2025/01/02 by Orr Moshe Shalit, Eli Shamovich, Shalit, Orr +1 · 1 citation
Computer Science · Mathematics · #15A22 #46L52 #47A10 #47A13 #Advanced Topics in Algebra #FOS: Mathematics #Functional Analysis (math.FA) #Holomorphic and Operator Theory #Matrix Theory and Algorithms #Operator Algebras (math.OA)
paper · pdf · doi:10.48550/arxiv.2501.01325
openalex publication_date 2025/01/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
With every operator space structure E on ℂd, we associate a spectral radius function ρE on d-tuples of operators. For a d-tuple X = (X1, …, Xd) ∈ Mn(ℂd) of matrices we show that ρE(X)<1 if and only if X is jointly similar to a tuple in the open unit ball of Mn(E), that is, there is an invertible matrix S such that ‖S-1X S‖Mn(E)<1, where S-1 X S =(S-1 X1 S, …, S-1 Xd S). When E is the row operator space, for example, our spectral radius coincides with the joint spectral radius considered by Bunce, Popescu, and others, and we recover the condition for a tuple of matrices to be simultaneously similar to a strict row contraction. When E is the minimal operator space min(ℓ^∞d), our spectral radius ρE is related to the joint spectral radius considered by Rota and Strang but differs from it and has the advantage that ρE(X)<1 if and only if X is simultaneously similar to a tuple of strict contractions. We show that for a nc rational function f with descriptor realization (A,b,c), the spectral radius ρE(A)<1 if and only the domain of f contains a neighborhood of the noncommutative closed unit ball of the operator space dual E^* of E.