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.

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