Journal of Operator Theory

Volume 52, Issue 1, Summer 2004 pp. 21-37.

A decomposition theorem for generators of strongly continuous groups on Hilbert spaces

Authors: Markus Haase
Author institution: Abt. Angewandte Analysis, Universitaet Ulm, Helmholzstrasse 18, 98069 Ulm, Germany

Summary: For the generator $A$ of a strongly continuous group on a Hilbert space, we modify Liapunov's method of changing the scalar product to obtain a decomposition $A = B + C$ with $B$ skew-adjoint and $C$ bounded and selfadjoint (with respect to the new scalar product). This yields a new proof of the fact that $A$ has bounded $H^\infty$--calculi on vertical strips. Furthermore we show that, with respect to the new scalar product, $A^2$ can be obtained by a closed sectorial form in the sense of Kato.

Keywords: $C_0$-group, Liapunov's direct method, $H^\infty$-calculus, cosine function, square root problem

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