Commuting normal operators and joint numerical range

Abstract

Let $\mathcal{H}$ be a complex Hilbert space and let $\mathcal{B}(\mathcal{H})$ be the algebra of all bounded linear operators on $\mathcal{H}$. For a positive integer $k$ less than the dimension of $\mathcal{H}$ and $\mathbf{A}=(A_1,\dots ,A_m)\in \mathcal{B}(\mathcal{H}{)}^m$, the joint $k$-numerical range $W_k(\mathbf{A})$ is the set of $({\alpha }_1,\dots ,{\alpha }_m)\in {\mathbb{C}}^m$ such that ${\alpha }_i={\sum }_{j=1}^k⟨A_i{x}_j,x_j⟩$ for an orthonormal set $\{x_1,\dots ,x_k\}$ in $\mathcal{H}$. Relations between the geometric properties of $W_k(\mathbf{A})$ and the algebraic and analytic properties of $A_1,\dots ,A_m$ are studied. It is shown that there is $k\in \mathbb{N}$ such that $W_k(\mathbf{A})$ is a polyhedral set, i.e., the convex hull of a finite set, if and only if $A_1,\dots ,A_k$ have a common reducing subspace $\mathbf{V}$ of finite dimension such that the compression of $A_1,\dots ,A_m$ on the subspace $\mathbf{V}$ are diagonal operators $D_1,\dots ,D_m$ and $W_k(\mathbf{A})=W_k(D_1,\dots ,D_m)$. Characterization is also given to $\mathbf{A}$ such that the closure of $W_k(\mathbf{A})$ is polyhedral. The conditions are related to the joint essential numerical range of $\mathbf{A}$. These results are used to study $\mathbf{A}$ such that (a) $\{A_1,\dots ,A_m\}$ is a commuting family of normal operators, or (b) $W_k(A_1,\dots ,A_m)$ is polyhedral for every positive integer $k$. It is shown that conditions (a) and (b) are equivalent for finite rank operators but it is no longer true for compact operators. Characterizations are given for compact operators $A_1,\dots ,A_m$ satisfying (a) and (b), respectively. Results are also obtained for general non-compact operators.

The paper is combined the earlier paper "The Joint k-numerical range of operators" arXiv:2105.04621 with additional results

References in corpus (1)

• Joint numerical ranges and compressions of powers of operators