A transference principle for general groups and functional calculus on UMD spaces

Haase, Markus

doi:10.1007/s00208-009-0347-3

Cited by 22 publications

(56 citation statements)

References 29 publications

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“…One can obtain nontrivial functional calculus results for C 0 -groups even when X is not a Hilbert space. For example, it was shown in [25] that if −iA generates a In [29] a similar statement was obtained on general Banach spaces, where one restricts the calculus to real interpolation spaces between X and the domain of A.…”

Section: Introductionmentioning

confidence: 58%

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Functional calculus forC0-groups using type and cotype

Rozendaal

2018

The Quarterly Journal of Mathematics

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We study the functional calculus properties of generators of C 0groups under type and cotype assumptions on the underlying Banach space. In particular, we show the following. Let −iA generate a C 0 -group on a Banach space X with type p ∈ [1, 2] and cotype q ∈ [2, ∞). Then f (A) : (X, D(A)) 1 p − 1 q ,1 → X is bounded for each bounded holomorphic function f on a sufficiently large strip. As a corollary of this result, for sectorial operators we quantify the gap between bounded imaginary powers and a bounded H ∞ -calculus in terms of the type and cotype of the underlying Banach space. For cosine functions we obtain similar results as for C 0 -groups. We extend our theorems to R-bounded operator-valued calculi, and we give an application to the theory of rational approximation of C 0 -groups.2010 Mathematics Subject Classification. Primary 47A60; Secondary 47D03, 46B20, 42A45.

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Section: Introductionmentioning

confidence: 58%

“…To prove Theorem 1.1 we use transference principles going back to [8,11,13]. The transference technique has been applied to functional calculus theory in [14], [30] and [25,26,28,29]. In [29] an interpolation version of the transference principle for unbounded groups from [25] was established.…”

Section: Introductionmentioning

confidence: 99%

Functional calculus forC0-groups using type and cotype

Rozendaal

2018

The Quarterly Journal of Mathematics

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“…The equivalence of (i) and (iii) is well-known see, e.g., [28,Theorem 7.4.7 and Corollary 7.4.6] and [32,Section 5] for related matters in the case of arbitrary UMD spaces.…”

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confidence: 91%

Resolvent characterisation of generators of cosine functions and C 0-groups

Król

2013

J. Evol. Equ.

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Abstract. The paper provides new characterisations of generators of cosine functions and C 0 -groups on UMD spaces and their applications to some classical problems in cosine function theory. In particular, we show that on UMD spaces, generators of cosine functions and C 0 -groups can be characterised by means of a complex inversion formula. This allows us to provide a strikingly elementary proof of Fattorini's result on square root reduction for cosine function generators on UMD spaces. Moreover, we give a cosine function analogue of McIntosh's characterisation of the boundedness of the H ∞ functional calculus for sectorial operators in terms of square function estimates. Another result says that the class of cosine function generators on a Hilbert space is exactly the class of operators which possess a dilation to a multiplication operator on a vector-valued L 2 space. Finally, we prove a cosine function analogue of the Gomilko-Feng-Shi characterisation of C 0 -semigroup generators and apply it to answer in the affirmative a question by Fattorini on the growth bounds of perturbed cosine functions on Hilbert spaces.

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“…Here the boundedness of the functional calculus is crucial, and the algebra of functions it is defined on reflects its quality. Connections between growth conditions on the group U(t) and functional calculus have been investigated in several articles [2,10,14]: First, if X is a Hilbert space, then B has a bounded C 0 (R) calculus (and consequently is of scalar type) if and only if U(t) is a uniformly bounded group. This has been extended by Boyadzhiev and deLaubenfels [2] to arbitrary groups.…”

Section: Introductionmentioning

confidence: 99%

Functional calculus and dilation for C 0-groups of polynomial growth

Kriegler

2012

Semigroup Forum

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Let U(t) = e itB be a C 0 -group on a Banach space X.which is a Banach algebra. It is shown that U(t) ≤ C(1 + |t|) α for all t ∈ R if and only if the generator B has a bounded E α ∞ functional calculus, under some minimal hypotheses, which exclude simple counterexamples. A third equivalent condition is that U(t) admits a dilation to a shift group on some space of functions R → X. In the case U(t) = A it with some sectorial operator A, we use this calculus to show optimal bounds for fractions of the semigroup generated by A, resolvent functions and variants of it. Finally, the E α ∞ calculus is compared with Besov functional calculi as considered in Cowling et al. (J. Aust. Math. Soc., Ser. A, 60(1): 1996) and Kriegler (Spectral multipliers, R-bounded homomorphisms, and analytic diffusion semigroups. PhD-thesis).

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A transference principle for general groups and functional calculus on UMD spaces

Cited by 22 publications

References 29 publications

Functional calculus forC0-groups using type and cotype

Functional calculus forC0-groups using type and cotype

Resolvent characterisation of generators of cosine functions and C 0-groups

Functional calculus and dilation for C 0-groups of polynomial growth

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