The main result of this note, Theorem 2, is the following: a Borel measure on the space of infinite Hermitian matrices, that is invariant under the action of the infinite unitary group and that admits welldefined projections onto the quotient space of “corners” of finite size, must be finite. A similar result, Theorem 1, is also established for unitarily invariant measures on the space of all i...

A well-known result on spectral variation of a Hermitian matrix due to Mirsky is the following: Let A and à be two n×n Hermitian matrices, and let λ1, . . . , λn and λ̃1, . . . , λ̃n be their eigenvalues arranged in ascending order. Then ∣∣∣∣∣∣diag (λ1 − λ̃1, . . . , λn − λ̃n)∣∣∣∣∣∣ ≤ ∣∣∣∣∣∣A− Ã∣∣∣∣∣∣ for any unitarily invariant norm ||| · |||. In this paper, we generalize this to the perturbation ...

Let A = UP be a polar decomposition of an n×n complex matrix A. Then for every unitarily invariant norm ||| · |||, it is shown that ||| |UP − PU |||| ≤ |||A∗A−AA∗||| ≤ ‖UP + PU‖ |||UP − PU |||, where ‖·‖ denotes the operator norm. This is a quantitative version of the wellknown result that A is normal if and only if UP = PU . Related inequalities involving self-commutators are also obtained.

Whenever we have a group acting on a class of functions by translation, the bispectrum offers a principled and lossless way of representing such functions invariant to the action. Unfortunately, computing the bispectrum is often costly and complicated. In this paper we propose a unitarily equivalent, but easier to compute set of invariants, which we call the skew spectrum. For functions on homo...

Recently, the supersymmetry method was extended from Gaussian ensembles to arbitrary unitarily invariant matrix ensembles by generalizing the Hubbard–Stratonovich transformation. Here, we complete this extension by including arbitrary orthogonally and unitary–symplectically invariant matrix ensembles. The results are equivalent to, but the approach is different from the superbosonization formul...