On majorization, factorization, and range inclusion of operators on Hilbert space

Abstract

The purpose of this note is to show that a close relationship exists between the notions of majorization, factorization, and range inclusion for operators on a Hilbert space. Although fragments of these results are to be found scattered throughout the literature (usually buried in proofs), it does not seem to have been noticed how nicely they fit together to yield our theorems. We will also make an attempt at extending our result to the case of unbounded operators in the hope that it might be useful in establishing existence theorems for linear partial differential equations. The author wishes to acknowledge that he discovered these relations in the study of an unpublished manuscript of deBranges and Rovnyak. Also, we acknowledge our indebtedness to P. Halmos for several conversations on this subject and note, in particular, that it was he who first noticed the equivalence of (1) and (3) in Theorem 1. The Hilbert space considered can be either real or complex. We use [1 ] as our basic reference and use the definitions and notation therein with the following exception. For an operator A on the Hilbert space we will denote the range and null space of A by range [A] and null [A ], respectively.