a-Numerical range on $C^*$-algebras

Abstract

Let $\mathfrak {A}$ be a unital $C^*$-algebra and $\mathfrak {A}'$ be its topological dual space. Let a be a positive element in $\mathfrak {A}$, and set $\mathscr {S}_a(\mathfrak {A}):= \left\{ f\in \mathfrak {A}': f\ge 0, f(a)=1\right\} .$ The a-numerical range and a-numerical radius of any element $x\in \mathfrak {A}$ are defined by

$$\begin{aligned} V_a(x):= \left\{ f(ax): f\in \mathscr {S}_a(\mathfrak {A})\right\} , \end{aligned}$$

and

$$\begin{aligned} v_a(x):=\sup \left\{ \left| z\right| :z\in V_a(x)\right\} , \end{aligned}$$

respectively. In this paper, we establish some permanence properties of the a-numerical range and a-numerical radius of elements in $\mathfrak {A}$. In particular, we investigate when the a-numerical range of an element of $\mathfrak {A}$ is closed, and provide explicit formulas for the a-numerical radius of the so-called a-hermitian elements of $\mathfrak {A}$. Furthermore, given a positive operator A on a complex Hilbert space ${\mathscr {H}}$, we study and investigate the relationship between the algebraic and spatial A-numerical ranges of bounded linear operators on ${\mathscr {H}}$.