Functional Calculus for a Pair of Almost Commuting Selfadjoint Operators

AbstractLet (A, B) be a pair of almost commuting selfadjoint operators on Hilbert space. The aim of this paper is to construct a functional calculus defined on a class of functions two variables and satisfying natural properties (linearity and multiplicativity modulo the trace class). The problem is to make this class as big as possible. It is proved that it is impossible to construct such a functional calculus on the class of all continuously differentiable functions. On the other hand we find a class C of functions in two real variables and construct a functional calculus on this class which extends functional calculi constructed earlier. It is proved that for this functional calculus the trace formula of Helton-Howe holds

A bootstrap method for sum-of-poles approximations

A bootstrap method is presented for finding efficient sum-of-poles approximations of causal functions. The method is based on a recursive application of the nonlinear least squares optimization scheme developed in (Alpert et al. in SIAM J. Numer. Anal. 37:1138–1164, 2000), followed by the balanced truncation method for model reduction in computational control theory as a final optimization step. The method is expected to be useful for a fairly large class of causal functions encountered in engineering and applied physics. The performance of the method and its application to computational physics are illustrated via several numerical examples

Right invertible multiplication operators and stable rational matrix solutions to an associate Bezout equation, I. the least squares solution.

In this paper a state space formula is derived for the least squares solution X of the corona type Bezout equation G(z)X(z) =

Functions of triples of noncommuting self-adjoint operators under perturbations of class Sp

In this paper we study properties of functions of triples of not necessarily commuting self-adjoint operators. The main result of the paper shows that unlike in the case of functions of pairs of self-adjoint operators there is no Lipschitz type estimates in any Schatten-von Neumann norm Sp, 1 ≥ p ≥ ∞, for arbitrary functions in the Besov class B1 ∞, 1(ℝ3). In other words, we prove that for p ∈ [1,∞], there is no constant K > 0 such that the inequality ||f(A1,B1,C1) - f(A2,B2,C2)||Sp ≤K||f||B1 ∞,1 max {||A1-A2||Sp, ||B1-B2||Sp, ||C1 - C2||Sp} holds for an arbitrary function f in B1 ∞,1 (ℝ3) and for arbitrary finite rank self-adjoint operators A1, B1, C1, A2, B2 and C2. © 2017 American Mathematical Society

Functions of triples of noncommuting self-adjoint operators under perturbations of class Sp

In this paper we study properties of functions of triples of not necessarily commuting self-adjoint operators. The main result of the paper shows that unlike in the case of functions of pairs of self-adjoint operators there is no Lipschitz type estimates in any Schatten-von Neumann norm Sp, 1 ≥ p ≥ ∞, for arbitrary functions in the Besov class B1 ∞, 1(ℝ3). In other words, we prove that for p ∈ [1,∞], there is no constant K > 0 such that the inequality ||f(A1,B1,C1) - f(A2,B2,C2)||Sp ≤K||f||B1 ∞,1 max {||A1-A2||Sp, ||B1-B2||Sp, ||C1 - C2||Sp} holds for an arbitrary function f in B1 ∞,1 (ℝ3) and for arbitrary finite rank self-adjoint operators A1, B1, C1, A2, B2 and C2. © 2017 American Mathematical Society

Functions of commuting contractions under perturbation

The purpose of the paper is to obtain estimates for differences of functions of two pairs of commuting contractions on Hilbert space. In particular, Lipschitz type estimates, Hölder type estimates, Schatten–von Neumann estimates are obtained. The results generalize earlier known results for functions of self-adjoint operators, normal operators, contractions and dissipative operators. © 2019 WILEY-VCH Verlag GmbH & Co. KGaA, Weinhei

Multiple operator integrals, Haagerup and Haagerup-like tensor products, and operator ideals

We study Schatten-von Neumann properties of multiple operator integrals with integrands in the Haagerup tensor product of L∞ spaces. We obtain sharp, best possible estimates. This allowed us to obtain sharp Schatten-von Neumann estimates in the case of Haagerup-like tensor products. © 2017 London Mathematical Society