Reducing Subspaces of Complex Symmetric Operators

Abstract

An operator T on a separable, infinite dimensional, complex Hilbert space $${\mathcal {H}}$$ is called complex symmetric if T has a symmetric matrix representation relative to some orthonormal basis for $${\mathcal {H}}$$ . This paper aims to describe reducing subspaces of complex symmetric operators from the view point of approximation. In particular, given a complex symmetric operator T, $$1\le n\le \aleph _0$$ and $$\varepsilon >0$$ , it is proved that there exists a compact operator K with $$\Vert K\Vert <\varepsilon $$ such that $$T+K$$ is complex symmetric and has exactly n minimal reducing subspaces.