Schmidt Representation of Bilinear Operators on Hilbert Spaces

TL;DR

This paper extends the Schmidt representation to bilinear operators on Hilbert spaces, establishing conditions under which such a representation exists, especially focusing on the role of singular values for compact operators.

Contribution

It introduces the concept of singular values for bilinear operators and proves the existence of Schmidt representation under specific conditions on these singular values.

Findings

Schmidt representation exists for compact bilinear operators with ordered singular values.

The concept of singular values is fundamental for the representation.

The results are established for real Hilbert spaces.

Abstract

Current work defines Schmidt representation of a bilinear operator $T: H_1 \times H_2 \rightarrow K$, 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.