# 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 language English 49-84 36 Journal of functional analysis 271 1 https://doi.org/10.1016/j.jfa.2016.04.011 Published - 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

@article{10c9e6c9bbe84acc9ea1624ab31962b3,
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|).",
author = "F.L. Schwenninger",
year = "2016",
month = "7",
doi = "10.1016/j.jfa.2016.04.011",
language = "English",
volume = "271",
pages = "49--84",
journal = "Journal of functional analysis",
issn = "0022-1236",
publisher = "Academic Press Inc.",
number = "1",

}

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

Research output: Contribution to journalArticleAcademicpeer-review

TY - JOUR

T1 - On measuring unboundedness of the H∞-calculus for generators of analytic semigroups

AU - Schwenninger, F.L.

PY - 2016/7

Y1 - 2016/7

N2 - 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|).

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|).

U2 - 10.1016/j.jfa.2016.04.011

DO - 10.1016/j.jfa.2016.04.011

M3 - Article

VL - 271

SP - 49

EP - 84

JO - Journal of functional analysis

JF - Journal of functional analysis

SN - 0022-1236

IS - 1

ER -