On the spectra theory of operator bundles

Abstract

This paper primarily discusses the spectral theory of quadratic bundles of the form

Au-?Bu-?²Cu = ? ?H,

where H is an arbitrary Hilbert space and the linear operators A, B, C :H ->H are Hermitian and continuous. The main result is a Parseval equality and expansion theorem which does not rely on the concept of two-fold completeness as is usually required in the study of quadratic bundles. An extension of the theory to polynomial bundles is indicated.