Strictly singular operator

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] The class of all strictly singular operators is quite nice in the sense that it forms a norm-closed operator ideal.

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

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