Explicit representation of compact linear operators in Banach spaces via polar sets

Tom 214 / 2013

David E. Edmunds, Jan Lang Studia Mathematica 214 (2013), 265-278 MSC: Primary 46B50, 47A75; Secondary 47A80, 47A58. DOI: 10.4064/sm214-3-5


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.


  • David E. EdmundsDepartment of Mathematics
    University of Sussex
    Pevensey I
    Brighton, BN1 9QH, United Kingdom
  • Jan LangDepartment of Mathematics
    The Ohio State University
    100 Math Tower
    231 West 18th Avenue
    Columbus, OH 43210-1174, U.S.A.

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