Commutants and spectral properties of operators in hilbert spaces

Abstract

This thesis is devoted to the study of commutants and spectral properties of operators in Hilbert spaces. This is done via the following operator equations: AB =λBA, where λ ∈ℂ, AX = XB and AXB = X. In the operator equation AB = λ BA, conditions on A and B under which λ =1 are investigated. This indeed is a sufficient condition for the operators A and B to belong to the commutant of each other. In the operator equations AX = XB and AXB = X , conditions that ensure the existence of the operator equations C(A,B)X = 0 and R(A, B)X = 0 are given. Finally in the operator equation AB = λ BA, the equality of the general spectra and other subsets of the spectra namely essential and approximate point spectra of AB and BA or B and λ B, are established. This final bit justifies the spectral properties part of our thesis