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Integral representation of functions on sectors, functional calculus and norm estimates. (English) Zbl 1046.47027

Summary: We find an explicit integral representation for bounded holomorphic functions \(f(z)\) on sectors \(|\text{Arg} (z)|<\psi\) in terms of the kernel \(z(z+\lambda)^{-2}\) and present some applications to operator theory. Namely, given a sectorial operator \(A\), we define the functional calculus \(A\to f (A)\) and find pointwise estimates and moment type inequalities for \(\| f(A)x\|\). We show that sectorial operators have a bounded \(H^\infty\)-functional calculus on a dense subspace. We also find exact estimates for the norm \(\| e^{-\lambda A}\|\) of analytic semigroups.

MSC:

47B44 Linear accretive operators, dissipative operators, etc.
30E20 Integration, integrals of Cauchy type, integral representations of analytic functions in the complex plane
47A60 Functional calculus for linear operators
47D06 One-parameter semigroups and linear evolution equations
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