Joint k-numerical ranges of operators

Abstract

Let \(\mathcal{H} \) be a complex Hilbert space and \(\mathcal{B}(\mathcal{H}) \) the algebra of all bounded linear operators on \( \mathcal{H} \). For a positive integer k less than the dimension of \(\mathcal{H} \) and \({\bf A}=(A_1,...,AM)\in\mathcal{B}(\mathcal{H})^{m} \), the joint k-numerical range \(W_k(\bf A) \) is the set of \((\alpha_1,..,\alpha_m)\in \mathbb{C^m}\) such that \(\alpha_{i}={\Sigma^{k}_{j=1}} \langle A_{i}x{j},x{j }\rangle \) for an orthonormal set \(\{x_{1},...,x_{k}\} \) in \(\mathcal{H} \). The geometrical properties of \( W_k(\bf A) \) and their relations with the algebraic properties of \(\{A_{1},...,A_{m}\} \) are investigated in this paper. First, conditions for\(W_k(\bf A) \) to be convex are studied, and some results for finite dimensional operators are extended to the infinite dimensional case. An example of A is constructed such that \(W_k(\bf A) \) is not convex, but \(W_r(\bf A) \) is convex for all positive integer r not equal to k. Second, descriptions are given for the closure of \(W_k(\bf A) \) and the closure of conv\(W_k(\bf A) \) in terms of the joint essential numerical range of A for infinite dimensional operators\(A_{1},...,A_{m} \) . These lead to characterizations of \(W_k(\bf A) \)Wk(A) or conv\(W_k(\bf A) \) to be closed. Moreover, it is shown that conv \(W_k(\bf A) \) is closed whenever \(W_{k+1}(\bf A) \) or conv\(W_{k+1}(\bf A) \) is. These results are used to study the connection between the geometric properties of \( W_k(\bf A) \) and algebraic properties of \(A_{1},...,A_{m} \). For instance, \(W_k(\bf A) \) is a polyhedral set, i.e., the convex hull of a finite set, if and only if \(A_{1},...,A_{k} \) have a common reducing subspace V of finite dimension such that the compression of \(A_{1},...,A_{m} \) on the subspace V are diagonal operators \( D_{1},...,D_{m} \) and \( W_k(A)=W_{k}(D_1,...,D_m) \). Characterization is also given to A such that the closure of \(W_k(\bf A) \) is polyhedral. For finite rank operators the following two condition are equivalent: (1) \(A_{1},...,A_{m} \) is a commuting family of normal operators. (2) \(W_k(A)=W_{k}(A_1,...,A_m) \) is polyhedral for every positive integer k less than dim\(\mathcal{H} \). However, the two conditions are not equivalent for compact operators. Characterizations are given for compact operators \( A_{1},...,A_{m} \) satisfying (1) and (2), respectively. Results are also obtained for general non-compact operators. Open problems and future research topics are presented.