Cosine Families of Unbounded Operators

Abstract

The initial value problem for is shown to have a unique strong solution provided the spectrum of the linear time-independent operator –A exhibits a parabolic condensation along the negative real axis. The particular case when A is a nonsymmetric perturbation of a self-adjoint elliptic partial differential operator does not fit in the commonly known framework of strongly continuous cosine families because in this case the requirement of well-posedness is violated. Making use of Dunford's functional calculus we develop instead a concept of cosine families of unbounded linear operators which turns out to be the appropriate setup for the case above. We extend herewith results obtained in an earlier paper¹ by completely different methods.

Additional information

Notes on contributors

R. P. Gilbert

M. Sova