Abstract
Let f be a symmetric norm on \(\mathbb {R}^n\) 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 \(\Vert A\Vert _f = f(s_1(A), . . . , s_n(A))\), where \(s_k(A) = \mathrm{inf}\{\Vert A-X\Vert : X \in \mathcal {B}(\mathcal {H})\, \mathrm{has\, rank\, less\, than}\, k\}\) is the kth singular value of A. Basic properties of the norm \(\Vert \cdot\Vert _f\) 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 \(\Vert L(A)-L(B)\Vert _f =\Vert A-B\Vert _f\) for any \(A,B \in \mathcal {B}(\mathcal {H})\).
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Acknowledgment
Li is an affiliate member of the Institute for Quantum Computing, University of Waterloo; his research was partially supported by the Simons Foundation Grant 851334. The authors would like to thank Professor Ngai-Ching Wong for some inspiring discussions and comments, and also the anonymous referee for the helpful comments.
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Chan, JT., Li, CK. Unitarily invariant norms on operators. Acta Sci. Math. (Szeged) 88, 611–625 (2022). https://doi.org/10.1007/s44146-022-00042-x
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DOI: https://doi.org/10.1007/s44146-022-00042-x