Explicit representation of compact linear operators in Banach spaces via polar sets
Volume 214 / 2013
Studia Mathematica 214 (2013), 265-278
MSC: Primary 46B50, 47A75; Secondary 47A80, 47A58.
DOI: 10.4064/sm214-3-5
Abstract
We consider a compact linear map $T$ acting between Banach spaces both of which are uniformly convex and uniformly smooth; it is supposed that $T$ has trivial kernel and range dense in the target space. 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. This provides a Banach space version of the well-known Hilbert space result of E. Schmidt.
