Abstract
We construct a new bounded functional calculus for the generators of bounded semigroups on Hilbert spaces and generators of bounded holomorphic semigroups on Banach spaces. The calculus is a natural (and strict) extension of the classical Hille–Phillips functional calculus, and it is compatible with the other well-known functional calculi. It satisfies the standard properties of functional calculi, provides a unified and direct approach to a number of norm-estimates in the literature, and allows improvements of some of them.
Highlights
This work was partially supported financially by a Leverhulme Trust Visiting Research Professorship and an NCN grant UMO-2017/27/B/ST1/00078, and inspirationally by the ambience of the Lamb and Flag, Oxford
The so-called holomorphic functional calculus for sectorial operators became an indispensable tool in applications of operator theory to PDEs and harmonic analysis
The reproducing formula determines the function algebra for the extended functional calculus, and it is basic for the calculus construction
Summary
We define B to be the space of those holomorphic functions f on C+ such that f B0 := sup | f (x + i y)| d x < ∞. Since U is contained in the closed unit ball of H ∞(C+), which is compact in the topology of uniform convergence on compact sets, by Montel’s Theorem, it suffices to consider a sequence ( fn) in U which converges to a holomorphic function f uniformly on compact sets and to show that f ∈ U. There is a superficial resemblance between (2.3) and the definition of the Hardy space H 1(C+) on the half-plane C+, which consists of the holomorphic functions f on C+ such that f H1 := sup | f (x + i y)| d y < ∞. By a consequence of the Phragmén–Lindelöf principle [74, Theorem 5.63], it follows that lim|z|→∞ f (z) = lims→±∞ f b(i s).
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