On measuring unboundedness of the H∞-calculus for generators of analytic semigroups

Research output: Contribution to journalArticleAcademicpeer-review

Abstract

We investigate the boundedness of the H$^\infty$-calculus by estimating the bound b(ε) of the mapping H$^\infty$→B(X): f→f(A)T(ε) for ε near zero. Here, −A generates the analytic semigroup T and H$^\infty$ is the space of bounded analytic functions on a domain strictly containing the spectrum of A. We show that b(ε) =O(|logε|) in general, whereas b(ε) =O(1) for bounded calculi. This generalizes a result by Vitse and complements work by Haase and Rozendaal for non-analytic semigroups. We discuss the sharpness of our bounds and show that single square function estimates yield b(ε) =O(|logε|).
Original languageEnglish
Pages (from-to)49-84
Number of pages36
JournalJournal of functional analysis
Volume271
Issue number1
DOIs
Publication statusPublished - Jul 2016

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Analytic Semigroup
Calculus
Generator
Bounded Analytic Functions
Square Functions
Sharpness
Boundedness
Semigroup
Strictly
Complement
Generalise
Zero
Estimate

Cite this

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title = "On measuring unboundedness of the H∞-calculus for generators of analytic semigroups",
abstract = "We investigate the boundedness of the H$^\infty$-calculus by estimating the bound b(ε) of the mapping H$^\infty$→B(X): f→f(A)T(ε) for ε near zero. Here, −A generates the analytic semigroup T and H$^\infty$ is the space of bounded analytic functions on a domain strictly containing the spectrum of A. We show that b(ε) =O(|logε|) in general, whereas b(ε) =O(1) for bounded calculi. This generalizes a result by Vitse and complements work by Haase and Rozendaal for non-analytic semigroups. We discuss the sharpness of our bounds and show that single square function estimates yield b(ε) =O(|logε|).",
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On measuring unboundedness of the H∞-calculus for generators of analytic semigroups. / Schwenninger, F.L.

In: Journal of functional analysis, Vol. 271, No. 1, 07.2016, p. 49-84.

Research output: Contribution to journalArticleAcademicpeer-review

TY - JOUR

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AU - Schwenninger, F.L.

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AB - We investigate the boundedness of the H$^\infty$-calculus by estimating the bound b(ε) of the mapping H$^\infty$→B(X): f→f(A)T(ε) for ε near zero. Here, −A generates the analytic semigroup T and H$^\infty$ is the space of bounded analytic functions on a domain strictly containing the spectrum of A. We show that b(ε) =O(|logε|) in general, whereas b(ε) =O(1) for bounded calculi. This generalizes a result by Vitse and complements work by Haase and Rozendaal for non-analytic semigroups. We discuss the sharpness of our bounds and show that single square function estimates yield b(ε) =O(|logε|).

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