2021/05/10 by Jor-Ting Chan, Chan, Jor-Ting, Chi-Kwong Li +3 · 1 citation
Computer Science · Mathematics · #15A47 #15A60 #Advanced Topics in Algebra #FOS: Mathematics #Functional Analysis (math.FA) #Holomorphic and Operator Theory #Matrix Theory and Algorithms
paper · pdf · doi:10.48550/arxiv.2105.04621
openalex publication_date 2021/05/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let \mathcal B(\mathcal H) be the algebra of all bounded linear operators on the Hilbert space \mathcal H. For a positive integer k less than the dimension of \mathcal H and \mathbf A = (A1, …, Am)∈ \mathcal B(\mathcal H)m, the joint k-numerical range Wk(\mathbf A) is the set of vector (α1, …, αm) ∈\mathbb Cm such that αi = ∑j = 1k ⟨ Aixj, xj⟩ for an orthonormal set \x1, …, xk\ in \mathcal H. Geometrical properties of Wk(\mathbf A) and their relations with the algebraic properties of \A1, …, Am\ are investigated in this paper. For example, conditions for Wk(\mathbf A) to be convex are studied. Descriptions are given for the closure of Wk(\mathbf A) and the closure of \rm conv Wk(\mathbf A) in terms of the joint essential numerical range of \mathbf A for infinite dimensional operators A1, …, Am. Characterizations are obtained for Wk(\mathbf A) or \rm conv Wk(\mathbf A) to be closed. It is shown that Wk(\mathbf A) is a polyhedral set if and only if A1, …, Ak have a common reducing subspace \mathbf V of finite dimension such that the compression of A1, …, Am on the subspace \mathbf V are diagonal operators D1, …, Dm and Wk(\mathbf A) = Wk(D1, …, Dm). Similar results are obtained for \bf A such that the closure of Wk(\mathbf A) is polyhedral. Classifications are given for operators satisfying (1) \A1, …, Am\ is a commuting family of normal operators, or (2) Wk(A1, …, Am) is polyhedral for every positive integer k less than dim \mathcal H.