Schmidt Representation of Bilinear Operators on Hilbert Spaces

Abstract

Current work defines Schmidt representation of a bilinear operator $T:H_1\times H_2\to K$, where $H_1,H_2$ and $K$ are separable Hilbert spaces. Introducing the concept of singular value and ordered singular value, we prove that if $T$ is compact, and its singular values are ordered, then $T$ has a Schmidt representation on real Hilbert spaces. We prove that the hypothesis of existence of ordered singular values is fundamental.

27 pages