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.
