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Series representation of compact linear operators in Banach spaces

Edmunds D., Lang J.

Commentationes Mathematicae

2016

Vol 56, No. 1

17--27

Let p∈(1,∞) and I=(0,1); suppose that T:Lp(I)→Lp(I) is a~compact linear map with trivial kernel and range dense in Lp(I). It is shown that if the Gelfand numbers of T decay sufficiently quickly, then the action of T is given by a series with calculable coefficients. The special properties of Lp(I) enable this to be established under weaker conditions on the Gelfand numbers than in earlier work set in the context of more general spaces.