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A transference principle for general groups and functional calculus on UMD spaces. (English) Zbl 1184.47005

Summary: Let \(-iA\) be the generator of a \(C_0\)-group \((U(s)_{s\in\mathbb R}\) on a Banach space \(X\) and \(\omega>\theta(U)\), the group type of \(U\). We prove a transference principle that allows to estimate \(\|f(A)\|\) in terms of the \(L^p(\mathbb R,X)\)-Fourier multiplier norm of \(f(\cdot\pm i\omega)\). If \(X\) is a Hilbert space, this yields new proofs of important results of A.McIntosh [Proc.Cent.Math.Anal.Aust.Natl.Univ.14, 210–231 (1986; Zbl 0634.47016)] and K.N.Boyadzhiev and R.de Laubenfels [Proc.Am.Math.Soc.120, 127–136 (1994; Zbl 0820.47017)]. If \(X\) is a UMD space, one obtains a bounded \(H^\infty_1\)-calculus of \(A\) on horizontal strips. Related results for sectorial and parabola-type operators follow. Finally, it is proved that each generator of a cosine function on a UMD space has bounded \(H^\infty\)-calculus on sectors.

MSC:

47A60 Functional calculus for linear operators
47D06 One-parameter semigroups and linear evolution equations
44A40 Calculus of Mikusiński and other operational calculi
42A45 Multipliers in one variable harmonic analysis
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