Unitarily invariant Norms on Operators

2021/12/17 by Chan, Jor-Ting, Li, Chi-Kwong
#15A04 #15A60 #47B48 #FOS: Mathematics #Functional Analysis (math.FA)

paper · doi:10.48550/arxiv.2112.13656

Abstract

Let f be a symmetric norm on \mathbb Rⁿ and let \mathcal B(\mathcal H) be the set of all bounded linear operators on a Hilbert space \mathcal H of dimension at least n. Define a norm on \mathcal B(\mathcal H) by ‖A‖_f = f(s₁(A), …, s_n(A)), where s_k(A) = inf\‖A-X‖: X∈ \mathcal B(\mathcal H) \hbox has rank less than k\ is the kth singular value of A. Basic properties of the norm ‖⋅‖_f are obtained including some norm inequalities and characterization of the equality case. Geometric properties of the unit ball of the norm are obtained; the results are used to determine the structure of maps L satisfying ‖L(A)-L(B)‖_f=‖A - B‖_f for any A, B ∈ \mathcal B(\mathcal H).

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