A spectral theorem for bilinear compact operators in Hilbert spaces

Abstract

Current work defines Schur representation of a bilinear operator $$T: H \times H \rightarrow H$$ , where H is a separable Hilbert space. Introducing the concepts of self-adjoint bilinear operators, ordered eigenvalues and eigenvectors, we prove that if T is compact, self-adjoint, and its eigenvalues are ordered, then T has a Schur representation, thus obtaining a spectral theorem for T on real Hilbert spaces. We prove that the hypothesis of the existence of ordered eigenvalues is fundamental.