On the $A$-spectrum for $A$-bounded operator on von-Neumann algebras

Let $\mathfrak{M}$ be a von Neumann algebra and let $A$ be a nonzero positive element of $\mathfrak{M}$. By ${\sigma }_A(T)$ and $r_A(T)$ we denote the $A$-spectrum and the $A$-spectral radius of $T\in {\mathfrak{M}}^A$, respectively. In this paper, we show that $\sigma (PTP,P\mathfrak{M}P)\subseteq {\sigma }_A(T)$. Sufficient conditions for the equality ${\sigma }_A(T)=\sigma (PTP,P\mathfrak{M}P)$ to be true are presented. Also, 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 $\mathfrak{M}$. Next , we consider the relationship between elements $S$ and $T\in {\mathfrak{M}}^A$ that satisfy one of the following two conditions: (1) ${\sigma }_A(SX)={\sigma }_A(TX)$ for all $X\in {\mathfrak{M}}^A$, (2) $r_A(SX)=r_A(TX)$ for all $X\in {\mathfrak{M}}^A$. Finally, a Gleason-Kahane-Żelazko’s theorem for the $A$-spectrum is derived.% Finally, we introduce and study the notion of the $A$-approximate point spectrum for element of $X\in {\mathfrak{M}}^A$.