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

Abstract

We investigate the boundedness of the H ∞ -calculus by estimating the bound b(ε) of the mapping H ∞ → B(X): f → f (A)T (ε) for ε near zero. Here, −A generates the analytic semigroup T and H ∞ is the space of bounded analytic functions on a domain strictly containing the spectrum of A. We show that b(ε) = O(| log ε|) 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( | log ε|).

A Besov algebra calculus for generators of operator semigroups and related norm-estimates

“…The following corollary covers some ideas which are included in [78, Theorem 5.5.12, etc] using Peller's method, and in [44] 4), and a similar result for bounded holomorphic semigroups on Banach spaces was given by Schwenninger [67]. The results are immediate consequences of Lemma 3.2(2).…”

Functional calculus estimates for Tadmor–Ritt operators

“…Indeed, such an example can be constructed for an L 2 -space on the unit circle with suitable weight, see [26,Thm. 5.2], [42], and [44,Section 4.3]. In fact, the example in [42,Thm.…”

“…A careful study of this sketch reveals that it is based on a similar approach as in Bakaev's proof, which, with a sharper estimation and some additional work, actually yields (1.6). We will encounter a similar approach in the proof of Theorem 2.3, which was actually motivated by a result of the author for analytic semigroups, [42].…”

Functional Calculus for C 0-semigroups Using Infinite-dimensional Systems Theory

“…Moreover, the techniques here and in Vitse's work [15] require that the functions f are bounded, analytic on a half-plane. In [13] it is shown that the corresponding result is even true for functions f that are only bounded, analytic on sectors which are larger than the sectorality sector of the generator A.…”