Abstract
AbstractWe construct two bounded functional calculi for sectorial operators on Banach spaces, which enhance the functional calculus for analytic Besov functions, by extending the class of functions, generalising and sharpening estimates and adapting the calculus to the angle of sectoriality. The calculi are based on appropriate reproducing formulas, they are compatible with standard functional calculi and they admit appropriate convergence lemmas and spectral mapping theorems. To achieve this, we develop the theory of associated function spaces in ways that are interesting and significant. As consequences of our calculi, we derive several well-known operator norm estimates and provide generalisations of some of them.
Highlights
We construct two bounded functional calculi for sectorial operators on Banach spaces, which enhance the functional calculus for analytic Besov functions, by extending the class of functions, generalising and sharpening estimates and adapting the calculus to the angle of sectoriality
The calculi are based on appropriate reproducing formulas, they are compatible with standard functional calculi and they admit appropriate convergence lemmas and spectral mapping theorems
The study of functional calculus based on the algebra B was initiated in [57] for generators of bounded semigroups on Hilbert spaces and in [53] for generators of holomorphic semigroups
Summary
The theory of functional calculi forms a basis for the study of sectorial operators and semigroup generators. The study of functional calculus based on the algebra B was initiated in [57] for generators of bounded semigroups on Hilbert spaces and in [53] for generators of holomorphic semigroups These works adapted and extended the approach from [47] to a more demanding and involved setting of unbounded operators. Inspired by ideas in [47] for the discrete case, they extended the B-calculus for those operators to a strictly larger Banach algebra A in which B is continuously embedded Their extension is complementary to our extensions to the D and H-calculi for negative generators of bounded holomorphic C0-semigroups on Banach spaces. We are grateful to Loris Arnold for pointing out several defects in the original version of this article
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