Strictly singular operator

From Wikipedia, the free encyclopedia
  (Redirected from Strictly singular)
Jump to: navigation, search

In functional analysis, a branch of mathematics, a strictly singular operator is a bounded linear operator L from a Banach space X to another Banach space Y, such that it is not an isomorphism, and fails to be an isomorphism on any infinite dimensional subspace of X. Any compact operator is strictly singular, but not vice-versa.[1][2]

Every bounded linear map T:l_p\to l_q, for 1\le q, p < \infty, p\ne q, is strictly singular. Here, l_p and l_q are sequence spaces. Similarly, every bounded linear map T:c_0\to l_p and T:l_p\to c_0, for 1\le p < \infty, is strictly singular. Here c_0 is the Banach space of sequences converging to zero. This is a corollary of Pitt's theorem, which states that such T, for q < p, are compact.

References[edit]

  1. ^ N.L. Carothers, A Short Course on Banach Space Theory, (2005) London Mathematical Society Student Texts 64, Cambridge University Press.
  2. ^ C. J. Read, Strictly singular operators and the invariant subspace problem, Studia Mathematica 132 (3) (1999), 203-226. fulltext