We introduce the concept of Stieltjes integral of an operator-valued function with respect to the spectral measure associated with a normal operator. We give sufficient conditions for the existence of this integral and find bounds on its norm. The results obtained are applied to the Sylvester and Riccati operator equations. Assuming that the entry C is a normal operator, the spectrum of the ent...

We study the spectral norm of matrices W that can be factored as W = BA, where A is a random matrix with independent mean zero entries and B is a fixed matrix. Under the (4 + ε)-th moment assumption on the entries of A, we show that the spectral norm of such an m×n matrix W is bounded by √ m + √ n, which is sharp. In other words, in regard to the spectral norm, products of random and determinis...

We prove a conjecture of Viterbo from 2007 on the existence uniform bound Lagrangian spectral norm Hamiltonian deformations zero section in unit cotangent disk bundles, for bases given by compact rank one symmetric spaces $$S^n, {\mathbb {R}}P^n, {C}}P^n, {H}}P^n,$$ $$n\ge 1.$$ discuss generalizations and give applications, particular to $$C^0$$ symplectic topology. Our key method consists quan...

We present a proof of the irreversibility of renormalization group flows, i.e. the c–theorem for unitary, renormalizable theories in four (or generally even) dimensions. Using Ward identities for scale transformations and spectral representation arguments, we show that the c–function based on the trace of the energy-momentum tensor (originally suggested by Cardy) decreases monotonically along r...

Abstract For two given subspaces of C we investigate their distance and angle on the basis of a joint decomposition of the corresponding orthogonal projectors. Further attention is paid to the notions inclinedness, orthogonal incidence and minimal angle. Some formulas for the spectral norm of orthogonal projectors or functions thereof are derived. Also the proofs of results known from Hilbert s...

We present new examples illustrating the fact that optimal Hankel structured rank-1 approximation of matrices is usually different for Frobenius and spectral norm. Further, we compare our results to well-known Cadzow algorithm [3].

An algorithm for matrix approximation, when only some of its entries are taken into consideration, is described. The approximation constraint can be any whose approximated solution is known for the full matrix. For low rank approximations, this type of algorithms appears recently in the literature under different names, where it usually uses the Expectation-Maximization algorithm that maximizes...