SPECTRAL PROPERTIES OF COMMUTANTS OF UNBOUNDED SELF-ADJOINT OPERATORS WITH SIMPLE SPECTRA

ABSTRACT

An operator B  is a commutant of the unbounded Self-adjoint operator with simple spectra F  if BF⊆FB.  The properties of the commutant are determined by those of the operator F . In this article, we show that the spectrum of these commutants is a subset of the real number set. We also establish the effect of the spectral properties of the unbounded Self-adjoint operators with simple spectra to the spectrum of its commutant. Finally, we show that the spectral measure of the unbounded Self-adjoint operator with simple spectra is a scalar multiple of that of its commutant.