279 research outputs found
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,
is a self-adjoint operator, and 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
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
Unitary interpolants and factorization indices of matrix functions
For an bounded matrix function we study unitary
interpolants , i.e., unitary-valued functions such that , . We are looking for unitary interpolants for which
the Toeplitz operator is Fredholm. We give a new approach based on
superoptimal singular values and thematic factorizations. We describe
Wiener--Hopf factorization indices of in terms of superoptimal singular
values of and thematic indices of , where is a superoptimal
approximation of 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 of class
H^\be+C we study unitary interpolants of class .Comment: 20 page
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