Vol. 25, No. 3, 1968

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Spectral analysis of collectively compact, strongly convergent operator sequences

Philip Marshall Anselone and Theodore Windle Palmer

Vol. 25 (1968), No. 3, 423–431

A set of linear operators on a normed linear space is collectively compact if and only if the union of the images of the unit ball has compact closure. Bounded linear operators T and Tn, n = 1,2, , such that Tn T strongly and {Tn T} is collectively compact are investigated. The theory somewhat resembles that for Tn T∥→ 0. The spectrum of Tn is eventually contained in any neighborhood of the spectrum of T. If f(T) is defined by the operational calculus, then f(Tn) is eventually defined, f(Tn) f(T) strongly, and {f(Tn) f(T)} is collectively compact. If f(Tn) and f(T) are spectral projections, the corresponding spectral subspaces eventully have the same dimension. Other results compare eigenvalues and generalized eigenmanifolds of Tn and T.

Mathematical Subject Classification
Primary: 47.45
Received: 27 February 1967
Published: 1 June 1968
Philip Marshall Anselone
Theodore Windle Palmer