A distance formula for tuples of operators

2022/06/03 by Priyanka Grover, Sushil Singla, Grover, Priyanka +1 · 1 citation
Computer Science · Mathematics · #15A21 #15A27 #47A12 #47B35 #Advanced Topics in Algebra #FOS: Mathematics #Functional Analysis (math.FA) #Holomorphic and Operator Theory #Matrix Theory and Algorithms

paper · pdf · doi:10.48550/arxiv.2206.01503

openalex publication_date 2022/06/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

For a tuple of operators \boldsymbolA= (A₁, …, A_d), dist(\boldsymbolA, \mathbb C^d \boldsymbolI) is defined as min_\boldsymbolz ∈ \mathbb C^d ‖\boldsymbolA-zI‖ and var_x (\boldsymbolA) as ‖\boldsymbolA x‖²-∑_j=1^d |⟨ x| A_j x⟩|². For a tuple \boldsymbolA of commuting normal operators, it is known that dist(\boldsymbolA, \mathbb C^d \boldsymbolI)²=sup_‖x‖=1var_x (\boldsymbolA). We give an expression for the maximal joint numerical range of a tuple of doubly commuting matrices. Consequently, we obtain that the above distance formula holds for tuples of doubly commuting matrices. We also discuss some general conditions on the tuples of operators for this formula to hold. As a result, we obtain that it holds for tuples of Toeplitz operators as well.

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