Gamma-boundedness of $C_0$-semigroups and their $H^\infty$-functional calculi

TL;DR

This paper explores gamma-boundedness in $C_0$-semigroups and their $H^$-functional calculi, establishing new characterizations and extending classical results like Gearhart-Pruss to $K$-convex spaces.

Contribution

It introduces the notion of gamma-$H^$-bounded calculus, characterizes gamma-bounded $C_0$-semigroup generation in $K$-convex spaces, and extends the Gearhart-Pruss theorem.

Findings

Established connection between gamma-$H^$-bounded calculus and semigroup generation.

Provided a characterization of gamma-bounded $C_0$-semigroups in $K$-convex spaces.

Extended the Gearhart-Pruss theorem to $K$-convex spaces.

Abstract

We discuss the notion of $\gamma$-$H^{\infty}$-bounded calculus, strong $\gamma$-$m$-$H^{\infty}$-bounded calculus on half-plane and weak-$\gamma$-Gomilko-Shi-Feng condition and give a connection between them. Then we state a characterization of generation of $\gamma$-bounded $C_0$-semigroup in $K$-convex space, which leads to a version of Gearhart-Pr\"uss on $K$-convex space.

Equations

$$f(A)=\frac{1}{2\pi i}{\int }_{\partial R}f(\lambda )R(\lambda ,A)d\lambda .$$

$$R_{\alpha }=\{z\in \mathbb{C}:Re(z)>\alpha \}.$$

$$\forall \alpha <\omega ,\sup \{\Vert R(z,A)\Vert :Re(z)\le \alpha \}<\infty .$$

$$\mathcal{E}(R_{\alpha }):=\{f\in H^{\infty }(R_{\alpha }):\exists s>0,f(z)=O(|z{|}^{-(1+s)})\text{ as }|z|\to \infty \}.$$

$$f(A):=\frac{1}{2\pi i}{\int }_{-\infty }^{\infty }f(\delta +it)R(\delta +it,A)dt$$

$$f(A)=e(A{)}^{-1}(ef)(A),$$

$$e^{-tA}:=(e^{-tz})(A).$$

$$\Vert f(A)\Vert \le C{\Vert f\Vert }_{H^{\infty }(R_{\omega })}.$$

$${\int }_{\mathbb{R}}|⟨R(\alpha +it,A{)}^{m+1}x,y⟩|dt\le \frac{C}{(\omega -\alpha {)}^m}\Vert x\Vert \Vert y\Vert$$

$$\forall \lambda \in \mathbb{C}\setminus \overline{R_0},\Vert \lambda R(\lambda ,A)\Vert \le C.$$

$$(-\alpha ){\int }_{\mathbb{R}}|⟨R(\alpha +it,A{)}^{2}x,y⟩|dt$$

$$\le {\int }_{\mathbb{R}}\frac{-C\alpha }{(-\alpha {)}^2+t^2}dt\Vert x\Vert \Vert y\Vert$$

$$=C\pi \Vert x\Vert \Vert y\Vert ,$$

$$f^{(m)}(A)\in \mathcal{L}(X)\text{ and }\Vert f^{(m)}(A)\Vert \le \frac{C}{(\omega -\alpha {)}^m}{\Vert f\Vert }_{H^{\infty }(R_{\alpha })}.$$

$$\Vert f^{(m)}(A)\Vert \le C{\Vert f^{(m)}\Vert }_{H^{\infty }(R_{\omega })}\le \frac{KC}{(\omega -\alpha {)}^m}{\Vert f\Vert }_{H^{\infty }(R_{\alpha })}.$$

$$c\left\|x\right\|\leq\sup\bigl{\{}|\langle x,y\rangle|\,:\,y\in Z,\ \left\|y\right\|\leq 1\bigr{\}}.$$

$$X^{\odot }:=\{x\in X^{\ast },\Vert T_t^{\ast }x-x\Vert \underset{t\to 0}{⟶}0\}.$$

$$\left\|x\right\|^{\prime}:=\sup\big{\{}|\langle x,x^{\odot}\rangle|,\,x^{\odot}\in X^{\odot},\,\left\|x^{\odot}\right\|\leq 1\big{\}},$$

$${\Vert x\Vert }^{\prime }\le \Vert x\Vert \le M{\Vert x\Vert }^{\prime },x\in X.$$

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

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

$${\Vert \sum _{n=1}^N{\gamma }_n\otimes T_n{x}_n\Vert }_{G(Y)}\le C{\Vert \sum _{n=1}^N{\gamma }_n\otimes x_n\Vert }_{G(X)}.$$

$$c\left\|\big{(}\sum_{n=1}^{N}|x_{n}|^{2}\big{)}^{\frac{1}{2}}\right\|_{X}\leq\left\|\sum_{n=1}^{N}\gamma_{n}\otimes x_{n}\right\|_{G(X)}\leq C\left\|\big{(}\sum_{n=1}^{N}|x_{n}|^{2}\big{)}^{\frac{1}{2}}\right\|_{X},$$

$$\left\|\big{(}\sum_{n=1}^{N}|T_{n}x_{n}|^{2}\big{)}^{\frac{1}{2}}\right\|_{Y}\leq C\left\|\big{(}\sum_{n=1}^{N}|x_{n}|^{2}\big{)}^{\frac{1}{2}}\right\|_{X}.$$

$${\Vert F(\cdot )x\Vert }_{L^1(\Omega ;X)}\le K\Vert x\Vert ,x\in X.$$

$$T_{\varphi }^{F}x={\int }_{\Omega }\varphi (s)F(s)xd\mu (s),x\in X,$$

$$\mathcal{T}^{F}_{\infty}:=\big{\{}T^{F}_{\phi}\,:\,\phi\in L^{\infty}(\Omega),\,\left\|\phi\right\|_{L^{\infty}(\Omega)}\leq C\big{\}}$$

$$\sum _{n=1}^N|⟨T_n{x}_n,y_n^{\ast }⟩|\le C{\Vert \sum _{n=1}^N{\gamma }_n\otimes x_n\Vert }_{G(X)}{\Vert \sum _{n=1}^N{\gamma }_n\otimes y_n^{\ast }\Vert }_{G(Y^{\ast })}.$$

$$\sum _{n=1}^N|⟨T_n{x}_n,y_n⟩|\le C{\Vert \sum _{n=1}^N{\gamma }_n\otimes x_n\Vert }_{G(X)}{\Vert \sum _{n=1}^N{\gamma }_n\otimes y_n\Vert }_{G(Y)}.$$

$$⟨T_G^{\varphi }x,y⟩:={\int }_{\Omega }\varphi (s)⟨G(s)x,y⟩d\mu (s),\text{for x∈X and y∈Y},$$

$$\Big{\{}T_{G}^{\phi}\,:\,\phi\in L^{1}(\Omega),\;\left\|\phi\right\|_{L^{1}(\Omega)}\leq C\Big{\}}\subset\mathcal{L}(X,Y^{*})$$

Full text

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

1.   

INTRODUCTION

The $H^{\infty}$-functional calculus for a sectorial operator and a strip-type operator have played and important role in the spectral theory and evolutions equations [6]. The $H^{\infty}$-functional calculus for a half-plane type operator is a recent tool studied in [4]. As for sectorial and strip-type operators, it is natural to construct a holomorphic functional calculus for a half-plane type operator $A$ via the Dunford formula

[TABLE]

Here, $R$ is a half-plane and $f$ is a bounded analytic function on $R$ with good properties which ensure that $f(A)$ is bounded. This construction allows to define a notion of bounded $H^{\infty}$-functional calculus for a half-plane type operator. Contrary to bounded $H^{\infty}$-functional calculus for a sectorial or a strip-type operator, the bounded $H^{\infty}$-functional calculus for a half-plane type operator has no characterization with simple estimates, even on Hilbert space. However, a weaker notions of bounded $H^{\infty}$-functional calculus, called strong $m$-bounded functional calculus (Definition 2.7) turn out to be equivalent to a condition studied independently by Gomilko [5] and Shi and Feng [18] (Definition 2.5), called GFS condition in this paper. Furthermore, they show that this condition is sufficient for the generation of bounded $C_{0}$-semigroups. Therefore, the set (containing all sectorial operators of type $<\frac{\pi}{2}$) of all half-plane type operators which have GFS is included (equal when $X$ is Hilbert) in the set of all negative generators of bounded $C_{0}$-semigroup.

The aim of this paper is to study the generation of a $\gamma$-bounded $C_{0}$-semigroups on a Banach space $X$, that is, a $C_{0}$-semigroup $(T_{t})_{t\geqslant 0}$ on a Banach space $X$ such that the set $\{T(t)\,:\,t\geq 0\}$ is $\gamma$-bounded. A first step is to consider a stronger condition than the GFS condition (however equivalent on Hilbert spaces), that we call weak $\gamma$-Gomilko-Shi-Feng (Definition 3.4) and abreviate as the $W\gamma$-GFS condition. It turns out that the $W\gamma$-GFS condition is equivalent to a notion of $\gamma$-bounded strong $m$-bounded functional calculus. Furthermore the latter condition is sufficient for the generation of weak $\gamma$-bounded $C_{0}$-semigroups, and hence for the generation of $\gamma$-bounded $C_{0}$-semigroups when the underlying space is $K$-convex. Therefore when $X$ is $K$-convex, the set (containing all $\gamma$-sectorial operators of $\gamma$-type $<\frac{\pi}{2}$) of all operators which satisfy the $W\gamma$-GFS condition is equal to the set of all negative generators of $\gamma$-bounded $C_{0}$-semigroups. This last statement is contained in our main result (Theorem 4.1), which gives equivalence of the generation of $\gamma$-bounded $C_{0}$-semigroup with the $\gamma$-m-bounded functional calculus, as well as with some estimates of the resolvent of the negative generator and its adjoint. From this theorem, we deduce, when $X$ is $K$-convex space, a $\gamma$-bounded version of the Gearhart-Prüss Theorem (Corollary 5.3) and the following result: if $(T_{t})_{t\geq 0}$ is a bounded $C_{0}$-semigroup on a $K$-convex space and if the set $\{e^{-\delta t}T_{t}\,:\,t\geq 0\}$ is $\gamma$-bounded for one $\delta>0$, then it is $\gamma$-bounded for each $\delta>0$. Moreover it is possible to find an example of bounded $C_{0}$-semigroup such that $\{e^{-\delta t}T_{t}\,:\,t\geq 0\}$ is $\gamma$-bounded for all $\delta>0$ but not for $\delta=0$.

Now we describe the structure of the paper. Section 2 only contains preliminary results. We recall important results of [4], then we collect some results about $\gamma$-boundedness and weak $\gamma$-boundedness and generalized square functions. In Section 3, we discuss the $W\gamma$-GFS condition and strong $\gamma$-$m$-bounded $H^{\infty}$-functional calculus. We give our main result in Section 4, namely a version of Gomilko-Shi-Feng Theorem on $K$-convex spaces. Section 5 is devoted to some consequences of the results of Section 4, in particular we state a version of the Gearhart-Prüss Theorem in $K$-convex spaces. Finally, Section 6 is dedicated to an overview of the implications between the different notions of $H^{\infty}$-bounded functional calculus for half-plane type operator and the generation of bounded and $\gamma$-bounded $C_{0}$-semigroups.

2.   

BACKGROUND AND PRELIMINARY RESULTS

For any Banach spaces $X,Y$, we let $\mathcal{L}(X,Y)$ denote the space of all bounded linear operators from $X$ into $Y$. If $Y=X$, we write $\mathcal{L}(X)$ instead of $\mathcal{L}(X,X)$. If $A$ is a closed operator on $X$, we denote by $Dom(A)$, $\rho(A)$ and $\sigma(A)$ the domain, the resolvent set and the spectrum of $A$, respectively. When $\lambda\in\rho(A)$, we let $R(\lambda,A)=(\lambda I-A)^{-1}$ denote the corresponding resolvent operator.

2.1.   Half-plane type operator.

In this subsection we review the definitions of half-plane type operators and their functional calculi, following [4].

We fix a real number $\omega$. For any $\alpha\leq\omega$, we consider the right open half-plane

[TABLE]

Definition 2.1**.**

Let $A$ be a closed and densely defined operator on $X$. We will say that $A$ is of half-plane type $\omega$ if $\sigma(A)\subset\overline{R_{\omega}}$ and

[TABLE]

We will say that a $C_{0}$-semigroup $(T_{t})_{t\geq 0}$ is of type $\omega\in\mathbb{R}$ if there exists a constant $C>0$ such that $\left\|T_{t}\right\|\leq Ce^{-\omega t}$ for all $t\geq 0$. Note that such $\omega$ always exists [17, Theorem 2.2 section 1.2]. It follows from the Laplace formula (see e.g. [17, Formula (7.1) section 1.7]) that if $-A$ generates a $C_{0}$-semigroup $(T_{t})_{t\geq 0}$ of type $\omega$, then $A$ is an operator of half-plane type $\omega$ (in fact $A$ is even strong half-plane in the sense of [4, Definition 2.1]).

Throughout the rest of this subsection, we let $A$ be an operator of half-plane type $\omega$.

For any $\alpha\leq\omega$, let $H^{\infty}(R_{\alpha})$ be the space of all bounded analytic functions $f:R_{\alpha}\to\mathbb{C}$, equipped with the norm $\left\|f\right\|_{H^{\infty}(R_{\alpha})}:=\underset{z\in R_{\alpha}}{\sup}|f(z)|$. Then $H^{\infty}(R_{\alpha})$ is a Banach algebra.

Next we consider the auxiliary space

[TABLE]

Whenever $\alpha<\delta<\omega$ and $f\in\mathcal{E}(R_{\alpha})$, the integral

[TABLE]

is absolutely convergent in $\mathcal{L}(X)$. Further its value is independent of $\delta\in(\alpha,\omega)$. This is due to Cauchy’s theorem for vector-valued holomorphic functions.

If $f\in H^{\infty}(R_{\alpha})$, we can define a closed, densely defined, operator $f(A)$ by regularisation as follows (see [4] and [6] for more details). Let $\mu<\alpha$ and set $e(z):=(\mu-z)^{-2}$. Then $e\in\mathcal{E}(R_{\alpha})$, $ef\in\mathcal{E}(R_{\alpha})$ and $e(A)=R(\mu,A)^{2}$ is injective. Then $f(A)$ is defined by

[TABLE]

with $Dom(f(A))$ equal to the space of all $x\in X$ such that $[(ef)(A)](x)$ belongs to the range of $e(A)$ (= $Dom(A^{2})$). It turns out that this definition does not depend on the choice of $\mu$.

In particular, for any $t\geq 0$, the function $z\mapsto e^{-tz}$ belongs to $H^{\infty}(R_{\alpha})$, hence the above construction provides an operator

[TABLE]

The following proposition (see [4, Proposition 2.5]) gives an expected link between the generator of a $C_{0}$-semigroup and its exponential in the previous sense.

Proposition 2.2**.**

The operator $-A$ is the generator of a $C_{0}$-semigroup $(T_{t})_{t\geq 0}$ if and only if $e^{-tA}$ is a bounded operator for all $t\in[0,1]$ and $\underset{t\in[0,1]}{\sup}\left\|e^{-tA}\right\|<\infty$. In this case, we have $T_{t}=e^{-tA}$ for all $t\geq 0$.

Definition 2.3**.**

We say that $A$ has a bounded $H^{\infty}$-functional calculus of type $\omega$ if there exists $C>0$ such that for any $\alpha<\omega$ and for any $f\in H^{\infty}(R_{\alpha})$, $f(A)$ belongs to $\mathcal{L}(X)$ and

[TABLE]

Remark 2.4*.*

The reasoning at the beginning of [4, Section 5] and the so-called convergence lemma (see [4, Theorem 3.1]) show that to prove that an operator $A$ has a bounded $H^{\infty}$-functional calculus of type $\omega$, it suffices to prove an estimate (2) for any $f\in{\mathcal{E}}(R_{\alpha})$ and any $\alpha<\omega$.

As a direct consequence of Proposition 2.2, we see that if $A$ has a bounded $H^{\infty}$-functional calculus of type $\omega$, then $-A$ generates a $C_{0}$-semigroup of type $\omega$. The converse does not hold true, even on Hilbert space (see Section 6).

The following condition, called Gomilko-Shi-Feng condition (GFS), is important to connect $C_{0}$-semigroups and functional calculi.

Definition 2.5**.**

Let $m\geq 1$ be an integer. We say that $A$ satisfies $(GFS)_{m,\omega}$ if there exists a constant $C>0$ such that

[TABLE]

for any $x\in X$ and any $y\in X^{*}$.

Gomilko [5] and Shi-Feng [18] have shown the following two results: if $A$ has $(GFS)_{1,0}$, then $-A$ generates a bounded $C_{0}$-semigroup $(T_{t})_{t\geq 0}$; conversely if $X$ is a Hilbert space, the negative generator of a bounded $C_{0}$-semigroup satisfies $(GFS)_{1,0}$. This is now known as the Gomilko-Shi-Feng Theorem.

Remark 2.6*.*

It is easy to check that a sectorial operator of type $<\frac{\pi}{2}$ has $(GFS)_{1,0}$. We refer e.g. to [6] for information about sectorial operators. We recall that $A$ is a sectorial operator of type $<\frac{\pi}{2}$ if and only if $-A$ generates a bounded analytic semigroup and that in this case, there exists a constant $C>0$ such that

[TABLE]

This implies that for any $\alpha<0$ and any $t\in\mathbb{R}$, $\left\|R(\alpha+it,B)\right\|\leq\frac{C}{|\alpha+it|}$. Hence for any $\alpha<0$ and arbitrary $x\in X$ and $y\in X^{*}$, we have

[TABLE]

which proves the result.

It is noticed in [4, Lemma 5.4] that for any $\alpha<\beta<\omega$ and any $f\in H^{\infty}(R_{\alpha})$, the $m$-th derivative $f^{(m)}$ of $f$ belongs to $H^{\infty}(R_{\beta})$ for any integer $m\geq 1$. Thus it makes sense to define $f^{(m)}(A)$.

Definition 2.7**.**

Let $m\geq 1$ be an integer. We say that $A$ has a strong $m$-bounded functional calculus of type $\omega$ if there exists a constant $C>0$ such that for each $\alpha<\omega$ and each $f\in H^{\infty}(R_{\alpha})$,

[TABLE]

The following remarkable results are proved in [4, Theorem 6.4]: $A$ has $(GFS)_{m,\omega}$ if and only if $A$ has a strong $m$-bounded functional calculus of type $\omega$, if and only if $A$ has a strong $1$-bounded functional calculus of type $\omega$. (In particular, $(GFS)_{m,\omega}$ does not depend on $m$.) Further if $A$ has a strong $m$-bounded functional calculus of type $\omega$, then $-A$ generates a $C_{0}$-semigroup of type $\omega$. It is further shown in [4, Theorem 7.1] that if $X$ is a Hilbert space, then conversely, $A$ has a strong $m$-bounded functional calculus of type $\omega$ if $-A$ generates a $C_{0}$-semigroup of type $\omega$. This converse is wrong in general, see Section 6 for more on this.

The following is implicit in [4].

Proposition 2.8**.**

If $A$ has a bounded $H^{\infty}$-functional calculus of type $\omega$, then $A$ has a strong $m$-bounded functional calculus of type $\omega$, for any $m\geq 1$.

Proof.

According to [4, Lemma 5.4], there is a constant $K>0$, such that for any $\alpha<\omega$ and any $f\in H^{\infty}({R_{\alpha}})$, we have $\left\|f^{(m)}\right\|_{H^{\infty}(R_{\omega})}\leq\frac{K}{(\omega-\alpha)^{m}}\left\|f\right\|_{H^{\infty}(R_{\alpha})}$.

Assume that $A$ has a bounded $H^{\infty}$-functional calculus of type $\omega$. Then for any $f$ as above, $f^{(m)}(A)\in{\mathcal{L}}(X)$ and we have

[TABLE]

This proves that $A$ has a strong $m$-bounded functional calculus of type $\omega$. ∎

2.2.   The Sun Dual of a $C_{0}$-Semigroup

In this subsection we collect a few facts from [16] which are useful when dealing with non reflexive Banach spaces.

Let $X$ be a Banach space and let $Z\subset X^{*}$ be a closed subspace. We say that $Z$ is norming if there exists a constant $c>0$ such that for any $x\in X$,

[TABLE]

Let $(T_{t})_{t\geq 0}$ be a $C_{0}$-semigroup on $X$, with generator $-A$. It may happen that the dual semigroup $(T_{t}^{*})_{t\geq 0}$ is not strongly continuous on $X^{*}$. We denote by $X^{\odot}$ (pronounced $X$-sun) the set

[TABLE]

This set trivially satisfies $T^{*}_{t}(X^{\odot})\subset X^{\odot}$ for every $t\geq 0$. Moreover $X^{\odot}$ is a closed and weak*-dense subspace of $X^{*}$. Indeed we have $X^{\odot}=\overline{D(A^{*})}$.

We let $T^{\odot}_{t}$ denote the restriction of $T^{*}_{t}$ to $X^{\odot}$. Then by definition, $(T^{\odot}_{t})_{t\geq 0}$ is a $C_{0}$-semigroup on $X^{\odot}$. It is called the sun dual of $(T_{t})_{t\geq 0}$. Let $A^{\odot}$ be its negative generator. Then $A^{\odot}$* is the part of $A^{*}$ in $X^{\odot}$* (see [6, page 6] for definition).

We will use the following two results.

Theorem 2.9**.**

Let $(T_{t})_{t\geq 0}$ be a $C_{0}$-semigroup on $X$, with generator $-A$.

• (1)

We have $\rho(A)=\rho(A^{*})=\rho(A^{\odot})$ and $R(\lambda,A)^{*}y=R(\lambda,A^{*})y=R(\lambda,A^{\odot})y$ for all $\lambda\in\rho(A)$ and for any $y\in X^{\odot}$.

• (2)

The space $X^{\odot}$ is norming.

About (2), we note that more precisely, if we define

[TABLE]

and if we let $M:=\varlimsup_{t\rightarrow 0}\left\|T(t)\right\|$, then we have

[TABLE]

2.3.   $\gamma$-boundedness on Banach spaces.

In recent years, $\gamma$-boundedness and $\mathcal{R}$-boundedness played an important role in the operator valued harmonic analysis, multiplier theory and functional calculi (see [10] for more details). Throughout $X,Y$ denote arbitrary 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

[TABLE]

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

[TABLE]

denote the induced norm.

Definition 2.10**.**

Let $\mathcal{T}\subset\mathcal{L}(X,Y)$ 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:

[TABLE]

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 $\gamma(\mathcal{T})=\infty$.

Replacing the sequence $(\gamma_{k})_{k\geq 1}$ by a sequence of independent Rademacher variables $(\epsilon_{k})_{k\geq 1}$ in the above definition, we obtain the definition of $\mathcal{R}$-boundedness. It is well known that any $\mathcal{R}$-bounded set is $\gamma$-bounded, and that these notions are equivalent when $X$ has finite cotype (see [10, theorem 8.6.4] for a more general result). Furthermore if $X$ has cotype $2$ and $Y$ has type 2 (especially when $X=Y$ is an Hilbert space) then $\mathcal{R}$-boundedness, $\gamma$-boundedness and uniform boundedness are equivalent.

Assume that $X$ and $Y$ are Banach lattices with finite cotype. By the Khintchine-Maurey inequality [10, Theorem 7.2.13], there exist $c,C>0$ such that for every $x_{1},\ldots,x_{n}\in X$,

[TABLE]

and $Y$ satisfies a similar property. Hence a set $\mathcal{T}\subset\mathcal{L}(X,Y)$ is $\gamma$-bounded if and only if there exists $C\geq 0$ such that for all finite sequences $(T_{n})_{n=1}^{N}\subset\mathcal{T}$ and $(x_{n})_{n=1}^{N}\subset X$,

[TABLE]

We recall for further use that $\gamma$-boundedness is stable under the strong operator topology.

Proposition 2.11**.**

Let $\mathcal{T}\subset\mathcal{L}(X,Y)$ be a $\gamma$-bounded set. Then the closure $\overline{\mathcal{T}}^{so}$ of $\mathcal{T}$ in the strong operator topology is $\gamma$-bounded with $\gamma(\overline{\mathcal{T}}^{so})=\gamma(\mathcal{T})$.

We will need the following lemma, for which we refer to [10, Theorem 8.5.4].

Lemma 2.12** ($L^{\infty}$-integral means).**

Let $(\Omega,\mu)$ be a measure space and let $F:\Omega\rightarrow\mathcal{L}(X,Y)$ be an operator-valued function. Assume that $F(\cdot)x$ belongs to $L^{1}(\Omega;Y)$ for any $x\in X$ and that there exists a constant $K>0$ such that

[TABLE]

Then, for any $\phi\in L^{\infty}(\Omega)$, we can define a bounded operator $T^{F}_{\phi}\in\mathcal{L}(X,Y)$ by

[TABLE]

and for any $C>0$, the set

[TABLE]

is $\gamma$-bounded.

The adjoint set of a $\gamma$-bounded set may not be $\gamma$-bounded ([10, example 8.4.2]). Following [11], we introduce weaker notions to circumvent this difficulty.

Definition 2.13**.**

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

[TABLE] 2. 2)

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

[TABLE]

If $\mathcal{T}$ is $W\gamma$-bounded (respectively $W^{*}\gamma$-bounded) then the adjoint set $\mathcal{T}^{*}$ is $W\gamma$-bounded (respectively $W^{*}\gamma$-bounded) (see [9, Lemma 2.4]). It is clear that $\gamma$-boundedness implies $W\gamma$-boundedness, however the converse is false in general. Indeed take a $\gamma$-bounded set $\mathcal{T}$ such that $\mathcal{T}^{*}$ is not $\gamma$-bounded, then as $\mathcal{T}$ is $W\gamma$-bounded, $\mathcal{T}^{*}$ is also $W\gamma$-bounded but not $\gamma$-bounded.

Lemma 2.14** (weak $L^{1}$-integral means).**

Let $(\Omega,\mu)$ be a measure space, let $\mathcal{T}\subset\mathcal{L}(X,Y^{*})$ be a $W^{*}\gamma$-bounded set and let $G:\Omega\longrightarrow\mathcal{L}(X,Y^{*})$ be an operator-valued function such that $G$ takes values in $\mathcal{T}$ and for all $x\in X$ and $y\in Y$, the scalar function $s\longmapsto\langle G(s)x,y\rangle$ is measurable.

Then for any $\phi\in L^{1}(\Omega)$, one can define a bounded operator $T_{G}^{\phi}\in\mathcal{L}(X,Y^{*})$ by

[TABLE]

and for any $C>0$, the set

[TABLE]

is $W^{*}\gamma$-bounded.

Proof.

Let $\sigma$ be the topology on $\mathcal{L}(X,Y^{*})$ generated by the family of seminorms $\rho_{x,y}$ defined by $\rho_{x,y}(T)=|\langle T(x),y\rangle|$, for any $x\in X$ and $y\in Y$. The topology $\sigma$ is called weak∗ operator topology on $\mathcal{L}(X,Y^{*})$. It is clear that if $\mathcal{T}\subset\mathcal{L}(X,Y^{*})$ is $W^{*}\gamma$-bounded, then its closure $\overline{\mathcal{T}}^{\sigma}$ is also $W^{*}\gamma$-bounded. Moreover if $\mathcal{T}\subset\mathcal{L}(X,Y^{*})$ is $W^{*}\gamma$-bounded then the absolute convex hull of $\mathcal{T}$ is $W^{*}\gamma$-bounded as well.

With these two facts in hand, one can obtain the result by mimicking the proof of [10, Theorem 8.5.2]. Details are left to the reader. ∎

We will need the notion of $K$-convexity, for which we refer to [14] or [10]. We recall that a Banach space $X$ is $K$-convex if and only if there exists a constant $K>0$ such that for all $x_{1},\ldots,x_{N}\in X$ the following inequality holds:

[TABLE]

It turns out that $X$ is $K$-convex if and only if $X^{*}$ is $K$-convex. If this is the case, then according to [10, Corollary 7.4.6], there exists a constant $K^{\prime}>0$ such that for all $y_{1},\ldots,y_{n}\in X^{*}$,

[TABLE]

We recall that all UMD spaces are $K$-convex. In particular, $L^{p}$-spaces are $K$-convex for any $1<p<\infty$. Further any closed subspace of a $K$-convex space is $K$-convex. We also recall that any $K$-convex space has a finite cotype. In particular, a $K$-convex Banach space cannot contain $c_{0}$. A fundamental result on $K$-convexity is Pisier’s Theorem [10, Theorem 7.4.23] which asserts that $X$ is $K$-convex if and only if $X$ has non-trivial type.

We note that there exist non reflexive $K$-convex Banach spaces. It readily follows from (8) that if $Y$ is a $K$-convex space, then a set $\mathcal{T}\subset\mathcal{L}(X,Y)$ is $W\gamma$-bounded if and only if it is $\gamma$-bounded. Likewise using (9), we obtain that if $Y$ is $K$-convex, then a set $\mathcal{T}\subset\mathcal{L}(X,Y^{*})$ is $W^{*}\gamma$-bounded if and only if it is $\gamma$-bounded.

We now turn to the definition of $\gamma$-spaces, which play a fundamental role in this paper. Let $H$ be a Hilbert space. A linear operator $T:H\rightarrow X$ is called $\gamma$-summing if

[TABLE]

where the supremum is taken over all finite orthonormal system $\{h_{1},...,h_{n}\}$ in $H$. We let $\gamma_{\infty}(H;X)$ denote the space of all $\gamma$-summing operator 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 $X$. The spaces $\gamma_{\infty}(H;X)$ and $\gamma(H;X)$ do not coincide in general [10, Example 9.1.21] but when $X$ does not contain a copy of $c_{0}$ (in particular when $X$ is $K$-convex) then these spaces coincide.

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 opertor $\mathbb{I}_{f}:L^{2}(S)\to X$, given by

[TABLE]

where this integral is defined in the Pettis sense.

We let $\gamma(S;X)$ be the space of all measurable and weakly $L^{2}$ functions $f:S\rightarrow X$ such that $\mathbb{I}_{f}$ belongs to $\gamma(L^{2}(S);X)$. We endow it with $\left\|f\right\|_{\gamma(S;X)}:=\left\|\mathbb{I}_{f}\right\|_{\gamma(L^{2}(S);X)}$. A remarkable fact is the density of simple function in the set $\gamma(S;X)$ [10, Proposition 9.2.5].

Now we collect some important results, which will be useful in the next sections. We start with the so-called Multiplier Theorem [10, Theorem 9.5.1], a high ranking result involving the $\gamma$-boundedness. We state it under the assumption that $X$ does not contain $c_{0}$. Thus the following statement applies to $K$-convex spaces.

Theorem 2.15** ($\gamma$-Multiplier theorem).**

Let $X$ be a Banach space not containing $c_{0}$. Let $M:S\rightarrow\mathcal{L}(X)$ be a strongly mesurable function and assume that its range $\mathcal{M}:=\{M(s):s\in S\}$ is $\gamma$-bounded. Then for every function $\psi:S\rightarrow X$ in $\gamma(S;X)$, the function $M\psi:S\rightarrow X$ belongs to $\gamma(S;X)$, and we have

[TABLE]

The next result is an inequality of Hölder type [10, Theorem 9.2.14 (1)].

Theorem 2.16** ($\gamma$-Hölder inequality).**

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

[TABLE]

Now we give an extension result, for which we refer to [10, Theorem 9.6.1]. We identify the algebraic tensor product $H^{*}\otimes X$ with the space of finite rank bounded operator operators from $H$ into $X$ in the usual way, that is, we set $(h^{*}\otimes x)(h)=h^{*}(h)x$ for any $h\in H,\,h^{*}\in H^{*}$ and $x\in X$.

Theorem 2.17**.**

[Extension theorem] Let $H_{1}$ and $H_{2}$ be Hilbert spaces. For any bounded operator $U:H_{1}^{*}\rightarrow H_{2}^{*}$, the mapping

[TABLE]

taking $h^{*}\otimes x$ to $U(h^{*})\otimes x$ for any $h^{*}\in H_{1}^{*}$ and $x\in X$, has an unique extension to a bounded linear operator $\tilde{U}:\gamma(H_{1};X)\rightarrow\gamma(H_{2};X)$ of the same norm. Furthermore for all $T\in\gamma(H_{1};X)$,

[TABLE]

where ${}^{t}U$ denotes the Banach space adjoint of $U$.

To conclude this part, we apply the above principles to the Fourier-Plancherel transform $L^{2}(\mathbb{R})\to L^{2}(\mathbb{R})$. We identify the dual of $L^{2}(\mathbb{R})$ with $L^{2}(\mathbb{R})$ via the usual duality map provided by integration on $\mathbb{R}$.

Lemma 2.18**.**

For any $f\in L^{1}(\mathbb{R};X)$, let $\hat{f}$ be its Fourier transform defined by

[TABLE]

Let $\mathcal{F}:L^{2}(\mathbb{R})\rightarrow L^{2}(\mathbb{R})$ be the Fourier-Plancherel transform (which coincides with $f\mapsto\hat{f}$ on $L^{1}(\mathbb{R})\cap L^{2}(\mathbb{R})$). Let $\tilde{\mathcal{F}}:\gamma(L^{2}(\mathbb{R});X)\rightarrow\gamma(L^{2}(\mathbb{R});X)$ be its extension provided by Theorem 2.17. If $f\in\gamma(\mathbb{R};X)\cap L^{1}(\mathbb{R};X)$, then we have

[TABLE]

and further,

[TABLE]

Proof.

Obviously $\widehat{f}$ is measurable, and as $f$ is weakly $L^{2}$, $\widehat{f}$ is also weakly $L^{2}$. Indeed for $x\in X^{*}$ one has by Fourier-Plancherel theorem :

[TABLE]

It follows that $\mathbb{I}_{\widehat{f}}$ is well defined and bounded. Now let $g\in L^{2}(\mathbb{R})\cap L^{1}(\mathbb{R})$. Then by Fubini theorem, using equality ${}^{t}\mathcal{F}=\mathcal{F}$ and (10):

[TABLE]

By density and since $\tilde{\mathcal{F}}(\mathbb{I}_{f})$ and $\mathbb{I}_{\widehat{f}}$ are bounded, the equality $\tilde{\mathcal{F}}(\mathbb{I}_{f})=\mathbb{I}_{\widehat{f}}$ follows. Hence $\widehat{f}\in\gamma(\mathbb{R};X)$.

Finally since $(\sqrt{2\pi})^{-1}\mathcal{F}$ is an isometry, extension principle yields the equalities

[TABLE]

∎

3.   

STRONG $\gamma$-m-BOUNDED FUNCTIONAL CALCULUS

We will say that a $C_{0}$-semigroup $(T_{t})_{t\geq 0}$ on Banach space is of $\gamma$-type $\omega$ (resp. of $W\gamma$-type $\omega$) if the set $\{e^{\omega t}T_{t}\,:\,t\geq 0\}$ is $\gamma$-bounded (resp. $W\gamma$-bounded). If no such $\omega$ exists, we will say that $(T_{t})_{t\geq 0}$ has no $\gamma$-type (resp. no $W\gamma$-type).

Example 3.1*.*

It is easy to exhibit $C_{0}$-semigroups with no $\gamma$-type. Let $1\leq p<\infty$ and let $(S_{t})_{t\geq 0}$ be the right translation $C_{0}$-semigroup on $L^{p}(\mathbb{R})$, defined by $S_{t}(f)=f(\cdot-t)$ for any $t\geq 0$ and any $f\in L^{p}(\mathbb{R})$. Then, for $p\not=2$, $(S_{t})_{t\geq 0}$ has no $\gamma$-type. This follows from the well-known fact that $\{S_{t},\;t\in[0,1]\}$ is not $\gamma$-bounded. Indeed, assume that $p\in[1,2)$ and for $n\in\mathbb{N}$ and $i\in\{1,\ldots,n\}$, let $t_{i}^{n}=\frac{i-1}{n}$ and $f^{n}_{i}=\chi_{[0,1/n]}$. Then we have

[TABLE]

whereas

[TABLE]

Hence the inequality (5) cannot be true. The proof in the case $p\in(2,\infty)$ is similar.

The fact that the set $\{S_{t}\,:\,t\in[0,1]\}$ is not $\gamma$-bounded immediately implies that for any $\omega$, the set $\{e^{\omega t}S_{t}\,:\,t\in[0,1]\}$ is not $\gamma$-bounded, and hence $\{e^{\omega t}S_{t}\,:\,t\geq 0\}$ cannot be $\gamma$-bounded.

Throughout this section, we let $X$ be a Banach space. Then we let $A$ be a half-plane type operator of type $\omega$ on $X$.

The condition that $A$ has a bounded $H^{\infty}$-functional calculus of type $\omega$ can be rephrased by saying that the set

[TABLE]

is uniformly bounded. This formulation and Definition 2.7 motivate the following definitions.

Definition 3.2**.**

We say that $A$ has a $\gamma$-bounded (resp. a $W\gamma$-bounded) $H^{\infty}$-functional calculus of type $\omega$ if $A$ has a bounded $H^{\infty}$-functional calculus of type $\omega$ and the set

[TABLE]

is $\gamma$-bounded (resp. $W\gamma$-bounded). 2. 2)

Let $m\geq 1$ be an integer. We say that $A$ has a strong $\gamma$-$m$-bounded (resp. a strong $W\gamma$-$m$-bounded) functional calculus of type $\omega$ if $A$ has a strong $m$-bounded functional calculus of type $\omega$ and the set

[TABLE]

is $\gamma$-bounded (resp. $W\gamma$-bounded).

Remark 3.3*.*

To prove that the set

[TABLE]

it is enough to prove that

[TABLE]

This follows from the convergence lemma [4, Theorem 3.1], the argument in the proof of [4, Theorem 5.6 (a)], and Proposition 2.11. Details are left to the reader. 2. 2)

Likewise to prove that $A$ has a $\gamma$-bounded $H^{\infty}$-functional calculus of type $\omega$, it is enough to prove that the set

[TABLE]

is $\gamma$-bounded. 3. 3)

If $A$ has a $\gamma$-bounded $H^{\infty}$-functional calculus of type $\omega$, then it has a strong $\gamma$-$m$-bounded functional calculus of type $\omega$. Indeed consider the set

[TABLE]

and assume that $\Delta$ is $\gamma$-bounded. Let $\alpha<\omega$ and let $f\in H^{\infty}(R_{\alpha})$ with $\left\|f\right\|_{H^{\infty}(R_{\alpha})}\leq 1$. According to [4, Lemma 5.4], $f^{(m)}\in H^{\infty}(R_{\beta})$ for any $\beta\in(\alpha,\omega)$ and

[TABLE]

Consequently,

[TABLE]

Hence the above set is $\gamma$-bounded, which shows that $A$ has a strong $\gamma$-$m$-bounded functional calculus of type $\omega$.

The previous three statements hold as well with $W\gamma$-boundedness replacing $\gamma$-boundedness.

Recall the condition $(GFS)_{m,\omega}$ from Definition 2.5. We introduce the following stronger form.

Definition 3.4**.**

Let $m\geq 1$ be an integer. We say that $A$ has property $(W\gamma\text{-}GFS)_{m,\omega}$ if there exists a constant $C>0$ such that for any $N\in\mathbb{N}$, for any $\alpha_{1},\ldots,\alpha_{N}<\omega$, and for any $x_{1},\ldots,x_{N}\in X$ and $y_{1},\ldots,y_{N}\in X^{*}$, we have

[TABLE]

Clearly, $(W\gamma\text{-}GFS)_{m,\omega}$ implies $(GFS)_{m,\omega}$. Further if $X$ is a Hilbert space, then $(GFS)_{m,\omega}$ and $(W\gamma\text{-}GFS)_{m,\omega}$ are equivalent.

In the sequel we let $\mathbb{D}=\{z\in\mathbb{C}\,:\,|z|<1\}$ denote the open unit disc. The following statement is straightforward.

Lemma 3.5**.**

Let $m\geq 1$ be an integer. Assume that $A$ has property $(GFS)_{m,\omega}$. For any measurable function $\epsilon:\mathbb{R}\rightarrow\overline{\mathbb{D}}$ and for any $\alpha<\omega$, let

[TABLE]

denote the operator defined by

[TABLE]

Then, the operator $A$ has property $(W\gamma\text{-}GFS)_{m,\omega}$ if and only if the set

[TABLE]

is $W^{*}\gamma$-bounded.

Remark 3.6*.*

Arguing as in Remark 2.6, one shows that if $A$ is $\gamma$-sectorial of $\gamma$-type $<\frac{\pi}{2}$, then $A$ has property $W\gamma$-$(GFS)_{1,0}$. (We refer e.g. to [10] for information on $\gamma$-sectorial operators.)

Indeed assume that $A$ is $\gamma$-sectorial of $\gamma$-type $<\frac{\pi}{2}$. Then the set

[TABLE]

is $\gamma$-bounded. Next for any measurable $\epsilon:\mathbb{R}\rightarrow\overline{\mathbb{D}}$, we can write

[TABLE]

Since $\left\|t\mapsto(-\alpha)(\alpha+it)^{-2}\epsilon(t)\right\|_{L^{1}}\leq\pi$, Lemma 2.12 ensures that the set

[TABLE]

is $W^{*}\gamma$-bounded. Hence the result follows from the above Lemma 3.5.

Proposition 3.7**.**

Let $m\geq 1$ be an integer and assume that $A$ has property $(W\gamma\text{-}GFS)_{m,\omega}$. Then for any integer $1\leq p\leq m$, $A$ has property $(W\gamma\text{-}GFS)_{p,\omega}$.

Proof.

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

Suppose that $A$ has property $(W\gamma\text{-}GFS)_{m,\omega}$, with $m\geq 2$. Let $\epsilon:\mathbb{R}\rightarrow\overline{\mathbb{D}}$ be a measurable function and let $\alpha<\omega$. Applying [4, Proposition 6.3. (a)], we have

[TABLE]

for any $t\in\mathbb{R}$. Hence for any $x\in X$, for any $y\in X^{*}$ and for any $t\in\mathbb{R}$,

[TABLE]

We now integrate over $t$. Property $(GFS)_{m,\omega}$ ensures that we can apply Fubini’s theorem in the following computation:

[TABLE]

Let $\mathcal{T}$ be the set (12). By assumption, $\mathcal{T}$ is $W^{*}\gamma$-bounded hence by Lemma 2.14, the set

[TABLE]

is $W^{*}\gamma$-bounded. Since

[TABLE]

the above calculation shows that the set

[TABLE]

is included in $\Gamma$, hence is $W^{*}\gamma$-bounded. Hence, by Lemma 3.5, the operator $A$ has property $(W\gamma\text{-}GFS)_{m-1,\omega}$. ∎

We recalled that $(GFS)_{m,\omega}$ is equivalent to strong $m$-bounded functional calculus of type $\omega.$ The following theorem provides a similar statement in the context of $W\gamma$-boundedness.

Theorem 3.8**.**

The following assertions are equivalent for $m\geq 1$:

1. (i)

$A$* has $(W\gamma\text{-}GFS)_{m,\omega}$;* 2. (ii)

$A$* has a strong $W\gamma$-$1$-bounded functional calculus of type $\omega$;* 3. (iii)

$A$* has a strong $W\gamma$-$m$-bounded functional calculus of type $\omega$.*

Moreover, if $A$ satisfies these conditions, then $-A$ generates a $C_{0}$-semigroup of $W\gamma$-type $\omega$.

Proof.

$(i)\Rightarrow(ii)$: First, by Proposition 3.7, $A$ has $(W\gamma\text{-}GFS)_{1,\omega}$.

Let $\alpha_{1},\ldots,\alpha_{N}<\omega$, $\delta_{1}<\alpha_{1},\ldots,\delta_{N}<\alpha_{N}$, and $f_{1}\in\mathcal{E}(R_{\delta_{1}}),\ldots,f_{N}\in\mathcal{E}(R_{\delta_{N}})$ with $\left\|f_{k}\right\|_{H^{\infty}(R_{\alpha_{k}})}\leq 1$. Let $x_{1},\ldots,x_{N}\in X$ and $y_{1},\ldots,y_{N}\in X^{*}$. It follows from the proof of [4, Theorem 5.6 (a)]) that for any $k=1,\ldots,N$,

[TABLE]

Consequently,

[TABLE]

Since $\left\|f_{k}\right\|_{H^{\infty}(R_{\alpha_{k}})}\leq 1$ and $A$ has $(W\gamma\text{-}GFS)_{1,\omega}$, this yields an estimate

[TABLE]

According to Remark 3.3 (1), this implies that $A$ has a strong $W\gamma$-1-bounded functional calculus of type $\omega$.

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

[TABLE]

is $W\gamma$-bounded. For any $\alpha<\beta<\omega$ and $f\in H^{\infty}(R_{\alpha})$ with $\left\|f\right\|_{H^{\infty}(R_{\alpha})}\leq 1$, we have

[TABLE]

by [4, Lemma 5.4]. Hence

[TABLE]

Taking $\beta=\frac{\alpha+\omega}{2}$ in the above set, we obtain

[TABLE]

Hence the above is $W\gamma$-bounded. Thus $A$ has a strong $W\gamma$-$m$-bounded functional calculus of type $\omega$.

$(iii)\Rightarrow(i)$: Let $\alpha_{1},\ldots,\alpha_{N}<\omega$, $x_{1},\ldots,x_{N}\in X$ and $y_{1},\ldots,y_{N}\in X^{*}$. Arguing as in the proof of [4, Theorem 5.6 (b)], we consider $\beta_{k}\in(\alpha_{k},w)$ for any $k=1,\ldots,N$ and introduce measurable functions $\epsilon_{1},...,\epsilon_{N}:\mathbb{R}\to\overline{\mathbb{D}}$ such that

[TABLE]

for all $k=1,\ldots,N$ and all $t\in\mathbb{R}$. Next for any $R>0$ and any $Re(z)>\beta_{k}$, we set

[TABLE]

Since $R$ is finite, it is easy to show that $\phi_{k,R}(z)=O(|z|^{-2})$ as $|z|\rightarrow\infty$, and hence $\phi_{k,R}\in\mathcal{E}(R_{\alpha})$. Furthermore, it follows from the proof of [4, Theorem 5.6 (b)] that $\left\|\phi_{k,R}\right\|_{H^{\infty}(R_{\beta_{k}})}\leq 1$ and

[TABLE]

It therefore follows from $(iii)$ that we have an estimate

[TABLE]

Passing to the limit when $R\to\infty$, one obtains

[TABLE]

Now we choose $\beta_{k}=\frac{\omega+\alpha_{k}}{2}$ in the above estimate. We obtain the following inequality

[TABLE]

This shows that $A$ has $(W\gamma\text{-}GFS)_{m+1,\omega}$. Then by Proposition 3.7, $A$ has $(W\gamma\text{-}GFS)_{m,\omega}$.

Finally, assume that $(ii)$ holds true. In particular $A$ has a strong $m$-1-bounded functional calculus of type $\omega$ hence by Theorem [4, Theorem 6.4], $-A$ generates a $C_{0}$-semigroup $(T_{t})_{t\geq 0}$ of type $\omega$.

For any $t\geq 0$ and $\alpha<\omega$, let $\phi_{\alpha,t}(z)=e^{-tz}e^{t\alpha}$ for $z\in R_{\alpha}$. Then $\left\|\phi_{\alpha,t}\right\|_{H^{\infty}(R_{\alpha})}=1$. Hence by $(ii)$, the set

[TABLE]

Therefore the set

[TABLE]

We noticed in Proposition 2.2 that $e^{-tA}=T_{t}$ for any $t\geq 0$. Hence taking $\alpha=\omega-\frac{1}{t}$ for any $t>0$, we deduce that the set

[TABLE]

Hence $(T_{t})_{t\geq 0}$ is a $C_{0}$-semigroup of $W\gamma$-type $\omega$. ∎

Remark 3.9*.*

The above proof shows as well that if $A$ has a strong $\gamma$-1-bounded functional calculus of type $\omega$, then $-A$ generates a $C_{0}$-semigroup of $\gamma$-type $\omega$.

In the $K$-convex case, Theorem 3.8 can be strengthened as follows.

Theorem 3.10**.**

Assume that $X$ is $K$-convex and let $m\geq 1$ be an integer. The following assertions are equivalent:

1. (i)

There exist a norming subspace $Z\subset X^{*}$ and a constant $C>0$ such that for any $N\in\mathbb{N}$, for any $\alpha_{1},\ldots,\alpha_{N}<\omega$, for any $x_{1},\ldots,x_{N}\in X$ and for any $y_{1},\ldots,y_{N}\in Z$,

[TABLE] 2. (ii)

$A$* has a strong $\gamma$-$1$-bounded functional calculus of type $\omega$;* 3. (iii)

$A$* has a strong $\gamma$-$m$-bounded functional calculus of type $\omega$.*

Proof.

The proofs of $(ii)\Leftrightarrow(iii)$ and $(iii)\Rightarrow(i)$ are obvious by the equivalence of $W\gamma$-boundedness and $\gamma$-boundedness on a $K$-convex space, and Theorem 3.8.

Now assume $(i)$. By [10, Corollary 7.4.6.], there exists $M>0$ such that for all $N\in\mathbb{N}$ and $x_{1},\ldots,x_{N}\in X$ , we have

[TABLE]

The argument in the proof of Theorem 3.8 shows that for all $\alpha_{1},\ldots,\alpha_{N}<\omega$, $\delta_{1}<\alpha_{1},\ldots,\delta_{N}<\alpha_{N}$, $f_{1}\in\mathcal{E}(R_{\delta_{1}}),\ldots,f_{N}\in\mathcal{E}(R_{\delta_{N}})$ with $\left\|f_{k}\right\|_{H^{\infty}(R_{\alpha_{k}})}\leq 1$, $x_{1},\ldots,x_{N}\in X$ and $y_{1},\ldots,y_{N}\in Z$, we have an estimate

[TABLE]

Applying (13), this implies

[TABLE]

Hence $A$ has a $\gamma$-$1$-bounded functional calculus of type $\omega$. ∎

4.   

A SHI-FENG-GOMILKO THEOREM ON $K$-CONVEX SPACES

Let $(\Omega,\mu)$ be a measure space, let $N\in\mathbb{N}$ and let $(e_{1},\ldots,e_{N})$ denote the canonical basis of $l^{2}_{N}$. Let $l^{2}_{N}\overset{2}{\otimes}L^{2}(\Omega)$ denote the Hilbert space tensor product of $l^{2}_{N}$ and $L^{2}(\Omega)$. With the notation $\mathbb{N}_{N}=\{1,\ldots,N\}$, we have a unitary isomorphism

[TABLE]

Let $X$ be a Banach space. For any bounded operator $u:l^{2}_{N}\overset{2}{\otimes}L^{2}(\Omega)\longrightarrow X$, let $u_{k}:L^{2}(\Omega)\rightarrow X$ be defined by $u_{k}(f)=u(e_{k}\otimes f)$, for any $k=1,\ldots,N$. Then the mapping $u\mapsto\sum_{k}e_{k}\otimes u_{k}$ induces an algebraic isomorphism

[TABLE]

It is easy to check that $u:l^{2}_{N}\otimes L^{2}(\Omega)\rightarrow X$ belongs to $\gamma(L^{2}(\Omega\times\mathbb{N}_{N});X)$ if and only if $u_{k}$ belongs to $\gamma(L^{2}(\Omega);X)$ for any $k=1,\ldots,N$. This leads to an algebraic isomorphism

[TABLE]

Likewise, a function $f:\Omega\times\mathbb{N}_{N}\rightarrow X$ belongs to $\gamma(\Omega\times\mathbb{N}_{N};X)$ if and only if $f(\cdot,k)$ belongs to $\gamma(\Omega;X)$ for any $k=1,\ldots,N$.

Recall the sun dual $X^{\odot}$ from Subsection 2.2. We now state and prove the main result of this article.

Theorem 4.1**.**

Let $X$ be a $K$-convex Banach space and let $m\geq 1$ be an integer. Let $A$ be an operator of half-plane type $\omega$ on $X$. The following assertions are equivalent:

1. (i)

$-A$* generates a $C_{0}$-semigroup $(T_{t})_{t\geq 0}$ of $\gamma$-type $\omega$;* 2. (ii)

There exists a constant $C>0$ such that for all $N\in\mathbb{N}$, for all $x_{1},\ldots,x_{N}\in X$, for all $y_{1},\ldots,y_{N}\in\overline{dom(A^{*})}=X^{\odot}$, and for all $\alpha_{1},\ldots,\alpha_{N}<\omega$, the functions $(t,k)\mapsto\sqrt{\omega-\alpha_{k}}R(\alpha_{k}+it,A)x_{k}$ and $(t,k)\mapsto\sqrt{\omega-\alpha_{k}}R(\alpha_{k}+it,A^{*})y_{k}$ belong to $\gamma(\mathbb{R}\times\mathbb{N}_{N};X)$ and $\gamma(\mathbb{R}\times\mathbb{N}_{N};X^{*})$, respectively, and satisfy

[TABLE]

and

[TABLE] 3. (iii)

$A$* has a strong $\gamma$-$1$-bounded functional calculus of type $\omega$;* 4. (iv)

$A$* has a strong $\gamma$-$m$-bounded functional calculus of type $\omega$.*

Proof.

Without loss of generality, one may assume $\omega=0$.

The equivalence $(iii)\Leftrightarrow(iv)$ and the implication $(iv)\Rightarrow(i)$ follow from Theorems 3.8 and 3.10.

$(i)\Rightarrow(ii)$: Since $\mathcal{T}:=\{T_{t}\,:\,t\geq 0\}$ is $\gamma$-bounded, the strongly measurable function $M:\mathbb{R}_{+}\times\mathbb{N}_{N}\longrightarrow\mathcal{L}(X)$ defined by $M(t,k):=T_{t}$ has $\gamma$-bounded range $\mathcal{T}$. Then by Theorem 2.15, for each $\psi\in\gamma(\mathbb{R}_{+}\times\mathbb{N}_{N};X)$, the function $M\psi$ belongs to $\gamma(\mathbb{R}_{+}\times\mathbb{N}_{N};X)$, with

[TABLE]

Consider $\alpha_{1},\ldots,\alpha_{N}<0$ and $x_{1},\ldots,x_{N}\in X$, and define

[TABLE]

Then $\psi\in\gamma(\mathbb{R}_{+}\times\mathbb{N}_{N};X)$ and

[TABLE]

Indeed, $\psi=\sum_{k=1}^{N}h_{k}\otimes x_{k}$ where $h_{k}:\mathbb{R}^{+}\times\mathbb{N}_{N}\rightarrow\mathbb{R}$ is defined by

[TABLE]

Further $h_{1},\ldots,h_{N}$ are pairwise orthogonal with $\left\|h_{k}\right\|_{L^{2}(\mathbb{R}_{+}\times\mathbb{N}_{N})}=\left\|\sqrt{-\alpha_{k}}e^{\alpha_{k}t}\right\|_{L^{2}(\mathbb{R}_{+})}=\frac{1}{\sqrt{2}}$. Hence (17) follows from [10, Example 9.2.4].

Recall that for each $k\in\{1,\ldots,N\}$, we have

[TABLE]

Applying Lemma 2.18, we deduce that $(t,k)\longmapsto R(\alpha_{k}+it,A)x_{k}$ belongs to $\gamma(\mathbb{R}_{+}\times\mathbb{N}_{N};X)$, with

[TABLE]

Combining (16), (17) and (18), we actually obtain

[TABLE]

which proves (14).

Finally, since $X$ is $K$-convex, the set $\{T_{t}^{*}\,:\,t\geq 0\}$ is $\gamma$-bounded. Further $(T_{t}^{\odot})_{t\geq 0}$ is a $C_{0}$-semigroup on the sun dual $X^{\odot}$, with generator equal to $-A^{\odot}$. Then the above computations together with Theorem 2.9 lead to (15).

$(ii)\Rightarrow(iii)$: Let $\alpha_{1},\ldots,\alpha_{N}<0$, $x_{1},\ldots,x_{N}\in X$ and $y_{1},\ldots,y_{N}\in X^{\odot}$. Applying Theorem 2.16, one obtains

[TABLE]

Since $X^{\odot}$ is norming in $X$, it follows from the above estimate and Theorem 3.10 that $A$ has a strong $\gamma$-$1$-bounded calculus of type [math]. ∎

Remark 4.2*.*

Let $(S,\mu)$ be a measure space and let $E(S)$ be a $K$-convex Banach function space over $(S,\mu)$ (see [10, appendix F] for definition). Then $E(S)$ has finite cotype hence according to [10, Proposition 9.3.8], there exist $c>0$ and $C>0$ such that for each $f\in\gamma(\mathbb{R}\times\mathbb{N}_{N};E(S))$,

[TABLE]

Furthermore, the following equality holds,

[TABLE]

The space $E(S)^{*}$ satisfies similar properties. Hence using (19) and the Khintchine-Maurey inequality [10, Theorem 7.2.13], the condition $(ii)$ of Theorem 4.1 can be replaced by:

• $(ii)^{\prime}$

There exists a constant $C>0$ such that for all $N\in\mathbb{N}$, for all $x_{1},\ldots,x_{N}\in X$, for all $y_{1},\ldots,y_{N}\in\overline{dom(A^{*})}=X^{\odot}$, and for all $\alpha_{1},\ldots,\alpha_{N}<\omega$,

[TABLE]

and

[TABLE]

Thus $-A$ generates a $C_{0}$-semigroup of $\gamma$-type $\omega$ on $E(S)$ if and only if $(ii)^{\prime}$ holds true.

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

Remark 4.3*.*

In [8, Theorem 6.4], Haase and Rozendaal state that if $-A$ generates a $\gamma$-bounded $C_{0}$-semigroup on a Banach space $X$, then $A$ has a strong $m$-bounded functional calculus of type [math], for any $m\geq 1$. If $X$ is K-convex, this is a formal consequence of Theorem 4.1 and in this case, the latter is a strengthening of the Haase-Rozendaal theorem.

For general $X$, a proof of [8, Theorem 6.4] can be derived from the arguments in the proof of Theorem 4.1. Indeed assume that $-A$ generates a $\gamma$-bounded $C_{0}$-semigroup, consider $\alpha<0$ and let $x\in X$ and $y\in X^{\odot}$. The proof of Theorem 4.1 shows that $t\mapsto R(\alpha+it,A)x$ belongs to $\gamma(\mathbb{R};X)$ and that

[TABLE]

for some constant $C>0$ not depending either on $\alpha$ or $x$. Then let $\gamma^{\prime}(\mathbb{R};X^{*})$ be the space introduced in [12, Section 5]. Using [12, Remark 5.12, (S2)] instead of Theorem 2.15, one obtains in a similar manner that

[TABLE]

Finally, using [12, Remark 5.12, (S1)] instead of Theorem 2.16, one deduces from (20) and (21) that

[TABLE]

Since the sun dual $X^{\odot}$ is $w^{*}$-dense in $X^{*}$, this shows that $A$ has a strong $1$-bounded functional calculus of type [math] (and hence a strong $m$-bounded functional calculus of type [math] for any $m\geq 1$).

5.   

A GEARHART-PRÜSS THEOREM ON $K$-CONVEX SPACES

Let $A$ be a half-plane type operator on some Banach space $X$. Its abscissa of uniform boundedness $s_{0}(A)$ is defined by

[TABLE]

If $-A$ generates a $C_{0}$-semigroup $(T_{t})_{t\geq 0}$, then the exponential growth bound $\omega(A)$ if defined as the supremum of all $\omega\in\mathbb{R}$ such that $(T_{t})_{t\geq 0}$ is of type $\omega$, that is,

[TABLE]

We introduce $\gamma$-bounded analogues of these notions, as follows. First we set

[TABLE]

with the convention that $s_{0}^{\gamma}(A)=-\infty$ if no set $\{R(z,A)\,:\,Re(z)\leq\alpha\}$ is $\gamma$-bounded. Second, if $-A$ generates a $C_{0}$-semigroup $(T_{t})_{t\geq 0}$, and if the latter admits a $\gamma$-type, then we set

[TABLE]

By convention we set $\omega^{\gamma}(A)=-\infty$ if $(T_{t})_{t\geq 0}$ has no $\gamma$-type. See Example 3.1 for simple examples of such semigroups.

When $X$ is a Hilbert space, the Gearhart-Prüss Theorem [1, Theorem 5.2.1] asserts that $\omega(A)=s_{0}(A)$. The main purpose of this section is to give an analogous equality $\omega^{\gamma}(A)=s_{0}^{\gamma}(A)$ on $K$-convex Banach spaces.

It is obvious that $\omega^{\gamma}(A)\leq\omega(A)$ and $s_{0}^{\gamma}(A)\leq s_{0}(A)$. The next inequality is more significant.

Lemma 5.1**.**

Assume that $-A$ generates a $C_{0}$-semigroup. Then $\omega(A)\leq s_{0}^{\gamma}(A)$.

Proof.

Let $\omega<\omega_{0}<\omega(A)$. By assumption, there exists a constant $M>0$ such that $\left\|T_{s}\right\|\leq Me^{-\omega_{0}s}$ for any $s\geq 0$. Writing $e^{\omega s}T_{s}=e^{(\omega-\omega_{0})s}e^{\omega_{0}s}T_{s}$, we obtain that $s\mapsto e^{\omega s}T_{s}x$ belongs to $L^{1}((0,\infty),X)$ for any $x\in X$, with

[TABLE]

For any $\alpha\leq\omega$ and any $t\in\mathbb{R}$, we have

[TABLE]

Since $|e^{its}e^{(\alpha-\omega)s}|\leq 1$ for any $s>0$, we derive that the set

[TABLE]

is included in the set

[TABLE]

By Lemma 2.12 and (22), the above set is $\gamma$-bounded. Therefore the set (23) is $\gamma$-bounded. Hence $\omega<s_{0}^{\gamma}(A)$. Passing to the supremum, this yields $\omega(A)\leq s_{0}^{\gamma}(A)$. ∎

Summarizing, we have

[TABLE]

whenever $-A$ generates a $C_{0}$-semigroup.

Theorem 5.2**.**

Let $X$ be a $K$-convex Banach space. Let $-A$ be the generator of a $C_{0}$-semigroup of $\gamma$-type $\omega$ on $X$. Then $A$ has a strong $\gamma$-1-bounded functional calculus of type $s$ for each $s<s_{0}^{\gamma}(A)$.

Proof.

We fix some $s<s_{0}^{\gamma}(A)$. If $s\leq\omega$, then $A$ has a strong $\gamma$-1-bounded functional calculus of type $s$ by Theorem 4.1. Thus we may now assume that $\omega<s$.

Let $N\in\mathbb{N}$, let $x_{1},\ldots,x_{N}\in X$ and let $y_{1},\ldots,y_{N}\in X^{\odot}$. According to Theorem 4.1, an estimate (14) is satisfied for any $\alpha_{1},\ldots,\alpha_{N}<\omega$. Consider $s_{1},...,s_{N}<s$ and let $\alpha_{1},\ldots,\alpha_{N}<\omega$ be chosen such that

[TABLE]

By the resolvent identity, we have

[TABLE]

for any $k\in\mathbb{N}_{N}$ and any $t\in\mathbb{R}$. According to (25), this implies that

[TABLE]

Now define $M_{s_{1},...,s_{n}}:\mathbb{R}\times\mathbb{N}_{N}\rightarrow\mathcal{L}(X)$ by

[TABLE]

The range of $M_{s_{1},...,s_{n}}$ is included in the set

[TABLE]

which is independent of $s_{1},\cdots,s_{k}$. The latter set is $\gamma$-bounded, by the definition of $s_{0}^{\gamma}(A)$. Let $K>0$ denote its $\gamma$-bounded constant. Applying Theorem 2.15 and (14), we obtain that

[TABLE]

Since $X$ is $K$-convex, the set $\{I+(\omega-s)R(\alpha+it,A)^{*},\;\alpha\leq s\}$ is $\gamma$-bounded as well. Hence using (15) we obtain a similar estimate

[TABLE]

Now applying the implication “$(ii)\Rightarrow(iii)$” of Theorem 4.1, we obtain the desired result. ∎

Corollary 5.3**.**

Let $X$ be a $K$-convex Banach space and let $-A$ be the generator of a $C_{0}$-semigroup on $X$. If $\omega^{\gamma}(A)>-\infty$, then we have

[TABLE]

Proof.

If $\omega^{\gamma}(A)>-\infty$, then by Theorem 5.2, $A$ has a strong $\gamma$-1-bounded functional calculus of type $s$ for any $s<s_{0}^{\gamma}(A)$. According to Theorem 4.1, this implies that $(T_{t})_{t\geq 0}$ is of $\gamma$-type $s$ for any $s<s_{0}^{\gamma}(A)$. Thus $\omega^{\gamma}(A)\geq s_{0}^{\gamma}(A)$. Combining with (24), we obtain the result. ∎

Example 5.4*.*

Let $(\Omega,\mu)$ be a measure space and let $1<p<\infty$. According to [1, Theorem 5.3.6], if $-A$ is the generator of a positive $C_{0}$-semigroup $(T_{t})_{t\geq 0}$ on $L^{p}(\Omega)$, then $s_{0}(A)=\omega(A)$. If in addition $\omega^{\gamma}(A)>-\infty$ then the equalities $\omega^{\gamma}(A)=\omega(A)=s_{0}^{\gamma}(A)=s_{0}(A)$ hold by Corollary 5.3.

The equality $\omega^{\gamma}(A)=\omega(A)$ in Corollary 5.3 implies the following statement.

Corollary 5.5**.**

Let $(T_{t})_{t\geq 0}$ is a bounded $C_{0}$-semigroup on some $K$-convex Banach space. If there exists $\delta>0$ such that $\{e^{-\delta t}T_{t}\,:\,t\geq 0\}$ is $\gamma$-bounded, then $\{e^{-\delta t}T_{t}\,:\,t\geq 0\}$ is $\gamma$-bounded for any $\delta>0$.

Remark 5.6*.*

Let $(T_{t})_{t\geq 0}$ is a bounded $C_{0}$-semigroup on $X$, with generator $-A$. If $A$ is bounded (equivalently, if $(T_{t})_{t\geq 0}$ is uniformly continuous), then the property considered in the above statement is true, that is, $\{e^{-\delta t}T_{t},\;t\geq 0\}$ is $\gamma$-bounded for any $\delta>0$.

Indeed, consider $\delta>0$. Since $A$ is bounded and $\sigma(A)\subset\overline{R_{0}}$, there exists an open disk $D$ such that $\sigma(A+\delta)\subset D\subset R_{0}$. Let $\partial D$ be the boundary of $D$ oriented counterclockwise. Then by the Dunford-Riesz calculus, we have

[TABLE]

for any $t\geq 0$. Then a straightforward application of Lemma 2.12 shows that $\{e^{-\delta t}T_{t}\,:\,t\geq 0\}$ is $\gamma$-bounded.

We conclude this section with an observation and two questions. First we state a result that we recently obtained (with C. Le Merdy).

Theorem 5.7**.**

([2, Corollary 0.5]) Let $X$ be isomorphic to a separable Banach lattice with finite cotype such that $X$ is not isomorphic to an Hilbert space. Then there exists $A\in\mathcal{L}(X)$ such that $\{e^{-tA}\,:\,t\geq 0\}$ is bounded but not $\gamma$-bounded.

Combining this theorem with Remark 5.6, we obtain that Corollary 5.5 is sharp in the class of uniformly continuous semigroups. Namely on any $K$-convex separable Banach lattice not isomorphic to a Hilbert space (on $L^{p}$ for $1<p\not=2<\infty$, say) we obtain a uniformly continuous semigroup $(T_{t})_{t\geq 0}$ such that $\{e^{-\delta t}T_{t}\,:\,t\geq 0\}$ is $\gamma$-bounded for any $\delta>0$, $\{T_{t}\,:\,t\geq 0\}$ is bounded but $\{T_{t}\,:\,t\geq 0\}$ is not $\gamma$-bounded.

The assumption that $X$ is $K$-convex space in Corollary 5.5 is quite surprising. This leads to the following question:

Question 5.8*.*

Does the assumption $X$ is $K$-convex space in Corollary 5.5 can be dropped?

We recall (24) and the existence of $-A$ generating a $C_{0}$-semigroup $(T_{t})_{t\geq 0}$ such that $\omega(A)<s_{0}(A)$. So we ask

Question 5.9*.*

Does there exist an operator $A$ such that $-A$ which generates a $C_{0}$-semigroup $(T_{t})_{t\geq 0}$, satisfying $\omega(A)<s_{0}^{\gamma}(A)$ ?

6.   

AN OVERVIEW

Let $\omega\in\mathbb{R}$ and let $A$ be a half-plane type operator on some Banach space $X$. Either in [4] or in the present paper, the following six properties are considered:

1. (i)

$A$ has a bounded $H^{\infty}$-functional calculus of type $\omega$. 2. (ii)

$A$ has a strong $1$-bounded functional calculus of type $\omega$. 3. (iii)

$-A$ generates a $C_{0}$-semigroup of type $\omega$.

1. (i$)_{\gamma}$

$A$ has a $\gamma$-bounded $H^{\infty}$-functional calculus of type $\omega$. 2. (ii$)_{\gamma}$

$A$ has a strong $\gamma$-$1$-bounded functional calculus of type $\omega$. 3. (iii$)_{\gamma}$

$-A$ generates a $C_{0}$-semigroup of $\gamma$-type $\omega$.

The aim of this last section is to give an overview of the relations between these properties, at least on $K$-convex spaces. This will require the analysis of a specific example, see Proposition 6.1 below. In the above list, we have deliberately omitted the strong $m$-bounded and $\gamma$-$m$-bounded functional calculi.

It follows from Proposition 2.8 and [4, Theorem 6.4] that

[TABLE]

Likewise it follows from Remark 3.3 (3) and Remark 3.9 that

[TABLE]

The implication “$(ii)\Rightarrow(i)$” is wrong. Indeed it follows from either [3] or [15] that on any infinite dimensional Hilbert space $H$, there exists a bounded $C_{0}$-semigroup $(T_{t})_{t\geq 0}$ on $H$ whose negative generator $A$ does not have a bounded $H^{\infty}$-functional calculus of type [math]. Thus with $\omega=0$, $A$ satisfies (iii) and does not satisfy (i). Moreover (ii) and (iii) are equivalent on Hilbert space, by [4, Theorem 7.1]. This proves the result.

Since $\gamma$-boundedness and uniform boundedness are equivalent on Hilbert space, the above also shows that the implication “$(ii)_{\gamma}\Rightarrow(i)_{\gamma}$” is wrong.

The implication “$(iii)\Rightarrow(ii)$” is wrong. Indeed let $1<p\neq 2<+\infty$ and let $(T_{t})_{t\in\mathbb{R}}$ be the right translation group on $L^{p}(\mathbb{R})$, which is a bounded $C_{0}$-group. Let $-A$ denote its generator. It follows from Gomilko’s paper [5] that either $A$ or $-A$ does not have a strong $1$-bounded functional calculus of type [math].

We have shown in Theorem 4.1 that if $X$ is $K$-convex, then the implication “$(iii)_{\gamma}\Rightarrow(ii)_{\gamma}$” holds true. We do not know whether “$(iii)_{\gamma}\Rightarrow(ii)_{\gamma}$” holds true on any Banach space.

The implication “$(iii)_{\gamma}\Rightarrow(ii)$” holds true, by [8, Theorem 6.4] (see Remark 4.3 for more on this.)

We noticed above that $(iii)$ does not imply $(i)$ on Hilbert space. Consequently, The implication “$(iii)_{\gamma}\Rightarrow(i)$” is wrong.

The only remaining question is whether $(i)$ implies $(iii)_{\gamma}$. We are going to show that this is wrong on sufficently bad spaces, see Example 6.2 below.

For this purpose we introduce a class of $C_{0}$-(semi)groups of independent interest. Recall the Gaussian space $G(X)$ from Subsection 2.3. We will use the so-called ‘contraction principle’ [10, Theorem 6.1.13], which says that for any $x_{1},\ldots,x_{n}\in X$ and any $\alpha_{1},\ldots,\alpha_{n}\in\mathbb{C}$, we have

[TABLE]

We recall that $X$ has property $(\alpha)$ (see [10, section 7.5] for more details) if there exists a constant $C\geq 1$ such that for any finite family $(x_{ij})$ in $X$ and any finite family $(t_{ij})$ in $\mathbb{C}$, we have

[TABLE]

Banach spaces with property $(\alpha)$ have a finite cotype, thus $\mathcal{R}$-boundedness and $\gamma$-boundedness are equivalent on such spaces. We recall that the class of all Banach spaces with property $(\alpha)$ is stable under taking subspaces and that all Banach lattices with a finite cotype have property $(\alpha)$. In particular, for any $1\leq p<\infty$, $L^{p}$-spaces and their subspaces have property $(\alpha)$.

Let $(\xi_{k})_{k\geq 1}$ be a sequence of distinct points of $\mathbb{R}$. For any finite Gaussian sum $\sum_{k=1}^{n}\gamma_{k}\otimes x_{k}$, with $x_{1},\ldots,x_{n}\in X$, we let

[TABLE]

Then we have

[TABLE]

by (26). Since the finite Gaussian sums are dense in $G(X)$, each $T_{t}$ extends to a bounded linear operator on $G(X)$ (still denoted by $T_{t}$), with $\left\|T_{t}\right\|=1$. Furthermore $(T_{t})_{t\in\mathbb{R}}$ is a $C_{0}$-group. Indeed it is plain that for any finite Gaussian sum $z=\sum_{k=1}^{n}\gamma_{k}\otimes x_{k}$, $T_{t}(z)\to z$ when $t\to 0$. Then the strong continuity of $(T_{t})_{t\in\mathbb{R}}$ follows from the uniform boundedness of $(T_{t})_{t\in\mathbb{R}}$ and the density of the set of all finite Gaussian sums in $G(X)$.

Proposition 6.1**.**

Let $-A$ denote the generator of the $C_{0}$-group $(T_{t})_{t\in\mathbb{R}}$ defined by (28).

• (1)

$A$* has a bounded $H^{\infty}$-functional calculus of type [math].*

• (2)

The $C_{0}$-group $(T_{t})_{t\in\mathbb{R}}$ is $\gamma$-bounded if and only if $X$ has property $(\alpha)$.

Proof.

For any $b\in L^{1}(\mathbb{R})$, we let $\int_{\mathbb{R}}b(t)T_{t}dt\,\in\mathcal{L}(G(X))$ denote the operator defined by

[TABLE]

If $b\in L^{1}(\mathbb{R}_{+})$, we let $\mathcal{L}b$ denote the Laplace transform of $b$, that is,

[TABLE]

Obviously $\mathcal{L}b$ is continuous and bounded on $\overline{R_{0}}$ and its restriction to $R_{0}$ belongs to $H^{\infty}(R_{0})$.

For any $b\in L^{1}(\mathbb{R})$ and any $x_{1},\ldots,x_{n}\in X$, we have

[TABLE]

If $b\in L^{1}(\mathbb{R}_{+})$, this implies, using (26), that

[TABLE]

Now let $\alpha<0$ and let $f\in\mathcal{E}(R_{\alpha})$. According to [7, Lemma 5.1], there is a (necessarily unique) $b\in L^{1}(\mathbb{R}_{+})$ such that $f=\mathcal{L}b$ on $R_{0}$ and

[TABLE]

The estimate (30) therefore implies that

[TABLE]

This shows (1).

We now turn to the proof of (2). First assume that $X$ has property $(\alpha)$. Let $(t_{j})_{j}$ be a finite family of real numbers and for any $j$, let $z_{j}=\sum_{k}\gamma_{j}\otimes x_{jk}$ be a finite Gaussian sum. We have

[TABLE]

Applying (27) we deduce that

[TABLE]

Since the set of all finite Gaussian sums is dense in $G(X)$, this shows that $(T_{t})_{t\in\mathbb{R}}$ is $\gamma$-bounded.

Assume on the contrary that $X$ does not have property $(\alpha)$.  By (26), there exists a (necessarily unique) contractive, non degenerate, homomorphism

[TABLE]

such that

[TABLE]

for any $n\geq 1$ and any $x_{1},\ldots,x_{n}\in X$. We claim that the set

[TABLE]

is not $\gamma$-bounded. Indeed let $(t_{ik})$ and $(x_{ik})$ be finite families in $\mathbb{C}$ and $X$, respectively, and assume that $|t_{ik}|\leq 1$ for any $i,k$. There exist $f_{i}\in C_{0}(\mathbb{R})$ such that $f_{i}(\xi_{k})=t_{ik}$ and $\left\|f_{i}\right\|_{\infty}\leq 1$ for any $i,k$. Then

[TABLE]

If $S$ were $\gamma$-bounded, this would imply that the norm of the left hand side is dominated by the norm of $\sum_{i,k}\gamma_{i}\otimes\gamma_{k}\otimes x_{ik}$, which would imply property $(\alpha)$.

Note that the homomorphismm $w$ ‘extends’ $(T_{t})_{t\in\mathbb{R}}$ in the sense of [13, Definition 2.4]. Indeed, according to (29), we have

[TABLE]

for any $b\in L^{1}(\mathbb{R})$. Therefore if $(T_{t})_{t\in\mathbb{R}}$ were $\gamma$-bounded, then according to [13, Theorem 4.4], the above set $S$ would be $\gamma$-bounded. We just noticed that this does not hold true. Hence, $(T_{t})_{t\in\mathbb{R}}$ is not $\gamma$-bounded. ∎

Example 6.2*.*

Proposition 6.1 provides an example of an operator which satisfies $(i)$ without satisfying $(iii)_{\gamma}$, for $\omega=0$. Indeed, assume that $X$ does not have property $(\alpha)$ and let $(T_{t})_{t\in\mathbb{R}}$ (with generator $-A$) be given by the above proposition. Changing $T_{t}$ into $T_{-t}$, part (1) of Proposition 6.1 shows that $A$ and $-A$ have a bounded $H^{\infty}$-bounded functional calculus of type [math]. However by part (2) of Proposition 6.1, either $(T_{t})_{t\geq 0}$ or $(T_{-t})_{t\geq 0}$ is not $\gamma$-bounded.

We do not know if $(i)$ implies $(iii)_{\gamma}$ on non Hilbertian Banach spaces with property $(\alpha)$. In particular, we do not know if $(i)$ implies $(iii)_{\gamma}$ on $L^{p}$-spaces, for $1<p\not=2<\infty$.