On the Spectra of Left-Definite Operators

Abstract

If A is a self-adjoint operator that is bounded below in a Hilbert space H, Littlejohn and Wellman (J Diff Equ 181(2):280-339, 2002) showed that, for each r > 0, there exists a unique Hilbert space Hr and a unique self-adjoint operator Ar in Hr satisfying certain conditions dependent on H and A. The space Hr and the operator Ar are called, respectively, the rth left-definite space and rth left-definite operator associated with (H, A). In this paper, we show that the operators A, Ar , and As(r, s > 0) are isometrically isomorphically equivalent and that the spaces H, Hr , and Hs(r, s > 0) are isometrically isomorphic. These results are then used to repro- duce the left-definite spaces and left-definite operators. Furthermore, we will see that our new results imply that the spectra of A and Ar are equal, giving us another proof of this phenomenon that was first established in Littlejohn and Wellman (J Diff Equ 181(2):280-339, 2002).