Multiple operator integrals and higher operator derivatives

In this paper we consider the problem of the existence of higher derivatives of the function t\mapsto\f(A+tK), where \f is a function on the real line, $A$ is a self-adjoint operator, and $K$ is a bounded self-adjoint operator. We improve earlier results by Sten'kin. In order to do this, we give a new approach to multiple operator integrals. This approach improves the earlier approach given by Sten'kin. We also consider a similar problem for unitary operators.Comment: 24 page

On S. Mazur's problems 8 and 88 from the Scottish Book

The paper discusses Problems 8 and 88 posed by Stanislaw Mazur in the Scottish Book. It turns out that negative solutions to both problems are immediate consequences of the results of Section 5 of my paper "Estimates of functions of power bounded operators on Hilbert spaces", J. Operator Theory 7 (1982), 341-372. We discuss here some quantitative aspects of Problems 8 and 88 and give answers to open problems discussed in a recent paper by Pelczynski and Sukochev.Comment: 8 page

Unitary interpolants and factorization indices of matrix functions

For an $n\times n$ bounded matrix function $\Phi$ we study unitary interpolants $U$, i.e., unitary-valued functions $U$ such that $\hat U(j)=\hat\Phi(j)$, $j<0$. We are looking for unitary interpolants $U$ for which the Toeplitz operator $T_U$ is Fredholm. We give a new approach based on superoptimal singular values and thematic factorizations. We describe Wiener--Hopf factorization indices of $U$ in terms of superoptimal singular values of $\Phi$ and thematic indices of $\Phi-F$, where $F$ is a superoptimal approximation of $\Phi$ by bounded analytic matrix functions. The approach essentially relies on the notion of a monotone thematic factorization introduced in [AP]. In the last section we discuss hereditary properties of unitary interpolants. In particular, for matrix functions $\Phi$ of class H^\be+C we study unitary interpolants $U$ of class $QC$.Comment: 20 page

Super-optimal approximation by meromorphic functions.

Let G be a matrix function of type m × n and suppose that G is expressible as the sum of an H∞ function and a continuous function on the unit circle. Suppose also that the (k – 1)th singular value of the Hankel operator with symbol G is greater than the kth singular value. Then there is a unique superoptimal approximant to G in : that is, there is a unique matrix function Q having at most k poles in the open unit disc which minimizes s∞(G – Q) or, in other words, which minimizes the sequence with respect to the lexicographic ordering, where and Sj(·) denotes the jth singular value of a matrix. This result is due to the present authors [PY1] in the case k = 0 (when the hypothesis on the Hankel singular values is vacuous) and to S. Treil[T2] in general. In this paper we give a proof of uniqueness by a diagonalization argument, a high level algorithm for the computation of the superoptimal approximant and a recursive parametrization of the set of all optimal solutions of a matrix Nehari—Takagi problem