In 1957, Chandler Davis proved that unitarily invariant convex functions on the space of hermitian matrices are precisely those which are convex and symmetrically invariant on the set of diagonal matrices. We give a simple perturbation theoretic proof of this result. (Davis’ argument was also very short, though based on completely different ideas). Consider an orthogonally invariant function f ...

This is the first paper of a two-long series in which we study linear generalized inverses that minimize matrix norms. Such generalized inverses are famously represented by the Moore-Penrose pseudoinverse (MPP) which happens to minimize the Frobenius norm. Freeing up the degrees of freedom associated with Frobenius optimality enables us to promote other interesting properties. In this Part I, w...

Let D and A be unital and separable C∗-algebras; let D be strongly selfabsorbing. It is known that any two unital ∗-homomorphisms from D to A ⊗ D are approximately unitarily equivalent. We show that, if D is also K1-injective, they are even asymptotically unitarily equivalent. This in particular implies that any unital endomorphism of D is asymptotically inner. Moreover, the space of automorphi...

In their 1976 paper, Nagel and Rudin characterize the closed unitarily Möbius invariant spaces of continuous $$L^p$$ -functions on a sphere, for $$1\le p<\infty $$ . this paper we provide an analogous characterization weak*-closed $$L^\infty sphere. We also investigate algebras

We study global fluctuations for singular values of $M$-fold products several right-unitarily invariant $N\times N$ random matrix ensembles. As $N\to\infty$, we show the their height functions converge to an explicit Gaussian field, which is log-correlated $M$ fixed and has a white noise component $M\to\infty$ jointly with $N$. Our technique centers on multivariate Bessel generating these spect...

We study the backward shift operator on Hilbert spaces Hα (for α ≥ 0) which are norm equivalent to the Dirichlet-type spaces Dα. Although these operators are unitarily equivalent to the adjoints of the forward shift operator on certain weighted Bergman spaces, our approach is direct and completely independent of the standard Cauchy duality. We employ only the classical Hardy space theory and an...

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...