Collectively compact sets and the ergodic theory of semi-groups

M. V. Deshpande

Vol. 59 (1975), No. 2, 399–405

DOI: 10.2140/pjm.1975.59.399

Abstract

Let {T(t) : t ≧ 0} be a uniformly bounded semi-group of linear operators on a Banach space X such that 1 is an eigenvalue of each T(t) and T(a) is compact for some a > 0. Then the ergodic limit A(t) = lim_n→∞(1∕n){T(t) + T²(t) + ⋯ + Tⁿ(t)} exists for each t. In this paper it is proved that if each T(t),t > 0, is compact and 1 is, in a certain sense, an isolated eigenvalue of all T(t), then for t > 0, the dimension of the null space of T(t) − I is independent of t. Sufficient conditions are also obtained for the lim_t→∞T(t) to exist.

Mathematical Subject Classification

Primary: 47D05

Secondary: 47B05

Milestones

Received: 27 February 1974

Published: 1 August 1975