Vol. 43, No. 2, 1972

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Collectively compact and semi-compact sets of linear operators in topological vector spaces

M. V. Deshpande and N. E. Joshi

Vol. 43 (1972), No. 2, 317–326

A set of linear operators from one topological vector space to another is said to be collectively compact (resp. semi-compact) if and only if the union of images of a neighbourhood of zero (respectively every bounded set) is relatively compact. In this paper sufficient conditions for a set of operators to be collectively compact or semi-compact are obtained. It is proved that if Tn T asymptotically, where X is quasi-complete and Tn are W-compact then {Tn T} is collectively compact. The final section deals with collectively weakly compact sets. It is proved that in a reflexive locally convex space a family of continuous endomorphisms is collectively weakly compact if and only if

𝒦∗ = {K∗ : E∗s → E ∗w∗}

is collectively compact.

Mathematical Subject Classification 2000
Primary: 47B99
Received: 10 June 1971
Revised: 30 November 1971
Published: 1 November 1972
M. V. Deshpande
N. E. Joshi