On the A-spectrum for A-bounded operators on von-Neumann algebras

Let M be a von Neumann algebra. For a nonzero positive element A is an element of M, let P denote the orthogonal projection on the norm closure of the range of A and let sigma(A)(T) denote the A-spectrum of any T is an element of M-A. In this paper, we show that sigma(A)(T) is a non empty compact subset of C and that sigma(PTP, PMP) subset of sigma(A)(T) for any T is an element of M-A where sigma(PT P, PMP) is the spectrum of PT P in PMP. Sufficient conditions for the equality sigma(A)(T) = sigma(PTP, PMP) to be true are also presented. Moreover, we show that sigma(A)(T) is finite for any T is an element of M-A if and only if A is in the socle of M. Furthermore, we consider the relationship between elements S and T is an element of M-A that satisfy the condition sigma(A)(SX) = sigma(A)(T X) for all X is an element of M-A. Finally, a Gleason-Kahane-Zelazko's theorem for the A-spectrum is derived.