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Journal of Operator Theory

Volume 86, Issue 2, Fall 2021  pp. 331-354.

Perturbation by weakly continuous forms and semigroups on Hardy space

Authors:  Wolfgang Arendt (1), Isabelle Chalendar (2), Boitumelo Moletsane (3)
Author institution:(1) Institute of Applied Analysis, University of Ulm, Helmholtzstr. 18, D-89069 Ulm, Germany
(2) LAMA, Universite Gustave Eiffel, Universite Paris Est Creteil, CNRS, F-77454 Marne-la-Vallee, France
(3) University of the Witswatersrand, Johannesburg, South Africa

Summary: In the first part of the article perturbation of a closed form by a weakly continuous form is studied. This notion of weakly continuous perturbation is very handy and, as is shown in the article, leads to a new semigroup whose difference with the given semigroup consists of compact operators. We apply the results to elliptic operators on the Hardy space generalizing results from Semigroup Forum 95(2017), 281-292. A holomorphic semigroup operating on the Hardy space is obtained whose asymptotic behaviour is studied and which is compared with the semigroup generated by the elliptic operator with periodic boundary conditions on $L^2(0,2\pi)$.

Keywords: sesquilinear coercive form, essentially coercive forms, elliptic forms, selfadjoint operators, holomorphic semigroups, asymptotically compact semigroups, Hardy space

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