{N};X^{*})}.$
$\displaystyle\leq
C\left\|(x_{k})_{k\in\mathbb{N}_{N}}\right\|_{\gamma(\mathbb{N}_{N};X)}\left\|(x^{*}_{k})_{k\in\mathbb{N}_{N}}\right\|_{\gamma^{\prime}(\mathbb{N}_{N};X^{*})}.$

∎

######  Proposition 4.14.

Let $T\in B(X)$ with $(GFS)_{m}$. If $T$ has $(\gamma$-$GFS)_{m}$ then $T$ has
$(\gamma$-$GFS)_{p}$ for $1\leq p\leq m$.

###### Proof.

We proceed by induction, showing that if $m\geq 2$, then
$(\gamma\text{-}GFS)_{m}$ implies
$(\gamma\text{-}GFS)_{m-1}$.

Suppose that $T$ has property $(\gamma\text{-}GFS)_{m}$, with $m\geq 2$. Let
$\varepsilon:[0,2\pi)\rightarrow\overline{\mathbb{D}}$ be a measurable
function and let $r>1$. Then by the argument showing (2.7) and a change of
variable one has

$\displaystyle(r+1)(r-1)^{m-1}\int_{0}^{2\pi}\varepsilon(t)R(re^{it},T)^{m}dt$
$\displaystyle=r(m-1)\int_{1}^{+\infty}(r+1)(r-1)^{m-1}\int_{0}^{2\pi}\varepsilon(t)R(rue^{it},T)^{m+1}dtdu$
$\displaystyle=r(m-1)\int_{1}^{+\infty}\frac{(r+1)(r-1)^{m-1}}{(ru+1)(ru-1)^{m}}\int_{0}^{2\pi}(ru+1)(ru-1)^{m}\varepsilon(t)R(rue^{it},T)^{m+1}dtdu.$

Let $\mathcal{T}$ be the set (4.7). By assumption and Lemma 4.13,
$\mathcal{T}$ is $\gamma$-bounded hence by [8, Theorem 8.5.2.], the set

$\displaystyle\Gamma:=\big{\\{}\int_{1}^{\infty}$
$\displaystyle\phi(u)\int_{0}^{2\pi}(ru+1)(ru-1)^{m}\varepsilon(t)R(rue^{it},T)^{m+1}dtdu\,\big{|}\,$
$\displaystyle\varepsilon:[0,2\pi)\rightarrow\overline{\mathbb{D}}\text{
measurable},\,r>1,\,\phi\in
L^{1}((1,\infty)),\,\left\|\phi\right\|_{L^{1}}\leq 1\big{\\}}$

is $\gamma$-bounded. Since

$\left\|r(m-1)\frac{(r+1)(r-1)^{m-1}}{(r\cdot+1)(r\cdot-1)^{m}}\right\|_{L^{1}((1,\infty))}\leq
1,$

the above calculation shows that the set

$\big{\\{}(r+1)(r-1)^{m-1}\displaystyle\int_{0}^{2\pi}\varepsilon(t)R(re^{it},T)^{m}dt\,:\,\varepsilon:\mathbb{R}\rightarrow\overline{\mathbb{D}}\text{
measurable},\,r>1\big{\\}}$

is included in $\Gamma$, hence is $\gamma$-bounded. Hence, by Lemma 4.13, the
operator $T$ has property $(\gamma\text{-}GFS)_{m-1}$. ∎

### 4.3. ${m}$-derivative $\gamma$-bounded functional calculus

######  Definition 4.15.

Let $T\in B(X)$ with $\sigma(T)\subset\overline{\mathbb{D}}$. Then $T$ is said
to have ${m}$-derivative $\gamma$-bounded functional calculus if the set

$\big{\\{}(r-1)^{m}\phi^{(m)}(T):r>1,\,\phi\in
H^{\infty}(r\mathbb{D}),\,\left\|\phi\right\|_{H^{\infty}(r\mathbb{D})}\leq
1\big{\\}}$

is $\gamma$-bounded.

We give now the $\gamma$-bounded version of Theorem 2.10.

######  Theorem 4.16.

Let $T\in B(X)$ with $\sigma(T)\subset\overline{\mathbb{D}}$. The following
assertions are equivalent for $m\in\mathbb{N}$,

  1. (i)

T has $(\gamma$-$GFS)_{m}$;

  2. (ii)

T has ${1}$-derivative $\gamma$-bounded functional calculus;

  3. (iii)

T has ${m}$-derivative $\gamma$-bounded functional calculus.

###### Proof.

$(i)\Rightarrow(ii)$: First, by Proposition 4.14, $T$ has
$(\gamma\text{-}GFS)_{1}$. Let $N\in\mathbb{N}$ and $r_{1},\ldots,r_{N}>1$,
$\phi_{1}\in H^{\infty}(r_{1}\mathbb{D}),\ldots,\phi_{N}\in
H^{\infty}(r_{N}\mathbb{D})$ with
$\left\|\phi_{k}\right\|_{H^{\infty}(r_{k}\mathbb{D})}\leq 1$. Let
$x_{1},\ldots,x_{N}\in X$ and $x_{1}^{*},\ldots,x_{N}^{*}\in X^{*}$. Using
(2.2) and arguing as in the proof of $(i)\implies(ii)$ in Theorem 2.10, one
has

$\sum_{k=1}^{N}|\langle(r_{k}-1)\phi^{\prime}_{k}(T)x_{k},x^{*}_{k}\rangle|\leq\frac{1}{2\pi}\sum_{k=1}^{N}\left\|\phi_{k}\right\|_{H^{\infty}(r_{k}\mathbb{D})}\int_{0}^{2\pi}\big{|}\langle(r_{k}^{2}-1)R(r_{k}e^{it},T)^{2}x_{k},x^{*}_{k}\rangle\big{|}dt.$

Suppose
$\left\|(x_{k}^{*})_{k\in\mathbb{N}_{N}}\right\|_{\gamma^{\prime}(\mathbb{N}_{N};X^{*})}\leq
1$. Since $\left\|\phi_{k}\right\|_{H^{\infty}(r_{k}\mathbb{D})}\leq 1$ and
$T$ has $(\gamma\text{-}GFS)_{1}$, one obtains

$\sum_{k=1}^{N}|\langle(r_{k}-1)\phi^{\prime}_{k}(T)x_{k},x^{*}_{k}\rangle|\leq\frac{C}{2\pi}\left\|(x_{k})_{k\in\mathbb{N}_{N}}\right\|_{\gamma(\mathbb{N}_{N};X)}.$

Taking the supremum over $x_{1}^{*},\ldots,x_{N}^{*}$ this implies, thanks to
(4.3), that $T$ has ${1}$-derivative $\gamma$-bounded functional calculus.

$(ii)\Rightarrow(iii)$: It follows from the assumption that the set

$\Delta:=\Big{\\{}(\rho-1)\phi^{(m)}(T):1<\rho<r,\,\phi\in
H^{\infty}(r\mathbb{D}),\;\left\|\phi^{(m-1)}\right\|_{H^{\infty}(\rho\mathbb{D})}\leq
1\Big{\\}}$

is $\gamma$-bounded. For any $1<\rho<r$ and $\phi\in H^{\infty}(r\mathbb{D})$
with $\left\|\phi\right\|_{H^{\infty}(r\mathbb{D})}\leq 1$, by Lemma 2.7 one
has

$\left\|\frac{(r-\rho)^{m-1}}{(m-1)!}\phi^{(m-1)}\right\|_{H^{\infty}(\rho\mathbb{D})}\leq
1.$

Hence

$\bigg{\\{}\frac{(\rho-1)(r-\rho)^{m-1}}{(m-1)!}\phi^{(m)}(T):1<\rho<r,\,\phi\in
H^{\infty}(r\mathbb{D}),\;\left\|\phi\right\|_{H^{\infty}(r\mathbb{D})}\leq
1\bigg{\\}}\subset\Delta.$

Taking $\rho=\frac{r+1}{2}$ in the above set, we obtain

$\bigg{\\{}\frac{(r-1)^{m}}{2^{m}(m-1)!}\phi^{(m)}(T):r>1,\,\phi\in
H^{\infty}(r\mathbb{D}),\;\left\|\phi\right\|_{H^{\infty}(r\mathbb{D})}\leq
1\bigg{\\}}\subset\Delta.$

Hence the above set is $\gamma$-bounded. Thus $T$ has ${m}$-derivative
$\gamma$-bounded functional calculus.

$(iii)\Rightarrow(i)$: Let $r_{1},\ldots,r_{N}>1$,
$1<\rho_{1}<r_{1},\ldots,1<\rho_{N}<r_{N}$, $x_{1},\ldots,x_{N}\in X$ and
$x^{*}_{1},\ldots,x^{*}_{N}\in X^{*}$. Let us introduce measurable functions
$\varepsilon_{1},...,\varepsilon_{N}:[0,2\pi)\to\overline{\mathbb{D}}$ such
that

$|\langle
R((r_{k})e^{it},T)^{m+2}x_{k},x_{k}^{*}\rangle|=\varepsilon_{k}(t)\langle
R((r_{k})e^{it},T)^{m+2}x_{k},x_{k}^{*}\rangle.$

for all $k=1,\ldots,N$ and all $t\in[0,2\pi)$. Next we set

$\phi_{k}(z):=\frac{(r_{k}+\rho_{k})(r_{k}-\rho_{k})}{2\pi}\int_{0}^{2\pi}\frac{\varepsilon_{k}(t)}{(r_{k}e^{it}-z)^{2}}dt,\quad
z\in r_{k}\mathbb{D}.$

The computations in the proof of Theorem 2.10 show that
$\left\|\phi_{k}\right\|_{H^{\infty}(\rho_{k}\mathbb{D})}\leq 1$ and

$\phi_{k}^{(m)}(T)=\frac{(m+1)!(r_{k}+\rho_{k})(r_{k}-\rho_{k})}{2\pi}\int_{0}^{2\pi}\varepsilon_{k}(t)R(r_{k}e^{it},T)^{m+2}dt.$

Therefore it follows from $(iii)$ that we have the estimate

$\displaystyle\sum_{k=1}^{N}$
$\displaystyle\frac{(m+1)!(\rho_{k}-1)^{m}(r_{k}-\rho_{k})(r_{k}+1)}{2\pi}\int_{0}^{2\pi}|\langle
R(r_{k}e^{it},T)^{m+2}x_{k},x^{*}_{k}\rangle|dt$
$\displaystyle\leq\sum_{k=1}^{N}\frac{(m+1)!(\rho_{k}-1)^{m}(r_{k}-\rho_{k})(r_{k}+\rho_{k})}{2\pi}\int_{0}^{2\pi}\varepsilon_{k}(t)\langle
R(r_{k}e^{it},T)^{m+2}x_{k},x^{*}_{k}\rangle dt$
$\displaystyle=\sum_{k=1}^{N}\big{|}\langle(\rho_{k}-1)^{m}\phi_{k}^{(m)}(T)x_{k},x^{*}_{k}\rangle\big{|}dt$
$\displaystyle\leq
C\left\|(x_{k})_{k\in\mathbb{N}_{N}}\right\|_{\gamma(\mathbb{N}_{N};X)}\left\|(x^{*}_{k})_{k\in\mathbb{N}_{N}}\right\|_{\gamma^{\prime}(\mathbb{N}_{N};X^{*})}.$

The last inequality comes from Theorem 4.3. Now we choose
$\rho_{k}=\frac{r_{k}+1}{2}$ in the above estimate. We obtain the following
inequality

$\displaystyle\sum_{k=1}^{N}\int_{\mathbb{R}}|\langle(r_{k}+1)(r_{k}-1)^{m+1}R(r_{k}e^{it},T)^{m+2}x_{k},x^{*}_{k}\rangle|dt$
$\displaystyle\leq\frac{2^{m+2}\pi
C}{(m+1)!}\left\|(x_{k})_{k\in\mathbb{N}_{N}}\right\|_{\gamma(\mathbb{N}_{N};X)}\left\|(x^{*}_{k})_{k\in\mathbb{N}_{N}}\right\|_{\gamma^{\prime}(\mathbb{N}_{N};X^{*})}.$

This shows that $T$ has $(\gamma\text{-}GFS)_{m+1}$. Then by Proposition 4.14,
$T$ has $(\gamma\text{-}GFS)_{m}$. ∎

######  Definition 4.17.

Let $T\in B(X)$. If $T$ satisfies one of the three conditions of Theorem 4.16
then we will say that $T$ is a $\gamma$-GFS operator.

### 4.4. Characterization of power $\gamma$-bounded on Banach space $X$

In the following, the space $L^{2}((0,2\pi))$ will be equipped with the norm

$\left\|f\right\|^{2}_{2}=\frac{1}{2\pi}\int_{0}^{2\pi}|f(s)|^{2}ds,\quad f\in
L^{2}((0,2\pi)).$

Therefore, the Fourier-Parseval operator

$\begin{array}[]{ccccc}\mathcal{F}&:&L^{2}([0,2\pi))&\to&l^{2}_{\mathbb{Z}}\\\
&&f&\mapsto&(c_{n}(f))_{n\in\mathbb{Z}},\\\ \end{array}$

is an isometry. Here $c_{n}(f)$ is the $n$-th Fourier coefficient defined by

$c_{n}(f)=\frac{1}{2\pi}\int_{0}^{2\pi}f(t)e^{-int}dt.$

We give a characterization of power $\gamma$-bounded operators.

######  Theorem 4.18.

Let $X$ be a Banach space. Following assertions are equivalent :

  1. (i)

The operator $T$ is power $\gamma$-bounded;

  2. (ii)

The spectrum set $\sigma(T)$ is included in $\overline{\mathbb{D}}$ and there
exists a constant $C>0$ such that for all $N\in\mathbb{N}$,
$r_{1},\ldots,r_{N}>1$, $x_{1},\ldots,x_{N}\in X$ and
$x^{*}_{1},\ldots,x^{*}_{N}\in X^{*}$ the functions
$(t,k)\mapsto\sqrt{r_{k}^{2}-1}R(r_{k}e^{it},T)x_{k}$ and
$(t,k)\mapsto\sqrt{r_{k}^{2}-1}R(r_{k}e^{it},T^{*})x^{*}_{k}$ are in
$\gamma([0,2\pi)\times\mathbb{N}_{N};X)$ and
$\gamma^{\prime}([0,2\pi)\times\mathbb{N}_{N};X^{*})$ respectively, and
satisfy

(4.8)
$\left\|(t,k)\mapsto\sqrt{r_{k}^{2}-1}R(r_{k}e^{it},T)x_{k}\right\|_{\gamma([0,2\pi)\times\mathbb{N}_{N};X)}\leq
C\left\|(x_{k})_{k\in\mathbb{N}_{N}}\right\|_{\gamma(\mathbb{N}_{N};X)}$

and

(4.9)
$\left\|(t,k)\mapsto\sqrt{r_{k}^{2}-1}R(r_{k}e^{it},T^{*})x^{*}_{k}\right\|_{\gamma^{\prime}([0,2\pi)\times\mathbb{N}_{N};X^{*})}\leq\left\|(x_{k}^{*})_{k\in\mathbb{N}_{N}}\right\|_{\gamma^{\prime}(\mathbb{N}_{N};X^{*})};$

  3. (iii)

$T$ is a $\gamma$-GFS operator.

###### Proof.

$(iii)\implies(i)$ is Proposition 4.12.

$(i)\implies(ii)$ : Recall that for each $k\in\mathbb{N}_{N}$ one has

$R(r_{k}e^{it},T)x_{k}=\sum_{n=0}^{\infty}\frac{T^{n}x_{k}}{(r_{k}e^{it})^{n+1}}=\sum_{n=0}^{\infty}\frac{T^{n}x_{k}}{r_{k}^{n+1}}e^{-i(n+1)t}.$

We apply Lemma 4.5 with $H=l^{2}_{N}\overset{2}{\otimes}L^{2}((0,2\pi))$,
$K=l^{2}_{N}\overset{2}{\otimes}l^{2}_{\mathbb{Z}}$ and $U:H\rightarrow K$
defined by

$U\big{(}\sum_{k=1}^{N}e_{k}\otimes
f_{k}\big{)}=\sum_{k=1}^{N}e_{k}\otimes(c_{n}(f_{k}))_{n\in\mathbb{Z}},\quad
f_{1},\ldots,f_{N}\in L^{2}((0,2\pi)).$

We obtain that $(n,k)\mapsto\sqrt{r_{k}^{2}-1}\frac{T^{n}x_{k}}{r_{k}^{n+1}}$
belongs to $\gamma(\mathbb{N}\cup\\{0\\}\times\mathbb{N}_{N};X)$ and

$\left\|(t,k)\mapsto\sqrt{r_{k}^{2}-1}R(r_{k}e^{it},T)x_{k}\right\|_{\gamma([0,2\pi)\times\mathbb{N}_{N};X)}=\left\|(n,k)\mapsto\sqrt{r_{k}^{2}-1}\frac{T^{n}x_{k}}{r_{k}^{n+1}}\right\|_{\gamma(\mathbb{N}\cup\\{0\\}\times\mathbb{N}_{N};X)}.$

But, one has by (4.2)

$\left\|(n,k)\mapsto\sqrt{r_{k}^{2}-1}\frac{T^{n}x_{k}}{r_{k}^{n+1}}\right\|_{\gamma(\mathbb{N}\cup\\{0\\}\times\mathbb{N}_{N};X)}=\left\|\sum_{k=1}^{N}\sqrt{r_{k}^{2}-1}\sum_{n=0}^{\infty}\frac{1}{r_{k}^{n+1}}\gamma_{k,n}\otimes
T^{n}x_{k}\right\|_{G(X)},$

where $(\gamma_{k,n})_{k\in\mathbb{N}_{N},n\in\mathbb{N}\cup\\{0\\}}$ is a
family of independant complex valued standard Gaussian variables. Using the
$\gamma$-boundedness of $\\{T^{n}:n\in\mathbb{N}\\}$ it follows

$\left\|\sum_{k=1}^{N}\sqrt{r_{k}^{2}-1}\sum_{n=0}^{\infty}\frac{1}{r_{k}^{n+1}}\gamma_{k,n}\otimes
T^{n}x_{k}\right\|_{G(X)}\leq
C_{\gamma}\left\|\sum_{k=1}^{N}\sqrt{r_{k}^{2}-1}\sum_{n=0}^{\infty}\frac{1}{r_{k}^{n+1}}\gamma_{k,n}\otimes
x_{k}\right\|_{G(X)}.$

Furthermore
$\left\|\Bigg{(}\sqrt{r_{k}^{2}-1}\frac{1}{r_{k}^{n+1}}\Bigg{)}_{n\in\mathbb{N}\cup\\{0\\}}\right\|_{l^{2}}=1$
for each $k\in\mathbb{N}_{N}$ therefore

$\Big{(}\sum_{n=0}^{\infty}\sqrt{r_{k}^{2}-1}\frac{1}{r_{k}^{n+1}}\gamma_{k,n}\Big{)}_{k\in\mathbb{N}_{N}}$

is a sequence of independent standard Gaussian variables, which implies

$\left\|\sum_{k=1}^{N}\sqrt{r_{k}^{2}-1}\sum_{n=0}^{\infty}\frac{1}{r_{k}^{n+1}}\gamma_{k,n}\otimes
x_{k}\right\|_{G(X)}=\left\|(x_{k})_{k\in\mathbb{N}_{N}}\right\|_{\gamma(\mathbb{N}_{N};X)}.$

Hence one obtains (4.8).

Likewise, using Lemma 4.5 we obtain
$(n,k)\mapsto\sqrt{r_{k}^{2}-1}\frac{{T^{*}}^{n}x_{k}}{r_{k}^{n+1}}$ belongs
to $\gamma^{\prime}(\mathbb{N}\cup\\{0\\}\times\mathbb{N}_{N};X^{*})$ and

$\left\|(t,k)\mapsto\sqrt{r_{k}^{2}-1}R(r_{k}e^{it},T^{*})x^{*}_{k}\right\|_{\gamma^{\prime}([0,2\pi)\times\mathbb{N}_{N};X^{*})}=\left\|(n,k)\mapsto\sqrt{r_{k}^{2}-1}\frac{{T^{*}}^{n}x^{*}_{k}}{r_{k}^{n+1}}\right\|_{\gamma^{\prime}(\mathbb{N}\cup\\{0\\}\times\mathbb{N}_{N};X^{*})}.$

Furthermore, one has by (4.5)

$\left\|(n,k)\mapsto\sqrt{r_{k}^{2}-1}\frac{{T^{*}}^{n}x^{*}_{k}}{r_{k}^{n+1}}\right\|_{\gamma^{\prime}(\mathbb{N}\cup\\{0\\}\times\mathbb{N}_{N};X^{*})}=\lim\limits_{M\rightarrow\infty}\left\|(n,k)\mapsto\sqrt{r_{k}^{2}-1}\frac{{T^{*}}^{n}x^{*}_{k}}{r_{k}^{n+1}}\right\|_{\gamma^{\prime}(\mathbb{N}_{M}\cup\\{0\\}\times\mathbb{N}_{N};X^{*})}.$

But, by (4.4)

(4.10)
$\displaystyle\left\|(n,k)\mapsto\sqrt{r_{k}^{2}-1}\frac{{T^{*}}^{n}x^{*}_{k}}{r_{k}^{n+1}}\right\|_{\gamma^{\prime}(\mathbb{N}_{M}\cup\\{0\\}\times\mathbb{N}_{N};X^{*})}$
$\displaystyle=\sup\Big{\\{}\Big{|}\sum_{n=1}^{M}\sum_{k=1}^{N}\big{\langle}x_{n,k},\sqrt{r_{k}^{2}-1}\frac{{T^{*}}^{n}x^{*}_{k}}{r_{k}^{n+1}}\big{\rangle}\Big{|}:\left\|(x_{n,k})_{n\in\mathbb{N}_{M}\cup\\{0\\},k\in\mathbb{N}_{N}}\right\|_{\gamma(\mathbb{N}_{M}\cup\\{0\\}\times\mathbb{N}_{N},X)}\leq
1\Big{\\}},$

and for $M\in\mathbb{N}$, using Theorem 4.3,

$\displaystyle\Big{|}\sum_{n=1}^{M}\sum_{k=1}^{N}\langle
x_{n,k},\sqrt{r_{k}^{2}-1}\frac{{T^{*}}^{n}x^{*}_{k}}{r_{k}^{n+1}}\rangle\Big{|}$
$\displaystyle=\Big{|}\sum_{n=1}^{M}\sum_{k=1}^{N}\langle\sqrt{r_{k}^{2}-1}\frac{{T}^{n}x_{n,k}}{r_{k}^{n+1}},x_{k}^{*}\rangle\Big{|}$
$\displaystyle\leq\left\|\sqrt{r_{k}^{2}-1}\frac{{T}^{n}x_{n,k}}{r_{k}^{n+1}}\right\|_{{\gamma(\mathbb{N}_{M}\cup\\{0\\}\times\mathbb{N}_{N},X)}}\left\|(x_{k}^{*})_{k\in\mathbb{N}_{N}}\right\|_{\gamma^{\prime}(\mathbb{N}_{N};X^{*})}.$

By previous computations which leads to (4.8), for each $M\in\mathbb{N}$ one
has

$\left\|\sqrt{r_{k}^{2}-1}\frac{{T}^{n}x_{n,k}}{r_{k}^{n+1}}\right\|_{\gamma(\mathbb{N}_{M}\cup\\{0\\}\times\mathbb{N}_{N};X)}\leq
C_{\gamma}\left\|(x_{k})_{k\in\mathbb{N}_{N}}\right\|_{\gamma(\mathbb{N}_{N};X)}.$

This implies

$\left\|(n,k)\mapsto\sqrt{r_{k}^{2}-1}\frac{{T^{*}}^{n}x^{*}_{k}}{r_{k}^{n+1}}\right\|_{\gamma^{\prime}(\mathbb{N}_{M}\cup\\{0\\}\times\mathbb{N}_{N};X^{*})}\leq
C_{\gamma}\left\|(x_{k}^{*})_{k\in\mathbb{N}_{N}}\right\|_{\gamma^{\prime}(\mathbb{N}_{N};X^{*})}.$

Hence one obtains (4.9).
$(ii)\implies(iii)$ : Let $N\in\mathbb{N}$, $r_{1},\ldots,r_{N}>1$,
$x_{1},\ldots,x_{N}\in X$ and $x^{*}_{1},\ldots,x^{*}_{N}\in X^{*}$. One has

$\displaystyle\sum_{k=1}^{N}$
$\displaystyle\int_{\mathbb{R}}|\langle(r_{k}^{2}-1)R(r_{k}e^{it},T)^{2}x_{k},x_{k}^{*}\rangle|dt$
$\displaystyle=\sum_{k=1}^{N}\int_{\mathbb{R}}|\langle(r_{k}^{2}-1)R(r_{k}e^{it},T)x_{k},R(r_{k}e^{it},T)^{*}x_{k}^{*}\rangle|dt$
$\displaystyle=\left\|(t,k)\mapsto\langle\sqrt{r_{k}^{2}-1}R(r_{k}e^{it},T)x_{k},\sqrt{r_{k}-1}R(r_{k}e^{it},T)^{*}x_{k}^{*}\rangle\right\|_{L^{1}((0,2\pi)\times\mathbb{N}_{N})}.$

By Theorem 4.3 and assumptions (4.8) and (4.9),

$\displaystyle\left\|(t,k)\mapsto\langle\sqrt{r_{k}^{2}-1}R(r_{k}e^{it},T)x_{k},\sqrt{r_{k}^{2}-1}R(r_{k}e^{it},T)^{*}x_{k}^{*}\rangle\right\|_{L^{1}((0,2\pi)\times\mathbb{N}_{N})}$
$\displaystyle\leq\left\|\sqrt{r_{k}^{2}-1}R(r_{k}e^{it},T)x_{k}\right\|_{\gamma((0,2\pi)\times\mathbb{N}_{N};X)}\left\|\sqrt{r_{k}^{2}-1}R(r_{k}e^{it},T)^{*}x_{k}^{*}\right\|_{\gamma^{\prime}((0,2\pi)\times\mathbb{N}_{N};X^{*})}$
$\displaystyle\leq
C^{2}\left\|(x_{k})_{k\in\mathbb{N}_{N}}\right\|_{\gamma(\mathbb{N}_{N};X)}\left\|(x_{k})_{k\in\mathbb{N}_{N}}\right\|_{\gamma^{\prime}(\mathbb{N}_{N};X^{*})}.$

Hence $T$ has $(\gamma$-$GFS)_{1}$ and therefore it is a $\gamma$-GFS
operator. ∎

######  Remark 4.19.

When $X$ is $K$-convex (4.9) can be replaced by

$\left\|(t,k)\mapsto\sqrt{r_{k}^{2}-1}R(r_{k}e^{it},T^{*})x^{*}_{k}\right\|_{\gamma([0,2\pi)\times\mathbb{N}_{N};X^{*})}\leq\left\|(x_{k}^{*})_{k\in\mathbb{N}_{N}}\right\|_{\gamma(\mathbb{N}_{N};X^{*})}.$

Moreover let $(S,\mu)$ be a measure space and let $E(S)$ be a $K$-convex
Banach function space over $(S,\mu)$ (see [8, appendix F] for definition).
Then $E(S)$ has finite cotype hence according to [8, Proposition 9.3.8], there
exist $c>0$ and $C>0$ such that for each
$f\in\gamma([0,2\pi)\times\mathbb{N}_{N};E(S))$,

(4.11)
$c\left\|f\right\|_{E(S;L^{2}([0,2\pi)\times\mathbb{N}_{N}))}\leq\left\|f\right\|_{\gamma([0,2\pi)\times\mathbb{N}_{N};E(S))}\leq
C\left\|f\right\|_{E(S;L^{2}([0,2\pi)\times\mathbb{N}_{N}))}.$

Furthermore, the following equality holds,

$\left\|f\right\|_{E(S;L^{2}([0,2\pi)\times\mathbb{N}_{N}))}=\left\|\Big{(}\int_{0}^{2\pi}\sum_{k=1}^{N}|f(\cdot,k)|^{2}\Big{)}^{\frac{1}{2}}\right\|_{E(S)}.$

The space $E(S)^{*}$ satisfies similar properties. Hence using (4.11) and the
Khintchine-Maurey inequality [8, Theorem 7.2.13], $(ii)$ in Theorem 4.18 can
be replaced by:

  * $(ii)^{\prime}$

The spectrum set $\sigma(T)$ is included in $\overline{\mathbb{D}}$ and there
exists a constant $C>0$ such that for all $N\in\mathbb{N}$,
$r_{1},\ldots,r_{N}>1$, $x_{1},\ldots,x_{N}\in X$ and
$x^{*}_{1},\ldots,x^{*}_{N}\in X^{*}$

$\left\|\Big{(}\sum_{k=1}^{N}\int_{0}^{2\pi}(r_{k}^{2}-1)|R(r_{k}e^{it},T)x_{k}|^{2}dt\Big{)}^{\frac{1}{2}}\right\|_{E(S)}\leq
C\left\|\big{(}\sum_{k=1}^{N}|x_{k}|^{2}\big{)}^{\frac{1}{2}}\right\|_{E(S)}$

and

$\left\|\Big{(}\sum_{k=1}^{N}\int_{0}^{2\pi}(r_{k}^{2}-1)|R(r_{k}e^{it},T^{*})x^{*}_{k}|^{2}dt\Big{)}^{\frac{1}{2}}\right\|_{E(S)^{*}}\leq
C\left\|\big{(}\sum_{k=1}^{N}|x^{*}_{k}|^{2}\big{)}^{\frac{1}{2}}\right\|_{E(S)^{*}}.$

Thus $T\in B(E(S))$ is power $\gamma$-bounded if and only if $(ii)^{\prime}$
holds true.

In particular the above applies when $E(S)=L^{p}(S)$ for some $1<p<\infty$.

### 4.5. Peller calculus

We denote by $H^{p}$ the classical Hardy space over the unit disk. Let
$\mathcal{A}(\mathbb{D})$ be the set of all function
$F:\overline{\mathbb{D}}\rightarrow\mathbb{C}$ such that there exist two
sequences $(f_{k})_{k\in\mathbb{N}}$ of $C(\mathbb{T})$ and
$(h_{k})_{k\in\mathbb{N}}$ in $H^{1}$ satisfying

(4.12)
$\sum_{k=1}^{\infty}\left\|f_{k}\right\|_{\infty}\left\|h_{k}\right\|_{1}<\infty$

and

(4.13) $\forall z\in\overline{\mathbb{D}},\quad
F(z)=\sum_{k=1}^{\infty}f_{k}\star h_{k}(z).$

We endow $\mathcal{A}(\mathbb{D})$ with the norm

$\left\|F\right\|_{\mathcal{A}}=\inf\big{\\{}\sum_{k=1}^{\infty}\left\|f_{k}\right\|_{\infty}\left\|h_{k}\right\|_{1}\big{\\}}$

where the infimum runs over all sequences $(f_{k})_{k\in\mathbb{N}}$ of
$C(\mathbb{T})$ and $(h_{k})_{k\in\mathbb{N}}$ of $H^{1}$ satifsying (4.12)
and (4.13). It is known (see [13, Lemma 3.6.]) that with this norm
$\mathcal{A}(\mathbb{D})$ is a Banach algebra for pointwise multiplication.

Let $H$ be a Hilbert space and let $T\in B(H)$ be a power-bounded operator.
According to [13, Theorem 3.5.] or [14, Proposition 4.11] there exists $C>0$
such that for each polynomial $P$ one has

$\left\|P(T)\right\|\leq C\left\|P\right\|_{\mathcal{A}}.$

Now since the set of polynomial is dense in $\mathcal{A}(\mathbb{D})$ the
bounded algebra homomorphism $P\rightarrow P(T)$ extends to a bounded algebra
homomorphism from $\mathcal{A}(\mathbb{D})$ into $B(H)$. Our aim is now to
give a $\gamma$-version of this result.

######  Theorem 4.20.

Let $X$ be a Banach space. Let $T\in B(X)$ be a power $\gamma$-bounded
operator. Then the set

$\\{P(T):P\text{ is a polynomial with }\left\|P\right\|_{\mathcal{A}}\leq
1\\}$

is $\gamma$-bounded.

###### Proof.

We adapt an argument from [14, Proposition 4.16.]. First we show that the set

$\Gamma=\\{f\star(uv)(T):f\in C(\mathbb{T}),u,v\text{ are
polynomials},\left\|f\right\|_{\infty}\left\|u\right\|_{2}\left\|v\right\|_{2}\leq
1\\}$

is $\gamma$-bounded. Let $N\in\mathbb{N}$, $f_{1},\ldots,f_{N}\in
C(\mathbb{T})$ and let $u_{1},\ldots,u_{N}$, $v_{1},\ldots,v_{N}$ be
polynomials with
$\left\|f_{k}\right\|_{\infty}\left\|u_{k}\right\|_{2}\left\|v_{k}\right\|_{2}\leq
1$ for each $k\in\mathbb{N}_{N}$, and let $x_{1},\ldots,x_{N}\in X$,
$x^{*}_{1},\ldots,x^{*}_{N}\in X^{*}$. One has, for each $k\in\mathbb{N}_{N}$

$\langle\big{(}f_{k}\star(u_{k}v_{k})\big{)}(T)x_{k},x^{*}_{k}\rangle=\frac{1}{2\pi}\int_{0}^{2\pi}f_{k}(e^{is})\langle
u_{k}(e^{-is}T)x_{k},v_{k}(e^{-is}T)^{*}x^{*}_{k}\rangle ds.$

It follows, using Theorem 4.3, that

$\displaystyle\sum_{k=1}^{N}|\langle\big{(}f_{k}\star$
$\displaystyle(u_{k}v_{k})\big{)}(T)x_{k},x^{*}_{k}\rangle|\leq\frac{1}{2\pi}\sum_{k=1}^{N}\int_{0}^{2\pi}\frac{|f_{k}(e^{is})|}{\left\|f_{k}\right\|_{\infty}\left\|u_{k}\right\|_{2}\left\|v_{k}\right\|_{2}}|\langle
u_{k}(e^{-is}T)x_{k},v_{k}(e^{-is}T)^{*}x^{*}_{k}\rangle|ds$
$\displaystyle\leq\sum_{k=1}^{N}\int_{0}^{2\pi}\big{|}\big{\langle}\frac{u_{k}(e^{-is}T)x_{k}}{\left\|u_{k}\right\|_{2}},\frac{v_{k}(e^{-is}T)^{*}x_{k}^{*}}{\left\|v_{k}\right\|_{2}}\big{\rangle}\big{|}ds$
$\displaystyle=\left\|(s,k)\mapsto\langle\frac{u_{k}(e^{-is}T)x_{k}}{\left\|u_{k}\right\|_{2}},\frac{v_{k}(e^{-is}T)^{*}x_{k}^{*}}{\left\|v_{k}\right\|_{2}}\big{\rangle}\right\|_{L^{1}((0,2\pi)\times\mathbb{N}_{N})}$
$\displaystyle\leq\frac{1}{2\pi}\left\|(s,k)\mapsto\frac{u_{k}(e^{-is}T)x_{k}}{\left\|u_{k}\right\|_{2}}\right\|_{\gamma((0,2\pi)\times\mathbb{N}_{N};X)}\left\|(s,k)\mapsto\frac{v_{k}(e^{-is}T)^{*}x^{*}_{k}}{\left\|v_{k}\right\|_{2}}\right\|_{\gamma^{\prime}((0,2\pi)\times\mathbb{N}_{N};X^{*})}.$

By Lemma 4.5 and the $\gamma$-boundedness of $\\{T^{n}:n\in\mathbb{N}\\}$ one
obtains

$\displaystyle\left\|(s,k)\mapsto\frac{u_{k}(e^{-is}T)x_{k}}{\left\|u_{k}\right\|_{2}}\right\|_{\gamma((0,2\pi)\times\mathbb{N}_{N};X)}$
$\displaystyle=\left\|(n,k)\mapsto
c_{n}(u_{k})T^{n}\Big{(}\frac{x_{k}}{\left\|u_{k}\right\|_{2}}\Big{)}\right\|_{\gamma(\mathbb{N}\cup\\{0\\}\times\mathbb{N}_{N};X)}$
$\displaystyle=\left\|\sum_{n=0}^{\infty}\sum_{k=1}^{N}\gamma_{n,k}\otimes
T^{n}\Big{(}\frac{c_{n}(u_{k})x_{k}}{\left\|u_{k}\right\|_{2}}\Big{)}\right\|_{G(X)}$
$\displaystyle\leq
C_{\gamma}\left\|\sum_{n=0}^{\infty}\sum_{k=1}^{N}\gamma_{n,k}\otimes\frac{c_{n}(u_{k})x_{k}}{\left\|u_{k}\right\|_{2}}\right\|_{G(X)}.$

For each $k\in\mathbb{N}_{N}$,
$\displaystyle\frac{\left\|(c_{n}(u_{k}))_{n\in\mathbb{N}\cup\\{0\\}}\right\|_{l^{2}}}{\left\|u_{k}\right\|_{2}}=1$,
therefore

$\left\|\sum_{n=0}^{\infty}\sum_{k=1}^{N}\gamma_{n,k}\otimes\frac{c_{n}(u_{k})x_{k}}{\left\|u_{k}\right\|_{2}}\right\|_{G(X)}\leq\left\|(x_{k})_{k\in\mathbb{N}_{N}}\right\|_{\gamma(\mathbb{N}_{N};X)}.$

Then arguing as in the proof of Theorem 4.18,

$\left\|(s,k)\mapsto\frac{v_{k}(e^{-is}T)^{*}x^{*}_{k}}{\left\|v_{k}\right\|_{2}}\right\|_{\gamma((0,2\pi)\times\mathbb{N}_{N};X^{*})}\leq
C_{\gamma}\left\|(x^{*}_{k})_{k\in\mathbb{N}_{N}}\right\|_{\gamma^{\prime}(\mathbb{N}_{N};X^{*})}.$

This shows that $\Gamma$ is $\gamma$-bounded.

Now if $P=\sum_{k\in\mathbb{N}}f_{k}\star h_{k}$ is a polynomial with
$f_{k}\in C(\mathbb{T})$, $h_{k}\in H^{1}$ and
$\sum_{k\in\mathbb{N}}\left\|f_{k}\right\|_{\infty}\left\|h_{k}\right\|_{1}\leq
1$ then using the facts that for each $k\in\mathbb{N}$, $h_{k}$ can be written
as a product $h_{k}=u_{k}v_{k}$ with $u_{k},v_{k}\in H^{2}$ and
$\left\|h_{k}\right\|_{1}=\left\|u_{k}\right\|_{2}\left\|v_{k}\right\|_{2}$
and that the set of polynomials is dense in $H^{2}$ one obtains that
$P\in\overline{absconv(\Gamma)}^{\left\|\cdot\right\|}$. Finally one has the
following inclusion

$\\{P(T):P\text{ is a polynomial with }\left\|P\right\|_{\mathcal{A}}\leq
1\\}\subset\overline{absconv(\Gamma)}^{\left\|\cdot\right\|},$

and the latter set is $\gamma$-bounded thanks to Proposition 4.2 whence the
desired result. ∎

######  Remark 4.21.

Since the set $\\{P(T):P\text{ is a polynomial with
}\left\|P\right\|_{\mathcal{A}}\leq 1\\}$ is $\gamma$-bounded it is uniformly
bounded. Therefore, since polynomials are dense in $\mathcal{A}(\mathbb{D})$,
the homomorphism

$u:P\mapsto P(T)$

extends to a bounded algebra homomorphism from $\mathcal{A}(\mathbb{D})$ into
$B(X)$. Theorem 4.20 implies that this homomorphism is $\gamma$-bounded, that
is $\\{u(f):f\in\mathcal{A}(\mathbb{D}):\left\|f\right\|_{\mathcal{A}}\leq
1\\}$ is $\gamma$-bounded.

## References

  * [1] L. Arnold.  $\gamma$-boundedness of $c_{0}$-semigroups and their $h^{\infty}$-functional calculi.  Studia Mathematica, 254:77–108, 2020.
  * [2] L. Arnold and C. Le Merdy.  New counterexamples on ritt operators, sectorial operators and $\mathcal{R}$-boundedness.  Bulletin of the Australian Mathematical Society, pages 498–506, 2019.
  * [3] C. Batty, A. Gomilko, and Y. Tomilov.  A Besov algebra calculus for generators of operator semigroups and related norm-estimates.  Mathematische Annalen, 2019.
  * [4] C. Batty, M. Haase, and J. Mubeen.  The holomorphic functional calculus approach to operator semigroups.  Acta Sci. Math. (Szeged), 79 : 289-212, 2013.
  * [5] H. Cartan.  Elementary Theory of Analytic Functions of One Or Several Complex Variables.  Dover books on mathematics. Dover Publications, 1995.
  * [6] A.M. Gomilko.  Conditions on the generator of a uniformly bounded $\textsc{C}_{0}$-semigroup.  Funct. anal. Appl. 33, pages 294–296, 1999.
  * [7] A.M. Gomilko and P.A. Cojuhari.  On the characterization of scalar type spectral operators.  Studia Mathematica 194 (2), pages 121–132, 2008.
  * [8] T Hytönen, J. van Neerven, M. Veraar, and L. Weis.  Analysis in Banach Spaces. Volume II: Probabilistic Methods and Operator Theory, volume 2.  Springer, 2018.
  * [9] N. Kalton and L. Weis.  The $\textsc{H}^{\infty}$-functional calculus and square function estimates. in nigel j.kalton selecta.  1:715–771, 2016.
  * [10] C. Le Merdy.  $\gamma$-bounded representations of amenable groups.  Advances in Mathematics 224, pages 1641–1671, 2010.
  * [11] C. Le Merdy.  $\textsc{H}^{\infty}$functional calculus and square function estimates for Ritt operators.  Revista Matematica Iberoamericana, 30:1149–1190, 2012.
  * [12] A. Lebow.  A power-bounded operator that is not polynomially bounded.  Michigan Math. J., 15:397–399, 1968.
  * [13] V. Peller.  Estimates of functions of power bounded operators on Hilbert spaces.  J.Operator Theory, 7:341–372, 1982.
  * [14] G. Pisier.  Similarity Problems and Completely Bounded Maps.  Springer-Verlag Berlin Heidelberg, 2 edition, 2001.
  * [15] D.-H. Shi and Feng D.X.  Characteristic conditions of the generation of $\textsc{C}_{0}$-semigroups in a hilbert space.  J. Math. Anal. Appl. 247 : 356-376, 2000.
  * [16] S.J. White.  Norm-estimates for functions of semigroups of operators, volume PhD thesis,Univ. of Edinburgh.  https://www.era.lib.ed.ac.uk/handle/1842/11552, 1989.