Given two chains of subspaces in C, we study the set of unitary matrices that map the subspaces in the first chain onto the corresponding subspaces in the second chain, and minimize the value ‖U− In‖ for various unitarily invariant norms ‖ · ‖ on Cn×n. In particular, we give formula for the minimum value ‖U − In‖, and describe the set of all the unitary matrices in the set attaining the minimum...

We give a characterization of all the unitarily invariant norms on finite von Neumann algebra acting separable Hilbert space. The is analogous to Neumann’s for $$n\times n$$ complex matrices and in Fang et al. (J Funct Anal 255(1):142–183, 2008) $$II_{1}$$ factors.

This paper aims to present some inequalities for unitarily invariant norms. In section 2, we give a refinement of the Cauchy-Schwarz inequality for matrices. In section 3, we obtain an improvement for the result of Bhatia and Kittaneh [Linear Algebra Appl. 308 (2000) 203-211]. In section 4, we establish an improved Heinz inequality for the Hilbert-Schmidt norm. Finally, we present an inequality...

Given two chains of subspaces in C, the set of those unitary matrices is studied that map the subspaces in the first chain onto the corresponding subspaces in the second chain, and minimize the value ‖U − In‖ for various unitarily invariant norms ‖ · ‖ on Cn×n. In particular, a formula for the minimum value ‖U − In‖ is given, and the set of all the unitary matrices in the set attaining the mini...

A fundamental result of von Neumann’s identifies unitarily invariant matrix norms as symmetric gauge functions of the singular values. Identifying the subdifferential of such a norm is important in matrix approximation algorithms, and in studying the geometry of the corresponding unit ball. We show how to reduce many convex-analytic questions of this kind to questions about the underlying gauge...

We prove an inequality for unitarily invariant norms that interpolates between the Arithmetic-Geometric Mean inequality and the Cauchy-Schwarz inequality.

We show that for any unitarily invariant norm k k on M n (the space of n-by-n complex matrices) where denotes the Hadamard (entrywise) product. These results are a consequence of an inequality for absolute norms on C n kx yk 2 kx xk ky yk for all x; y 2 C n : (2) We also characterize the norms on C n that satisfy (2), characterize the unitary similarity invariant norms on M n that satisfy (1), ...

Let A,B be nonzero positive semidefinite matrices. We prove that ‖AB‖ ‖A‖ ‖B‖ ≤ ‖A + B‖ ‖A‖+ ‖B‖ , ‖A ◦B‖ ‖A‖ ‖B‖ ≤ ‖A + B‖ ‖A‖+ ‖B‖ for any unitarily invariant norm with ‖diag(1, 0, . . . , 0)‖ ≥ 1. Some related inequalities are derived. AMS classification: 15A60, 15A45