1. Introduction

The purpose of this article is twofold. The short part devotes to a generalization of this classical transference principle of Calderon, Coifman and Weiss. The major part gives applications of this new abstract result to discrete and continuous operator (semi) groups, in particular shall recover and generalize important results of Peller. In the classical transference principle(s) the objects under investigation are derived from the sequence of operators of the form.

${\displaystyle \underset{j}{\sum }{T}_{{\mu }^j}^j}={\displaystyle \underset{G}{\int }{\displaystyle \underset{j}{\sum }{T}^j\left(1+\epsilon \right){\mu }^j\left(d\left(1+\epsilon \right)\right)}}$ (1.1)

where G is a locally compact group and $T^j={\left(T^j\left(1+\epsilon \right)\right)}_{\left(1+\epsilon \right)\in G}:G\to \mathcal{L}\left(x\right)$ is a

strongly bounded continuous representation of G on a Banach space X. The integral (1.1) has to be understood in the strong sense, i.e.,

${\displaystyle \underset{j}{\sum }{T}_{{\mu }^j}^{j}x}={\displaystyle \underset{G}{\int }{\displaystyle \underset{j}{\sum }{T}^j\left(1+\epsilon \right)x{\mu }^j\left(d\left(1+\epsilon \right)\right)}}\left(x\in X\right)$

And ${\mu }^j$ are scalar measure that renders the meaningful expression. Since such sequence of operators occurs in a variety of situations, the applications of transference principles are manifold, and the literature on this topic is vast.

Originally, Calder’n [2] considered representations on $L^{\epsilon +1}$ induced by a G-flow of measure-preserving transformations of the underlying measure space. H is considerations that were motivated by ergodic theory and his aim was to obtain maximal inequalities. Subsequently, Coifman and Weiss [3] [4] shifted the focus to norm estimates and were able to drop Calder’n’s assumption of an underlying measure-preserving G-flow towards general G-representations on $L^{\epsilon +1}$ -spaces. Berkson, Gillespie and Muhly [5] were able to generalize the method towards general Banach spaces X. However, the representations considered in these works were still (uniformly) bounded. In the continuous one-parameter case (i.e., $G=ℝ$ ) Blower [6] showed that the original proof method could fruitfully be applied also to non-bounded representations. However, his result was in a sense

“local” and did not take into account the growth rate of the group ${\left(T^j\left(1+\epsilon \right)\right)}_{\left(1+\epsilon \right)\in ℝ}$

at infinity. In [7] we discovered Blower’s result and in [8] we could refine it towards a “global” transference result for strongly continuous one-parameter groups.

Markus Haase [1] showed a developing method of generating transference results and showed that the known transference principles, (the classical Berkson-Gillespie-Muhly result and the central results of [8] ) are special instances of it. The method has three important new features. Firstly, it allows to pass from groups to semigroups. More precisely, consider closed sub-semigroups S of a locally compact group G together with a strongly continuous representation $T^j:S\to \mathcal{L}\left(X\right)$ on a Banach space, and try to estimate the norms of sequence of operators of the form

${\displaystyle \underset{j}{\sum }{T}_{{\mu }^j}^j}={\displaystyle \underset{\left(1+\epsilon \right)}{\int }{\displaystyle \underset{j}{\sum }{T}^j\left(1+\epsilon \right){\mu }^j\left(d\left(1+\epsilon \right)\right)}}$ (1.2)

by means of the transference method. The second feature is the role of weights in the transference procedure, somehow hidden in the classical version. Thirdly, the account brings to light the formal structure of the transference argument. in the first step one establishes a factorization of the sequence of operators (1.2) over a convolution (i.e., Fourier multiplier) operator on a space of X-valued functions on G, then, in a second step, one uses this factorization to estimate the series norms, and finally, one may vary the parameters to optimize the obtained inequalities, So one can briefly subsume our method under the scheme.

factorize-estimate-optimize

where use one particular way of constructing the initial factorization. One reason for the power of the method lies in choosing different weight in the factorization, allowing for the optimization in the last step. The second reason lies in the purely formal nature of the factorization: this allows to re-interpret the same factorization involving different function spaces.

Devoted to applications of the transference method. These applications deal exclusively with the cases $S=\mathbb{Z},{\mathbb{Z}}_{+}$ , and $S=ℝ,{ℝ}_{+}$ which we for short call the discrete and the continuous case, respectively. However, let us point out that the general transference method works even for sub-semigroups of non-abelian groups.

To clarify what kind of applications we have in mind, let us look at the discrete case first. Here the semigroup consists of the powers ${\left({\left(T^j\right)}^n\right)}_{n\in {\mathbb{N}}_0}$ of one

single bounded sequence of operators $T^j$ , and the derived sequence of operators (1.2) take the form

$\underset{n\ge 0}{\overset{}{{\displaystyle \sum }}}{\left(\beta +\epsilon \right)}_n\underset{j}{{\displaystyle \sum }}{\left(T^j\right)}^n$

for some (complex) scalar sequence $\beta +\epsilon ={\left({\left(\beta +\epsilon \right)}_n\right)}_{n\ge 0}$ . In order to avoid convergence questions, suppose that $\left(\beta +\epsilon \right)$ is a finite sequence, hence

$\widehat{\left(\beta +\epsilon \right)}\left(z\right)=\underset{n\ge 0}{\overset{}{{\displaystyle \sum }}}{\left(\beta +\epsilon \right)}_n{\left(z\right)}^n$

is a complex polynomial. One usually writes

$\widehat{\left(\beta +\epsilon \right)}\underset{j}{{\displaystyle \sum }}\left(T^j\right)=\underset{n\ge 0}{\overset{}{{\displaystyle \sum }}}{\left(\beta +\epsilon \right)}_n\underset{j}{{\displaystyle \sum }}{\left(T^j\right)}^n$

and is interested in continuity properties of the functional calculus

$ℂ\left[z\right]\to \mathcal{L}\left(X\right),f_j\leftrightarrow f_j\left(T^j\right).$

That is, one looks for a function algebra norm ${\Vert \cdot \Vert }_{\left(A_j\right)}$ on $ℂ\left[z\right]$ that allows an estimates of the form

$\underset{j}{{\displaystyle \sum }}\Vert f_j\left(T^j\right)\Vert \lesssim \Vert \underset{j}{{\displaystyle \sum }}{f}_j{}_{\mathcal{A}}\Vert \left(f_j\in ℂ\left[z\right]\right)$ (1.3)

(The symbol $\lesssim$ is short for $<\epsilon$ for some unspecified constant $\epsilon >-1$ , see also the Terminology-paragraph at the end of the introduction). A rather trivial instance of (1.3) is based on the estimates

$\underset{j}{{\displaystyle \sum }}\Vert f_j\left(T^j\right)\Vert =\underset{j}{{\displaystyle \sum }}\Vert \underset{n\ge 0}{\overset{}{{\displaystyle \sum }}}{\left(\beta +\epsilon \right)}_n{\left(T^j\right)}^n\Vert \le \underset{n\ge 0}{\overset{}{{\displaystyle \sum }}}\left|{\left(\beta +\epsilon \right)}_n\right|\Vert \underset{j}{{\displaystyle \sum }}{\left(T^j\right)}^n\Vert$

Defining the positive sequence $\left(1+\epsilon \right)={\left({\left(1+\epsilon \right)}_n\right)}_n$ by ${\left(1+\epsilon \right)}_n:=\underset{j}{{\displaystyle \sum }}\Vert {\left(T^j\right)}^n\Vert$ , hence have

$\underset{j}{{\displaystyle \sum }}\Vert f_j\left(T^j\right)\Vert \le {\Vert \underset{j}{{\displaystyle \sum }}{f}_j\Vert }_{\left(1+\epsilon \right)}=\underset{n\ge 0}{\overset{}{{\displaystyle \sum }}}\left|{\left(\beta +\epsilon \right)}_n\right|{\left(1+\epsilon \right)}_n$ (1.4)

and by the submultiplicativity ${\left(1+\epsilon \right)}_{n+m}\le {\left(1+\epsilon \right)}_n{\left(1+\epsilon \right)}_m$ one sees that ${\Vert \cdot \Vert }_{\left(1+\epsilon \right)}$ is a functional gebra (semi)norm on $ℂ\left[z\right]$ .

The “functional calculus” given by (1.4) is tailored to the sequence of operators $T^j$ and uses no other information than the growth of the power of $T^j$ . The central question is: under which conditions can one obtain better estimates for ${\displaystyle {\sum }_j\Vert f_j\left(T^j\right)\Vert }$ i.e., in terms of weaker function norms? The conditions have

in mind may involve $T^j$ (or better: the semigroup ${\left({\left(T^j\right)}^n\right)}_{n\ge 0}$ ) or the underlying

Banach space. To recall a famous example: von Neumann’s inequality [9] states that if $X=H$ is a Hilbert space and ${\displaystyle {\sum }_j\Vert T^j\Vert }\le 1$ (i.e., $T^j$ is a contraction), then

$\underset{j}{{\displaystyle \sum }}\Vert f_j\left(T^j\right)\Vert \le {\Vert \underset{j}{{\displaystyle \sum }}{f}_j\Vert }_{\infty }\text{for}\text{every}{f}_j\in ℂ\left[z\right]$ (1.5)

where ${\displaystyle {\sum }_j{\Vert f_j\Vert }_{\infty }}$ are the series norms of $f_j$ in the Banach algebra $\mathcal{A}=H^{\infty }\left(\mathbb{D}\right)$ of bounded analytic functions on the open unit disc $\mathbb{D}$ .

Von Neumann’s result is optimal in the trivial sense that the estimate (1.5) of course implies that $T_j$ are contraction, but also in the sense that one cannot improve the estimate without further conditions: If $H=L^2\left(\mathbb{D}\right)$ and $\left(T^j\left(h\right)\right)\left(z\right)=zh\left(z\right)$ is multiplication with the complex coordinate, then ${\displaystyle {\sum }_j\Vert f_j\left(T^j\right)\Vert }={\displaystyle {\sum }_j{\Vert f_j\Vert }_{\infty }}$ for any $f_j\in C\left[z\right]$ . A natural question then is to ask which sequence of operators satisfy the slightly weaker estimates

$\underset{j}{{\displaystyle \sum }}\Vert f_j\left(T^j\right)\Vert \le {\Vert \underset{j}{{\displaystyle \sum }}{f}_j\Vert }_{\infty }\left(f_j\in ℂ\left[z\right]\right)$

(called “polynomial boundedness of $T^j$ ”). On a general Banach space this may fail even for a contraction: simply take $X={\mathcal{l}}^1\left(\mathbb{Z}\right)$ and $T^j$ the shift sequence of operators, given by ${\left(T^{j}X\right)}_n=X_{n+1}$ , $n\in \mathbb{Z}$ , $x\in {\mathcal{l}}^1\left(\mathbb{Z}\right)$ . On the other hand, Lebow [10] has shown that even on a Hilbert space polynomial boundedness of sequence of operators $T^j$ may fail if it is only assumed to be power-bounded,

i.e., if one has merely ${\sup }_{n\in \mathbb{N}}\Vert {\displaystyle {\sum }_j{\left(T^j\right)}^n}\Vert <\infty$ instead of $\Vert {\displaystyle {\sum }_j{\left(T^j\right)}^n}\Vert \le 1$ . The

class of power-bounded sequence of operators on Hilbert spaces is notoriously enigmatic, and it can be considered one of the most important problems in sequence of operators theory to find good functional calculus estimates for this class.

Let us shortly comment on the continuous case. Here one is given a strongly continuous (in short: $C_0$ ) semigroup ${\left(T^j\left(1+\epsilon \right)\right)}_{\epsilon \ge -1}$ of the sequence of operators on a Banach space X, and one considers integrals of the form

${\displaystyle \underset{{ℝ}_{+}}{\int }{\displaystyle \underset{j}{\sum }{T}^j\left(1+\epsilon \right){\mu }^j\left(d\left(1+\epsilon \right)\right)}}$ (1.6)

where assume for simplicity that the support of the measure ${\mu }^j$ are bounded. Shall use only basic results from semigroup theory, and refer to [11] [12] for further

information. The sequence of generators of the semigroup ${\left(T^j\left(1+\epsilon \right)\right)}_{\epsilon >-1}$ is in

general unbounded, closed and densely defined the sequence of operators $-A_j$ satisfying

$\underset{j}{{\displaystyle \sum }}{\left(\left(1+\epsilon \right)+A_j\right)}^{-1}={\displaystyle \underset{0}{\overset{\infty }{\int }}{\text{e}}^{-\left(1+\epsilon \right)\left(1+\epsilon \right)}{\displaystyle \underset{j}{\sum }{T}^j\left(1+\epsilon \right)d\left(1+\epsilon \right)}}$ (1.7)

for Re $\left(1+\epsilon \right)$ large enough. The sequence of generators are densely defined, i.e., its domain $\text{dom}\left(A_j\right)$ is dense in X. Exclusively deal with semigroups satisfying a polynomial growth ${\displaystyle {\sum }_j\Vert T^j\left(1+\epsilon \right)\Vert }\lesssim {\left(2+\epsilon \right)}^{\beta +\epsilon }$ for some $\epsilon >-\beta$ , and hence (1.7) holds at least for all Re $\epsilon >-1$ . One writes $T^j\left(1+\epsilon \right)={\text{e}}^{-\left(1+\epsilon \right)A_j}$ for $\epsilon >-1$ and, more generally,

${\displaystyle \underset{j}{\sum }\left(\mathcal{L}{\mu }^j\right)\left(A_j\right)}={\displaystyle \underset{{ℝ}_{+}}{\int }{\displaystyle \underset{j}{\sum }{T}^j\left(1+\epsilon \right){\mu }^j\left(d\left(1+\epsilon \right)\right)}}$

where

${\displaystyle \underset{j}{\sum }\left(\mathcal{L}{\mu }^j\right)\left(A_j\right)}={\displaystyle \underset{{ℝ}_{+}}{\int }{\text{e}}^{-z\left(1+\epsilon \right)}{\displaystyle \underset{j}{\sum }{\mu }^j\left(d\left(1+\epsilon \right)\right)}}$

is the Laplace transform of ${\mu }^j$ . So in the continuous case the Laplace transform takes the role of the Taylor series in the discrete case. Asking for good estimates for the sequence of operators of the form (1.6) is as asking for functional calculus estimates for the sequence of operators $A_j$ . The continuous version of von Neumann’s inequality states that if $X=H$ is a Hilbert space and if $\Vert {\displaystyle {\sum }_j{T}^j\left(1+\epsilon \right)}\Vert \le 1$ for all $\epsilon >-1$ (i.e., if $T^j$ are contraction semigroup), then

${\displaystyle {\sum }_j\Vert f_j\left(A_j\right)\Vert }\le {\Vert {\displaystyle {\sum }_j{f}_j}\Vert }_{\infty }\left(f_j\in \mathcal{L}{\mu }^j\right)$

where ${\Vert f_j\Vert }_{\infty }$ are the norms of $f_j$ in the Banach algebra $H^{\infty }\left({ℂ}_{+}\right)$ of bounded analytic functions on the open half place ${ℂ}_{+}:=\left\{z\in ℂ|\mathrm{Re}z>0\right\}$ , see ( [13] , Theorem 7.1.7).

There are similarities in the discrete and in the continuous case, but also characteristic differences. The discrete case is usually a little more general, shows more irregularities, and often it is possible to transfer results from the discrete to the continuous case. (However, this may become quite technical, and prefer direct proofs in the continuous case whenever possible.) In the continuous case, the role of power-bounded operators is played by bounded semigroups, and similar to the discrete case, the class of bounded semigroups on Hilbert spaces appears to be rather enigmatic. In particular, there is a continuous analogue of Lebow’s result due to Le Merdy [14] , cf. also ( [13] , Section 9.1.3). And there remain some notorious open questions involving the functional calculus, e.g., the power-boundedness of the Cayley transform of the generator, cf. [15] and the references therein.

The strongest results in the discrete case obtained so far can be found in there markable [16] by Peller from 1982. One of Peller’s results are that if $T^j$ is a power-bounded of sequence operators on a Hilbert space H, then

$\underset{j}{{\displaystyle \sum }}\Vert f_j\left(T^j\right)\Vert \lesssim {\Vert \underset{j}{{\displaystyle \sum }}{f}_j\Vert }_{B_{\infty ,1}^0}\left(f_j\in ℂ\left[z\right]\right)$

where is $B_{\infty ,1}^0\left(\mathbb{D}\right)$ is the so-called analytic Besov algebra on the disc .

In 2005, Vitse [17] made a major advance in showing that Peller’s Besov class estimate still holds true on general Banach spaces if the power-bounded sequence of operators $T^j$ is actually of Tadmor-Ritt type, i.e., satisfies the “analyticity condition”

${\sup }_{n\ge 0}\underset{j}{{\displaystyle \sum }}\Vert n\left({\left(T^j\right)}^{n+1}-{\left(T^j\right)}^n\right)\Vert <\infty .$

She moreover established in [18] an analogue for strongly continuous bounded analytic semigroups. Whereas Peller’s results rest on Grothendieck’s inequality (and hence are particular to Hilbert spaces) Vitse’s approach is based on repeated summation/integration by parts, possible because of the analyticity assumption.

Shall complement Vitse’s result by devising an entirely new approach, using the transference methods. In doing so, avoid Grothendieck’s inequality and reduce the problem to certain Fourier multipliers on vector-valued function spaces. By Plancherel’s identity, on Hilbert spaces these are convenient to estimate, but one can still obtain positive results on $L^{1+\epsilon }$ -spaces or on UMD spaces. The approach works simultaneously in the discrete and in the continuous case, and hence do not only recover Peller’s original result (Theorem 5.1) but only establish a complete continuous analogue (Theorm 5.3), conjectured in [18] . Moreover, we establish an analogue of the Besov-type estimates for $L^{1+\epsilon }$ -spaces and for UMD spaces (Theorem (5.7)). These results, however, are less satisfactory since the algebras of Fourier multipliers on the spaces $L^2\left(ℝ;X\right)$ and $L^2\left(\mathbb{Z};X\right)$ are not thoroughly understood if X is not a Hilbert space.

Show how the transference methods can also be used to obtain “γ-versions” of the Hilbert space results. The central notion here is the so-called γ-boundedness of sequence of operators family, a strengthening of operator norm boundedness. It is related to the notion of R-boundedness and plays a major role in Kalton and Weis’ work [19] on the $H^{\infty }$ -calculus. The “philosophy” behind this theory is that to each Hilbert space result based on Plancherel’s theorem there is a corresponding Banach space version, when operator norm boundedness is replaced by γ-boundedness.

Give evidence to this philosophy by showing how the transference results enables one to prove γ-versions of functional calculus estimates on Hilbert spaces. As examples, we recover the γ-version of a result of Boyadzhiev and deLaubenfels, first proved by Kalton and Weis in [19] (Theorem 6.5). Then derive γ-versions of the Besov calculus theorems in both the discrete and the continuous forms. The simple idea consists of going back to the original factorization in the transference method, but exchanging the function spaces on which the Fourier multiplier sequence of operators act from an $L^2$ -space into a γ-space. This idea is implicit in the original proof from [19] and has also been employed in a similar fashion recently by Le Merdy [20] .

Finally, discuss consequences of the estimates for full functional calculi and singular integrals for discrete and continuous semigroups. For instance, prove

that if ${\left(T^j\left(1+\epsilon \right)\right)}_{\left(\epsilon >-1\right)}$ is any strongly continuous semigroup on a UMD space X, then for all $0<a<a+\epsilon$ the principal value integral

${\lim }_{\epsilon \searrow 0}\underset{ϵ<\left|1-a\right|<a}{\overset{}{{\displaystyle \int }}}\frac{{\displaystyle {\sum }_j{T}^j\left(1+\epsilon \right)x}}{1-a}$

exists for all $x\in X$ . For $C_0$ -groups this is well-known, cf. [7] , but for semigroups which are not groups, this is entirely new.

Terminology: Use the common symbols $\mathbb{N}$ , $\mathbb{Z}$ , $ℝ$ , $ℂ$ for the sets of natural, integer, real and complex numbers. In our understanding 0 is not a natural number, and write

${\mathbb{Z}}_{+}:=\left\{n\in \mathbb{Z}|n\ge 0\right\}=\mathbb{N}\cup \left\{0\right\}\text{and}{ℝ}_{+}:=\left\{\left(1+\epsilon \right)\in ℝ|\epsilon >-1\right\}.$

Moreover, $:=\left\{z\in ℂ|\left|z\right|<1\right\}$ is the open unit disc, $\mathbb{T}:=\left\{z\in ℂ|\left|z\right|=1\right\}$ is the torus, and ${ℂ}_{+}:=\left\{z\in ℂ|\mathrm{Re}z>0\right\}$ is the open right half plane.

Use X, Y, Z to denote (complex) Banach spaces, and $A_j$ , B, C to denote closed possibly unbounded sequence of operators on them. By $\mathcal{L}\left(X\right)$ denote the Banach algebra of all bounded linear sequence of operators on the Banach space X, endowed with the ordinary sequence of operators norm. The domain, kernel and range of the sequence of operators $A_j$ are denoted by $\text{dom}\left(A_j\right)$ , $\text{ker}\left(A_j\right)$ and $\text{ran}\left(A_j\right)$ , respectively.

The Bochner space of equivalence classes of $1+\epsilon$ -integrable X-valued functions is denoted by $L^{\left(1+\epsilon \right)}\left(R;X\right)$ . If $\Omega$ is a locally compact space, then $M\left(\Omega \right)$ denotes the space of all bounded regular Borel measures on $\Omega$ . If ${\mu }^j\in M\left(\Omega \right)$ then $\sup {\mu }^j$ denotes its topological support. If $\Omega \subset ℂ$ is an open subset of the complex plane, $H^{\infty }\left(\Omega \right)$ denotes the Banach algebra of bounded holomorphic functions on Ω, endowed with the supremum norm

${\displaystyle {\sum }_j{\Vert f_j\Vert }_{H^{\infty }\left(\Omega \right)}}=\sup \left\{{\displaystyle {\sum }_j\left|f_j\left(z\right)\right|}|z\in \Omega \right\}$ .

Shall need notation and results from Fourier analysis as collected in [13] . In particular, use the symbol $\mathcal{F}$ for the Fourier transform acting on the space of (possibly vector-valued) tempered distributions on $ℝ$ , where agree that

${\displaystyle \underset{j}{\sum }\mathcal{F}{\mu }^j\left(1+\epsilon \right)}={\displaystyle \underset{ℝ}{\int }{\text{e}}^{-i\left(1+\epsilon \right)\left(1+\epsilon \right)}{\displaystyle \underset{j}{\sum }{\mu }^j\left(d\left(1+\epsilon \right)\right)}}$

is the Fourier transform of a bounded measure ${\mu }^j\in M\left(R\right)$ . A function $m\in L^{\infty }\left(ℝ\right)$ is called a bounded Fourier multiplier on $L^{\left(1+\epsilon \right)}\left(ℝ;X\right)$ if there is a constant $\epsilon >-1$ such that

${\displaystyle \underset{j}{\sum }{\Vert {\mathcal{F}}^{-1}\left(m\cdot \mathcal{F}{f}_j\right)\Vert }_{\left(1+\epsilon \right)}}\le \left(1+\epsilon \right){\Vert {\displaystyle \underset{j}{\sum }{f}_j}\Vert }_{\left(1+\epsilon \right)}$ (1.8)

holds true for all $f^j\in L^{\left(1+\epsilon \right)}\left(ℝ;X\right)\cap {\mathcal{F}}^{-1}\left(L^1\left(ℝ;X\right)\right)$ . The smallest $\left(1+\epsilon \right)$ that

can be chosen in (1.8) is denoted by ${\Vert \cdot \Vert }_{{\mathcal{M}}_{\left(1+\epsilon \right),x}}$ . This turns the space ${\mathcal{M}}_{\left(1+\epsilon \right),X}\left(ℝ\right)$ of all bounded Fourier multipliers on $L^{\left(1+\epsilon \right)}\left(ℝ;X\right)$ into a unital Banach algebra.

A Banach space X is a UMD space, if and only if the function $\left(1+\epsilon \right)\mapsto \text{sgn}\left(1+\epsilon \right)$ is a bounded Fourier multiplier on $L^2\left(ℝ;X\right)$ . Such spaces are the right ones to study singular integrals for vector-valued functions. In particular, by results of Bourgain, McConnel and Zimmermann, a vector-valued version of the classical Mikhlin theorem holds, see ( [13] , Appendix E.6) as well as Burkholder’s article [21] . Each Hilbert space is UMD, and if X is UMD, then $L^{\left(1+\epsilon \right)}\left(\Omega ,\Sigma ,{\mu }^j;X\right)$ is also UMD whenever $1<1+\epsilon <\infty$ and $\left(\Omega ,\Sigma ,{\mu }^j\right)$ is a measure space.

The Fourier transform of ${\mu }^j\in {\mathcal{l}}^1\left(\mathbb{Z}\right)$ are

${\displaystyle \underset{j}{\sum }\widehat{{\mu }^j}\left(z\right)}={\displaystyle \underset{n\in \mathbb{Z}}{\sum }{\displaystyle \underset{j}{\sum }{\mu }^j\left(n\right)z^n}}\left(z\in \mathbb{T}\right)$

Analogously to the continuous case, form the algebra ${\mathcal{M}}_{\left(1+\epsilon \right),X}\left(\mathbb{T}\right)$ of functions $m\in L^{\infty }\left(\mathbb{T}\right)$ which induce bounded Fourier multiplier sequence of operators on ${\mathcal{l}}^{\left(1+\epsilon \right)}\left(\mathbb{Z};X\right)$ , endowed with its natural norm.

Finally, given sets $A_j$ and two real-valued functions $f_j,g^j:A_j\to ℝ$ write

$f_j\left(a\right)\lesssim g^j\left(a\right)\left(a\in A_j\right)$

to abbreviate the statement that there is $\epsilon >-1$ such that $f_j\left(a\right)\le \left(1+\epsilon \right)g^j\left(a\right)$ for all $a\in A_j$ .

2. Transference Identities

Introduce the basic idea of transference. Let G be a locally compact group with left Haar measure ds. Let $S\subseteq G$ be a closed sub-semigroup of G and let

$T^j:S\to \mathcal{L}(X)$

be a strongly continuous representation of S on a Banach space X. Let ${\mu }^j$ be a (scalar) Borel measure on S such that

${\displaystyle \underset{S}{\int }{\displaystyle \underset{j}{\sum }\Vert T^j\left(1+\epsilon \right)\left|{\mu }^j\right|\left(d\left(1+\epsilon \right)\right)\Vert }}<\infty$

and let the sequence of operators $T_{{\mu }^j}^j\in \mathcal{L}\left(X\right)$ be defined by

${\displaystyle \underset{j}{\sum }{T}_{{\mu }^j}^{j}x}={\displaystyle \underset{S}{\int }{\displaystyle \underset{j}{\sum }{T}^j\left(1+\epsilon \right)x{\mu }^j\left(d\left(1+\epsilon \right)\right)}}\left(x\in X\right)$ (2.1)

The aim of transference is an estimate of ${\displaystyle {\sum }_j\Vert T_{{\mu }^j}^j\Vert }$ in terms of a convolution

sequence of operators involving ${\mu }^j$ . The idea to obtain such an estimate is, in a first step, purely formal.

For a (measurable) functions ${\phi }_j:S\to ℂ$ denote by ${\phi }_j{T}^j$ the pointwise products

$\left({\phi }_j{T}^j\right):S\to \mathcal{L}\left(X\right),\left(1+\epsilon \right)\mapsto {\phi }_j\left(1+\epsilon \right)T^j\left(1+\epsilon \right)$

and by ${\phi }_j{\mu }^j$ the measure

$\left({\phi }_j{T}^j\right)\left(d\left(1+\epsilon \right)\right)={\phi }_j\left(1+\epsilon \right){\mu }^j\left(d\left(1+\epsilon \right)\right)$

In the following do not distinguish between a function/measure defined on S and its extension to G by on G\S. Also, for Banach spaces $X,Y,Z$ and sequence of operators-valued functions $F:G\to \mathcal{L}\left(Z,Y\right)$ and $H:G\to \mathcal{L}\left(Y,X\right)$ define the convolution $H\ast F:G\to \mathcal{L}\left(Z,X\right)$ formally by

$\left(H\ast F\right)\left(1+\epsilon \right)={\displaystyle \underset{G}{\int }H\left(1+\epsilon \right)F\left({\left(1+\epsilon \right)}^{-1}\left(1+\epsilon \right)\right)d\left(1+\epsilon \right)}\left(\left(1+\epsilon \right)\in G\right)$ (2.2)

in the strong sense, as long as this is well defined. (Actually, as argue purely formally, at this stage do not bother too much about whether all things are well defined.) Instead, shall establish formulate first and then explore conditions under which they are meaningful.

The first lemma expresses the fact that a semigroup representation induces representations of convolution algebras on S (see, e.g., [1] ).

Lemma 2.1. Let G, S, $T^j$ , X as above and let ${\phi }_j={\left({\phi }_j\right)}_1+{\left({\phi }_j\right)}_2$ , ${\psi }_j={\left({\psi }_j\right)}_1+{\left({\psi }_j\right)}_2:S\to ℂ$ be functions. Then, formally,

$\left({\left({\phi }_j\right)}_1+{\left({\phi }_j\right)}_2\right)T^j\ast \left({\left({\psi }_j\right)}_1+{\left({\psi }_j\right)}_2\right)T^j=\left(\left({\left({\phi }_j\right)}_1+{\left({\phi }_j\right)}_2\right)\left({\left({\psi }_j\right)}_1+{\left({\psi }_j\right)}_2\right)\right)T^j$ .

Proof. Fix $\left(1+\epsilon \right)\in G$ . If $\left(1+\epsilon \right)\in G$ is such that $\left(1+\epsilon \right)\notin S\cap \left(1+\epsilon \right)S^{-1}$ then $\left({\left({\psi }_j\right)}_1+{\left({\psi }_j\right)}_2\right)\left(1+\epsilon \right)=0$ (in case $\left(1+\epsilon \right)\notin S$ or $\left({\left({\psi }_j\right)}_1+{\left({\psi }_j\right)}_2\right)\left({\left(1+\epsilon \right)}^{-1}\left(1+\epsilon \right)\right)=0$ (in case $\left(1+\epsilon \right)\notin \left(1+\epsilon \right)S^{-1}$ ). On the other hand, if $\left(1+\epsilon \right)\in S\cap \left(1+\epsilon \right)S^{-1}$ then $\left(1+\epsilon \right)$ , $\left(\frac{1+\epsilon }{\epsilon }\right)\left(1+\epsilon \right)\in S$ , which implies that $\left(1+\epsilon \right)\in S$ and $T^j\left(1+\epsilon \right)T^j\left(\left(\frac{1+\epsilon }{\epsilon }\right)\left(1+\epsilon \right)\right)=T^j\left(1+\epsilon \right)$ . Hence, formally

$\begin{array}{l}{\displaystyle \underset{j}{\sum }\left(\left({\left({\phi }_j\right)}_1+{\left({\phi }_j\right)}_2\right)T^j\ast \left({\left({\psi }_j\right)}_1+{\left({\psi }_j\right)}_2\right)T^j\right)\left(1+\epsilon \right)}\\ ={\displaystyle \underset{G}{\int }{\displaystyle \underset{j}{\sum }\left(\left({\left({\phi }_j\right)}_1+{\left({\phi }_j\right)}_2\right)T^j\right)\left(1+\epsilon \right)\left(\left({\left({\psi }_j\right)}_1+{\left({\psi }_j\right)}_2\right)T^j\right)\left(\left(\frac{1+\epsilon }{\epsilon }\right)\left(1+\epsilon \right)\right)d\left(1+\epsilon \right)}}\\ ={\displaystyle \underset{G}{\int }{\displaystyle \underset{j}{\sum }\left({\left({\phi }_j\right)}_1+{\left({\phi }_j\right)}_2\right)\left(1+\epsilon \right)\left({\left({\psi }_j\right)}_1+{\left({\psi }_j\right)}_2\right)\left(\left(\frac{1+\epsilon }{\epsilon }\right)\left(1+\epsilon \right)\right)}}\\ \times \left(T^j\left(1+\epsilon \right)T^j\left(\left(\frac{1+\epsilon }{\epsilon }\right)\left(1+\epsilon \right)\right)d\left(1+\epsilon \right)\right)\end{array}$

$\begin{array}{l}={\displaystyle \underset{S\cap \left(1+\epsilon \right)S^{-1}}{\int }{\displaystyle \underset{j}{\sum }\left({\left({\phi }_j\right)}_1+{\left({\phi }_j\right)}_2\right)\left(1+\epsilon \right)\left({\left({\psi }_j\right)}_1+{\left({\psi }_j\right)}_2\right)\left(\left(\frac{1+\epsilon }{\epsilon }\right)\left(1+\epsilon \right)\right)}}\\ \times \left(T^j\left(1+\epsilon \right)T^j\left(\left(\frac{1+\epsilon }{\epsilon }\right)\left(1+\epsilon \right)\right)d\left(1+\epsilon \right)\right)\end{array}$

$\begin{array}{l}={\displaystyle \underset{S\cap \left(1+\epsilon \right)S^{-1}}{\int }{\displaystyle \underset{j}{\sum }\left({\left({\phi }_j\right)}_1+{\left({\phi }_j\right)}_2\right)\left(1+\epsilon \right)\left({\left({\psi }_j\right)}_1+{\left({\psi }_j\right)}_2\right)\left(\left(\frac{1+\epsilon }{\epsilon }\right)\left(1+\epsilon \right)\right)}}\\ \times d\left(1+\epsilon \right)T^j\left(1+\epsilon \right)\end{array}$

$\begin{array}{l}={\displaystyle \underset{G}{\int }{\displaystyle \underset{j}{\sum }\left({\left({\phi }_j\right)}_1+{\left({\phi }_j\right)}_2\right)\left(1+\epsilon \right)\left({\left({\psi }_j\right)}_1+{\left({\psi }_j\right)}_2\right)\left({\left(1+\epsilon \right)}^{-1}\left(1+\epsilon \right)\right)}}\\ \times d\left(1+\epsilon \right)T^j\left(1+\epsilon \right)\\ ={\displaystyle \underset{j}{\sum }}\left(\left(\left({\left({\phi }_j\right)}_1+{\left({\phi }_j\right)}_2\right)\ast \left({\left({\psi }_j\right)}_1+{\left({\psi }_j\right)}_2\right)\right)T^j\right)\left(1+\epsilon \right)\end{array}$ ∎

For a function $F:G\to X$ and measures ${\mu }^j$ on G let us abbreviate

$\langle F,{\displaystyle \underset{j}{\sum }{\mu }^j}\rangle ={\displaystyle {\int }_{G}F\left(1+\epsilon \right){\displaystyle \underset{j}{\sum }{\mu }^j\left(d\left(1+\epsilon \right)\right)}}$

Defined in whatever weak sense. Stretch this notation to apply to all cases where it is reasonable. For example, ${\mu }^j$ could be a vector measure with values in $X^{\prime }$ or in $\mathcal{L}\left(X\right)$ .

The reflection $F^{\sim }$ of F is defined by

${\mathcal{F}}^{~}:G\to X,{\mathcal{F}}^{~}\left(1+\epsilon \right):=F\left(\left(\frac{1+\epsilon }{\epsilon }\right)\right).$

If $H:G\to \mathcal{L}\left(X\right)$ is the sequence of operators-valued function, write $H\ast F$ for the convolution $H\ast F$ is defined also by (2.2). Furthermore, let

$\left({\mu }^j\ast F\right)\left(1+\epsilon \right){\displaystyle {\int }_{G}F\left(\left(\frac{1+\epsilon }{\epsilon }\right)\left(1+\epsilon \right)\right){\mu }^{j}d\left(1+\epsilon \right)},\left(\left(1+\epsilon \right)\in G\right)$

which is in coherence with the definitions above if ${\mu }^j$ has density and scalars are identified with their induced dilation sequence of operators. ∎

The next lemma is almost a trivially.

Lemma 2.2. Let $H:G\to \mathcal{L}\left(X\right)$ , $F:G\to X$ and ${\mu }^j$ a measure on G. Then $\langle H\ast F,{\mu }^j\rangle =\langle H,{\mu }^j\ast F^{\sim }\rangle$ formally.

Proof. Writing out the brackets into integrals, it is just Fubini’s theorem:

$\begin{array}{l}\langle H\ast F,{\displaystyle \underset{j}{\sum }{\mu }^j}\rangle \\ ={\displaystyle \underset{G}{\int }{\displaystyle \underset{G}{\int }H\left(1+\epsilon \right)F\left(\left(\frac{1+\epsilon }{\epsilon }\right)\left(1+\epsilon \right)\right)d\left(1+\epsilon \right){\displaystyle \underset{j}{\sum }{\mu }^j\left(d\left(1+\epsilon \right)\right)}}}\\ ={\displaystyle \underset{G}{\int }{\displaystyle \underset{G}{\int }H\left(1+\epsilon \right)F\left({\left(1\left(\frac{1+\epsilon }{\epsilon }\right)\epsilon \right)}^{-1}\left(1+\epsilon \right)\right){\displaystyle \underset{j}{\sum }{\mu }^j\left(d\left(1+\epsilon \right)\right)d\left(1+\epsilon \right)}}}\\ ={\displaystyle \underset{G}{\int }H\left(1+\epsilon \right){\displaystyle \int F^{\sim }\left(\left(\frac{1+\epsilon }{\epsilon }\right)\left(1+\epsilon \right)\right){\displaystyle \underset{j}{\sum }{\mu }^j\left(d\left(1+\epsilon \right)\right)d\left(1+\epsilon \right)}}}\\ ={\displaystyle \underset{G}{\int }H\left(1+\epsilon \right)\underset{j}{{\displaystyle \sum }}\left({\mu }^j\ast F^{\sim }\right)\left(1+\epsilon \right)d\left(1+\epsilon \right)}=\langle H,{\displaystyle \underset{j}{\sum }{\mu }^j}\ast F^{\sim }\rangle \end{array}$ ∎

If combine Lemmas 2.1 and 2.2 obtain the following.

Proposition 2.3. Let S be a closed sub-semigroup of G and let $T^j:S\to \mathcal{L}\left(X\right)$ be a strongly continuous representation. Let ${\phi }_j,{\psi }_j:S\to ℂ$ and let ${\mu }^j$ be a measure on S. Then, writing $\eta :={\phi }_j\ast {\psi }_j$ ,

$T_{\eta {\mu }^j}^j=\langle T^j,\left({\phi }_j\ast {\psi }_j\right){\mu }^j\rangle =\langle {\phi }_j{T}^j,{\mu }^j\ast {\left({\psi }_j{T}^j\right)}^{~}\rangle$

formally. This result can be interpreted as a factorization of the sequence of operators $T_{\eta {\mu }^j}^j$ as

$\begin{array}{ccc}\text{Φ}\left(G;X\right)& \stackrel{L_{\mu }}{\to }& \text{Ψ}\left(G;X\right)\\ l\uparrow & & \downarrow P\\ X& \stackrel{T_{\eta \mu }}{\to }& X\end{array}$ (2.3)

$T_{\eta {\mu }^j}^j=P\circ L_{{\mu }^j}\circ \iota$ , where

1) $\iota$ maps $x\in X$ to the weighted orbit

$\left(\iota x\right)\left(1+\epsilon \right):={\psi }_j\left(\frac{1+\epsilon }{\epsilon }\right)T^j\left(\frac{1+\epsilon }{\epsilon }\right)x\left(\left(1+\epsilon \right)\in G\right)$

2) $L_{{\mu }^j}$ are the convolution sequence of operators with ${\mu }^j$

$L_{{\mu }^j}F:={\mu }^j\ast F;$

3) P maps an X-valued function on G back to an element of X by integrating against ${\phi }_j{T}^j$ :

$PF=\langle {\displaystyle \underset{j}{\sum }{\phi }_j{T}^j},F\rangle ={\displaystyle \underset{G}{\int }{\displaystyle \underset{j}{\sum }{\phi }_j\left(1+\epsilon \right)T^j\left(1+\epsilon \right)F\left(1+\epsilon \right)d\left(1+\epsilon \right)}},$

4) $\Phi \left(G;X\right)$ , $\Psi \left(G;X\right)$ are function spaces such that $\iota :X\to \Phi \left(G;X\right)$ and $P:\Psi \left(G;X\right)\to X$ are meaningful and bounded.

Call a factorization of the form (2.3) a transference identity. It induces a transference series estimates

${\displaystyle \underset{j}{\sum }{\Vert T_{\eta {\mu }^j}^j\Vert }_{\mathcal{L}\left(X\right)}}\le \Vert P\Vert {\Vert {\displaystyle \underset{j}{\sum }{L}_{{\mu }^j}}\Vert }_{\mathcal{L}\left(\Phi \left(G,X\right),\Psi \left(G,X\right)\right)}\Vert l\Vert$ (2.4)

3. Transference Principles for Groups

Shall explain that the classical transference principle of Berkson-Gillespie-Muhly [5] for uniformly bounded groups and the recent one for general C₀-groups [8] are instances of the explained technique (see, e.g., [1] ).

3.1. Unbounded C₀-Groups

Take $G=S=ℝ$ and let $U={\left(U\left(1+\epsilon \right)\right)}_{\left(1+\epsilon \right)\in ℝ}:ℝ\to \mathcal{L}\left(X\right)$ .

Be a strongly continuous representation on the Banach space X. Then U is exponentially bounded, i.e., its exponential type

$\theta \left(U\right):=\left\{\inf \epsilon >-1|\exists M\ge 0:U\left(1+\epsilon \right)\le M{\text{e}}^{\left(1+\epsilon \right)\left|(1+\epsilon )\right|}\left(\left(1+\epsilon \right)\in ℝ\right)\right\}$

is finite. Choose $\beta +\epsilon >\left(1+\epsilon \right)>\theta \left(U\right)$ and take a measure ${\mu }^j$ on $ℝ$ such that

${\left({\mu }^j\right)}_{\left(1+\epsilon \right)}:=\cosh \left(\left(1+\epsilon \right)\cdot \right){\mu }^j\in M(ℝ)$

is a finite measure. Then ${\displaystyle {\sum }_j{U}_{{\mu }^j}}={\displaystyle {\int }_{ℝ}\left({\displaystyle {\sum }_j{\mu }^j}\right)}$ is well-defined. It turns out [8] that one can factorize

$\eta =\frac{1}{\cosh \left(\left(1+\epsilon \right)\cdot \right)}={\phi }_j\ast {\psi }_j$

where ${\psi }_j=1/\text{cosh}\left(\left(\beta +\epsilon \right)\cdot \right)$ and $\cosh \left(\left(1+\epsilon \right)\cdot \right){\phi }_j=O\left(1\right)$ . Obtain ${\mu }^j=\eta {\pi }_{\left(1+\epsilon \right)}$ and, writing ${\left({\mu }^j\right)}_{\left(1+\epsilon \right)}$ for ${\mu }^j$ in Proposition 2.3,

$U_{{\mu }^j}=U_{\eta {\left({\mu }^j\right)}_{\left(1+\epsilon \right)}}={\phi }_j\langle U,{\left({\mu }^j\right)}_{\left(1+\epsilon \right)}\ast {\left({\phi }_{j}U\right)}^{\sim }\rangle =P\circ L_{{\left({\mu }^j\right)}_{\left(1+\epsilon \right)}}\circ \iota$ (3.1)

If $-i{A}_j$ is the sequence generators of U and $f_j=F{\mu }^j$ is the Fourier transform of ${\mu }^j$ , one writes

${\displaystyle \underset{j}{\sum }{f}_j\left(A_j\right)}={\displaystyle \underset{j}{\sum }{U}_{\left({\mu }^j\right)}}={\displaystyle \underset{ℝ}{\int }U\left(1+\epsilon \right){\displaystyle \underset{j}{\sum }{\mu }^j\left(d\left(1+\epsilon \right)\right)}}$

which is well-defined because the Fourier transform is injective. Applying the transference estimate (2.4) with $\varphi \left(ℝ;X\right)=\Psi \left(ℝ;X\right):=L^{\left(1+\epsilon \right)}\left(ℝ;X\right)$ as the function spaces as in [8] leads to the series estimates

${\displaystyle \underset{j}{\sum }\Vert f_j\left(A_j\right)\Vert }\le \frac{1}{2}({\Vert {\displaystyle \underset{j}{\sum }{f}_j\left(\cdot +i\left(1+\epsilon \right)\right)}\Vert }_{{\mathcal{M}}_{\left(1+\epsilon \right),X}\left(ℝ\right)}+{\Vert {\displaystyle \underset{j}{\sum }{f}_j\left(\cdot -i\left(1+\epsilon \right)\right)}\Vert }_{{\mathcal{M}}_{\left(1+\epsilon \right),X}(ℝ))}$

where ${\mathcal{M}}_{\left(1+\epsilon \right),X}\left(ℝ\right)$ denotes the space of all (scalar-valued) bounded Fourier multipliers on $L^{1+\epsilon }\left(ℝ;X\right)$ . In the case that X is a UMD space one can use the Mikhlin type result for Fourier multipliers on $L^{1+\epsilon }\left(ℝ;X\right)$ to obtain a generalization of the Hieber Pruss theorem [22] to unbounded groups, see ( [8] , Theorem 3.6).

If $\epsilon =1$ and $X=H$ , this Fourier multiplier norm coincides with the sup-norm by Plancherel’s theorem, and by the maximum principle one obtains the $H^{\infty }$ -series estimates

${\displaystyle \underset{j}{\sum }\Vert f_j\left(A_j\right)\Vert }\le {\Vert {\displaystyle \underset{j}{\sum }{f}_j}\Vert }_{H^{\infty }\left(St\left(1+\epsilon \right)\right)},$ (3.2)

where

$St\left(1+\epsilon \right):=\left\{z\in ℂ|\left|\mathrm{Im}z\right|<\left(1+\epsilon \right)\right\}$

Is the vertical strip of height $2\left(1+\epsilon \right)$ , symmetric about the real axis. This result is originally due to Boyadzhiev and De Laubenfels [23] and is closely related to McIntosh’s theorem on $H^{\infty }$ -calculus for sectorial operators with bounded imaginary powers from [24] , see ( [8] , Corollary 3.7) and ( [13] , Chapter 7).

3.2. Bounded Groups: The Classical Case

The classical transference principle, in the form put forward by Berkson, Gillespie and Muhly in [3] read as follows Let G be a locally compact amenable group,

let $U={\left(U\left(1+\epsilon \right)\right)}_{\left(1+\epsilon \right)\in G}$ be a uniformly bounded, strongly continuous representation of G on a Banach space X, and let $0<\epsilon <\infty$ , $\epsilon >0$ . Then

$\Vert {\displaystyle \underset{G}{\int }U\left(1+\epsilon \right){\displaystyle \underset{j}{\sum }{\mu }^j\left(d\left(1+\epsilon \right)\right)}}\Vert \le M^2{\Vert {\displaystyle \underset{j}{\sum }{L}_{\left({\mu }^j\right)}}\Vert }_{\mathcal{L}\left(L^{\left(1+\epsilon \right)}\left(G,X\right)\right)}$

for every bounded measures ${\mu }^j\in M\left(G\right)$ . (Here $:={\sup }_{\left(1+\epsilon \right)\in G}\Vert U\left(1+\epsilon \right)\Vert$ .)

Shall review its proof in the special case of $G=ℝ$ (but the general case is analogous using Følner’s condition, see ( [3] , p.10)). First, fix $n,N>0$ and suppose that $\text{supp}\left({\mu }^j\right)\subset \left[-N,N\right]$ . Then

$\eta ={\phi }_j\ast {\psi }_j=\frac{1}{2n}{1}_{\left[-n,n\right]}\ast 1_{\left[-N-n,N+n\right]}=1\text{on}\left[-N,N\right]$

So $\eta {\mu }^j={\mu }^j$ ; applying the transference estimate (2.4) with the function space $\varphi \left(ℝ;X\right)=\Psi \left(ℝ;X\right)\cdot L^{\left(1+\epsilon \right)}\left(ℝ;X\right)$ together with Holder’s inequality yields

$\begin{array}{c}{\displaystyle \underset{j}{\sum }\Vert T_{{\mu }^j}^j\Vert }\le M^2{\Vert {\phi }_j\Vert }_{\frac{1+\epsilon }{\epsilon }}{\Vert {\psi }_j\Vert }_{\left(1+\epsilon \right)}{\Vert L_{\left({\mu }^j\right)}\Vert }_{\mathcal{L}\left(L^{1+\epsilon }\left(ℝ,X\right)\right)}\\ =M^2{\left(2n\right)}^{-\left(\frac{1}{1+\epsilon }\right)}{\left(2N+2n\right)}^{\left(\frac{1}{1+\epsilon }\right)}\underset{}{{\displaystyle \sum }}{\Vert L_{\left({\mu }^j\right)}\Vert }_{\mathcal{L}\left(L^{1+\epsilon }\left(ℝ,X\right)\right)}\\ =M^2{\left(1+\frac{N}{n}\right)}^{\left(\frac{1}{1+\epsilon }\right)}{\displaystyle \underset{j}{\sum }{\Vert L_{\left({\mu }^j\right)}\Vert }_{\mathcal{L}\left(L^{1+\epsilon }\left(ℝ,X\right)\right)}}\end{array}$

Finally, let $n\to \infty$ and approximate a general ${\mu }^j\in M\left(ℝ\right)$ by measures of finite support.

Remark 3.1. This proof shows a feature to which pointed already in the introduction, but which was not represent in the case of unbounded groups treated above. Here, an additional optimization argument appears which is based on some freedom in the choice of the auxiliary functions ${\phi }_j$ and ${\psi }_j$ . Indeed, ${\phi }_j$ and ${\psi }_j$ can vary as long as ${\mu }^j=\left({\phi }_j\ast {\psi }_j\right){\mu }^j$ which amounts to ${\phi }_j\ast {\psi }_j=1$ on $\text{supp}\left({\mu }^j\right)$ .

Remark 3.2. A transference principle for bounded cosine functions instead of groups was for the first time established and applied in [25] .

4. A Transference Principle for Discrete and Continuous Operator Semigroup

Shall apply the transference method to the sequence of operators semi groups, i.e., strongly continuous representations of the semigroup ${ℝ}_{+}$ (continuous case) or ${\mathbb{Z}}_{+}$ (discrete case) (see, e.g., [1] ).

4.1. The Continuous Case

Let $T^j={\left(T^j\left(1+\epsilon \right)\right)}_{\epsilon >-1}$ be strongly continuous (i.e. C₀-) one-parameter semigroup on a (non-trivial) Banach space X. By standard semigroup theory [12] , $T^j$ is exponentially bounded, i.e., there exists $\epsilon >-1$ such that

${\displaystyle {\sum }_j\Vert T^j\left(1+\epsilon \right)\Vert }\le \left(1+\epsilon \right){\text{e}}^{-\left(1+\epsilon \right)\left(1+\epsilon \right)}$

for all $\epsilon >-1$ . Consider complex measures ${\mu }^j$ on ${ℝ}_{+}:=\left[0,\infty \right)$ such that

${\displaystyle \underset{0}{\overset{\infty }{\int }}{\displaystyle \underset{j}{\sum }\Vert T^j\left(1+\epsilon \right)\Vert \left|{\mu }^j\right|\left(d\left(1+\epsilon \right)\right)}}<\infty$

If ${\mu }^j$ are Laplace transformable and if $f_j=\mathcal{L}{\mu }^j$ are its Laplace (-Stieltjes) transform

${\displaystyle \underset{j}{\sum }\mathcal{L}{\mu }^j\left(z\right)}={\displaystyle \underset{0}{\overset{\infty }{\int }}{\text{e}}^{-z\left(1+\epsilon \right)}{\displaystyle \underset{j}{\sum }{\mu }^j\left(d\left(1+\epsilon \right)\right)}}$

then use (similar to the group case) the abbreviation

${\displaystyle \underset{j}{\sum }{f}_j\left(A_j\right)}={\displaystyle \underset{j}{\sum }{T}_{{\mu }^j}^j}={\displaystyle \underset{0}{\overset{\infty }{\int }}{\displaystyle \underset{j}{\sum }{T}^j\left(1+\epsilon \right){\mu }^j\left(d\left(1+\epsilon \right)\right)}}$

where $-A_j$ are the sequence of generators of the semigroup $T^j$ . The mapping $f_j\mapsto f_j\left(A_j\right)$ are well-defined since the Laplace transform is injective, and is called the Hille-Phillips functional calculus for $A_j$ , see ( [13] , Section 3.3) and ( [26] , Chapter XV).

Theorem 4.1. Let $\left(0<\epsilon <\infty \right)$ , $\epsilon >0$ . Then there is a constant $\epsilon >-1$ such that

${\displaystyle \underset{j}{\sum }\Vert T_{{\mu }^j}^j\Vert }\le \left(1+\epsilon \right)\left(1+\text{log}\left(a+\epsilon /a\right)\right)\left(1+\epsilon \right){\left(a+\epsilon \right)}^2{\Vert {\displaystyle \underset{j}{\sum }{L}_{{\mu }^j}}\Vert }_{\mathcal{L}\left(L^{\left(1+\epsilon \right)}\left(ℝ,X\right)\right)}$ (4.1)