# Journal of Operator Theory

Volume 62, Issue 2, Fall 2009 pp. 297-326.

The $a$-Drazin inverse and ergodic behaviour of semigroups and cosine operator functions**Authors**: P.L. Butzer (1) and J.J. Koliha (2)

**Author institution:**(1) Lehrstuhl A f\"ur Mathematik, RWTH Aachen, D-52056 Aachen, Germany

(2) Department of Mathematics and Statistics, The University of Melbourne, Melbourne VIC 3010, Australia

**Summary:**The paper introduces a special type of a Drazin-like inverse for closed linear operators that arises naturally in ergodic theory of operator semigroups and cosine operator functions. The Drazin inverse for closed linear operators defined by Nashed and Zhao and in a more general form by Koliha and Tran is not sufficiently general to be applicable to operator semigroups. The $a$-Drazin inverse is in general a closed, not necessarily bounded, operator. The paper gives applications of the inverse to partial differential equations.

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