Commutator inequalities via Schur products

For a self-adjoint unbounded operator D on a Hilbert space H, a bounded operator y on H and some complex Borel functions g(t) we establish inequalities of the type ||[g(D),y]|| \leq A|||y|| + B||[D,y]|| + ...+ X|[D, [D,...[D, y]...]]||. The proofs take place in a space of infinite matrices with operator entries, and in this setting it is possible to approximate the matrix associated to [g(D), y] by the Schur product of a matrix approximating [D,y] and a scalar matrix. A classical inequality of Bennett on the norm of Schur products may then be applied to obtain the results.Comment: 16 page

On the complete boundedness of the Schur block product

We give a Stinespring representation of the Schur block product, say (*), on pairs of square matrices with entries in a C*-algebra as a completely bounded bilinear operator of the form: A:=(a_{ij}), B:= (b_{ij}): A (*) B := (a_{ij}b_{ij}) = V* pi(A) F pi(B) V, such that V is an isometry, pi is a *-representation and F is a self-adjoint unitary. This implies an inequality due to Livshits and two apparently new ones on diagonals of matrices. ||A (*) B|| \leq ||A||_r ||B||_c operator, row and column norm; - diag(A*A) \leq A* (*) A \leq diag(A*A), and for all vectors f, g: | |^2 \leq < diag(AA*) g, g> .Comment: 10 p, revised, expanded and to appear in Proc. AM

Toeplitz separability, entanglement, and complete positivity using operator system duality

A new proof is presented of a theorem of L.~Gurvits, which states that the cone of positive block-Toeplitz matrices with matrix entries has no entangled elements. The proof of the Gurvits separation theorem is achieved by making use of the structure of the operator system dual of the operator system $C(S^1)^{(n)}$ of $n\times n$ Toeplitz matrices over the complex field, and by determining precisely the structure of the generators of the extremal rays of the positive cones of the operator systems C(S^1)^{(n)}\omin\B(\H) and C(S^1)_{(n)}\omin\B(\H), where \H is an arbitrary Hilbert space and $C(S^1)_{(n)}$ is the operator system dual of $C(S^1)^{(n)}$. Our approach also has the advantage of providing some new information concerning positive Toeplitz matrices whose entries are from \B(\H) when \H has infinite dimension. In particular, we prove that normal positive linear maps $\psi$ on \B(\H) are partially completely positive in the sense that $\psi^{(n)}(x)$ is positive whenever $x$ is a positive $n\times n$ Toeplitz matrix with entries from \B(\H). We also establish a certain factorisation theorem for positive Toeplitz matrices (of operators), showing an equivalence between the Gurvits approach to separation and an earlier approach of T.~Ando to universality

Completion of partial operator matrices

This work concerns completion problems for partial operator matrices. A partial matrix is an m-by-n array in which some entries are specified and the remaining are unspecified. We allow the entries to be operators acting between corresponding vector spaces (in general, bounded linear operators between Hilbert spaces). Graphs are associated with partial matrices. Chordal graphs and directed graphs with a perfect edge elimination scheme play a key role in our considerations. A specific choice for the unspecified entries is referred to as a completion of the partial matrix. The completion problems studied here involve properties such as: zero-blocks in certain positions of the inverse, positive (semi)definitness, contractivity, or minimum negative inertia for Hermitian operator matrices. Some completion results are generalized to the case of combinatorially nonsymmetric partial matrices. Several applications including a maximum entropy result and determinant formulae for matrices with sparse inverses are given.;In Chapter II we treat completion problems involving zero-blocks in the inverse. Our main result deals with partial operator matrices R, for which the directed graph is associated with an oriented tree. We prove that under invertibility conditions on certain principal minors, R admits a unique invertible completion F such that {dollar}(F\sp{lcub}-1{rcub})\sb{lcub}ij{rcub}{dollar} = 0 whenever {dollar}R\sb{lcub}ij{rcub}{dollar} is unspecified.;Chapter III treats positive semidefinite and Hermitian completions. In the case of partial positive operator matrices with a chordal graph, a maximum entropy principle is presented, generalizing the maximum determinant result in the scalar case. We obtain a linear fractional transform parametrization for the set of all positive semidefinite completions for a generalized banded partial matrix. We also give an inertia formula for Hermitian operator matrices with sparse inverses.;In Chapter IV prior results are applied to obtain facts about contractive and linearly constrained completion problems. The solution to a general n-by-n strong-Parrott type completion problem is the main result. We prove necessary and sufficient conditions for the existence of a solution as well as a cascade transform parametrization for the set of all solutions.;Chapter V extends the results in Chapter II and III to prove determinant formulae for matrices with sparse inverses. Several ideas from graph theory are used. An inheritance principle for chordal graphs is also presented

Spectral Properties of Heavy-Tailed Random Matrices

The classical Random Matrix Theory studies asymptotic spectral properties of random matrices when their dimensions grow to infinity. In contrast, the non-asymptotic branch of the theory is focused on explicit high probability estimates that we can obtain for large enough, but fixed size random matrices. This goal naturally brings into play some beautiful methods of high-dimensional probability and geometry, such as the concentration of measure phenomenon. One of the less understood random matrix models is a heavy-tailed model. This is the case when the matrix entries have distributions with slower tail decay than gaussian, e.g., with a few finite moments only. This work is devoted to the study of the heavy-tailed matrices and addresses two main questions: invertibility and regularization of the operator norm. First, the invertibility result of Rudelson and Vershynin is generalized from the case when the matrix entries are subgaussian to the case when only two finite moments are required. Then, it is shown that the operator norm of a matrix can be reduced to the optimal order O(sqrt(n)) if and only if the entries have zero mean and finite variance. We also study the constructive ways to perform such regularization. We show that deletion of a few large entries regularizes the operator norm only if all matrix entries have more than two finite moments. In the case with exactly two finite moments, we propose an algorithm that zeroes out a small fraction of the matrix entries to achieve the operator norm of an almost optimal order O(sqrt(n*ln ln n)) Finally, if in the latter case the matrix has scaled Bernoulli entries, we get a stronger regularization algorithm that provides a) O(sqrt(n))-operator norm of the resulting matrix and b) simple structure of the "bad" submatrix to be zeroed out.PHDMathematicsUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttps://deepblue.lib.umich.edu/bitstream/2027.42/146003/1/erebrova_1.pd