Vol. 59, No. 2, 1975

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Collectively compact sets and the ergodic theory of semi-groups

M. V. Deshpande

Vol. 59 (1975), No. 2, 399–405
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) = limn→∞(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 limt→∞T(t) to exist.

Mathematical Subject Classification
Primary: 47D05
Secondary: 47B05
Milestones
Received: 27 February 1974
Published: 1 August 1975
Authors
M. V. Deshpande