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

Let \(\mathfrak {M}\) be a von Neumann algebra. For a nonzero positive element \(A\in \mathfrak {M}\), 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 \(T\in \mathfrak {M}^A\). In this paper, we show that \(\sigma _A(T)\) is a non empty compact subset of \(\mathbb {C}\) and that \(\sigma (PTP, P\mathfrak {M}P)\subseteq \sigma _A(T)\) for any \(T\in \mathfrak {M}^A\) where \(\sigma (PTP, P\mathfrak {M}P)\) is the spectrum of PTP in \(P\mathfrak {M}P\). Sufficient conditions for the equality \(\sigma _A(T)=\sigma (PTP, P\mathfrak {M}P)\) to be true are also presented. Moreover, we show that \(\sigma _A(T)\) is finite for any \(T\in \mathfrak {M}^A\) if and only if A is in the socle of \(\mathfrak {M}\). Furthermore, we consider the relationship between elements S and \(T\in \mathfrak {M}^A\) that satisfy the condition \(\sigma _A(SX)=\sigma _A(TX)\) for all \(X\in \mathfrak {M}^A\). Finally, a Gleason–Kahane–Żelazko’s theorem for the A-spectrum is derived.