Operator Moment Problems Research Articles

Complete positivity and a quaternionic moment problem

The aim of this work is to study norm preserving extensions of completely positive and completely contractive maps on some spaces of fractions with values in a space of sesquilinear forms on a inner product space. The main result is stated in the case of non necessarily commutative C*-algebras with unit. Moreover, the main result is applied to characterize a “quaternionic” version of the operator moment problem. for this application, a special measure is introduced.

Schwartz operators

In this paper, we introduce Schwartz operators as a non-commutative analog of Schwartz functions and provide a detailed discussion of their properties. We equip them, in particular, with a number of different (but equivalent) families of seminorms which turns the space of Schwartz operators into a Fréchet space. The study of the topological dual leads to non-commutative tempered distributions which are discussed in detail as well. We show, in particular, that the latter can be identified with a certain class of quadratic forms, therefore making operations like products with bounded (and also some unbounded) operators and quantum harmonic analysis available to objects which are otherwise too singular for being a Hilbert space operator. Finally, we show how the new methods can be applied by studying operator moment problems and convergence properties of fluctuation operators.

Unbounded extensions and operator moment problems

Extension results, expressed in terms of complete boundedness, leading to necessary and sufficient conditions for the solvability of power moment problems with unbounded operator data are given. As an application, necessary and sufficient conditions for the existence of selfadjoint and normal extensions for some classes of commuting tuples of unbounded linear operators are obtained.

Model reduction using the Vorobyev moment problem

Given a nonsingular complex matrix $A\in{\mathbb C}^{N\times N}$ and complex vectors v and w of length N, one may wish to estimate the quadratic form w * A − 1 v, where w * denotes the conjugate transpose of w. This problem appears in many applications, and Gene Golub was the key figure in its investigations for decades. He focused mainly on the case A Hermitian positive definite (HPD) and emphasized the relationship of the algebraically formulated problems with classical topics in analysis - moments, orthogonal polynomials and quadrature. The essence of his view can be found in his contribution Matrix Computations and the Theory of Moments, given at the International Congress of Mathematicians in Zurich in 1994. As in many other areas, Gene Golub has inspired a long list of coauthors for work on the problem, and our contribution can also be seen as a consequence of his lasting inspiration. In this paper we will consider a general mathematical concept of matching moments model reduction, which as well as its use in many other applications, is the basis for the development of various approaches for estimation of the quadratic form above. The idea of model reduction via matching moments is well known and widely used in approximation of dynamical systems, but it goes back to Stieltjes, with some preceding work done by Chebyshev and Heine. The algebraic moment matching problem can for A HPD be formulated as a variant of the Stieltjes moment problem, and can be solved using Gauss-Christoffel quadrature. Using the operator moment problem suggested by Vorobyev, we will generalize model reduction based on matching moments to the non-Hermitian case in a straightforward way. Unlike in the model reduction literature, the presented proofs follow directly from the construction of the Vorobyev moment problem.

Phase space quantization and the operator moment problem

We consider questions related to a quantization scheme in which a classical variable f:Ω→R on a phase space Ω is associated with a (preferably unique) semispectral measure Ef, such that the moment operators of Ef are required to be of the form Γ(fk), with Γ a suitable mapping from the set of classical variables to the set of (not necessarily bounded) operators in the Hilbert space of the quantum system. In particular, we investigate the situation where the map Γ is implemented by the operator integral with respect to some fixed positive operator measure. The phase space Ω is first taken to be an abstract measurable space, then a locally compact unimodular group, and finally R2, where we determine explicitly the relevant operators Γ(fk) for certain variables f, in the case where the quantization map Γ is implemented by a translation covariant positive operator measure. In addition, we consider the question under what conditions a positive operator measure is projection valued.

Sebestyén moment problem: The multi-dimensional case

Given a family { h n } n ∈ Z + Ω \{h_{\mathbf {n}}\}_{\mathbf {n}\in \mathbb {Z}_+^\Omega } of vectors in a Hilbert space H \mathcal {H} we characterize the existence of a family of commuting contractions T = { T ω } w ∈ Ω \mathbf {T}=\{T_\omega \}_{w\in \Omega } on H \mathcal {H} having regular dilation and such that h n = T n h 0 , n ∈ Z + Ω . \begin{equation*} h_{\mathbf {n}}=\mathbf {T} ^{\mathbf {n}} h_{\mathbf {0}},\quad \mathbf {n}\in \mathbb {Z}_+^\Omega . \end{equation*} The theorem is a multi-dimensional analogue for some well-known operator moment problems due to Sebestyén in case | Ω | = 1 |\Omega |=1 or, recently, to Găvruţă and Păunescu in case | Ω | = 2 |\Omega |=2 .

The operator moment problem, vector continued fractions and an explicit form of the Favard theorem for vector orthogonal polynomials

We study the operator moment problem for a ( p + 2)-diagonal nonsymmetric matrix operator and give an explicit form of the Favard theorem for vector orthogonal polynomials. As a main tool we use the development of a system of resolvent functions of the operator in a vector continued fraction by a modified Jacobi-Perron algorithm.

Spectral functions of a symmetric linear relation with a directing mapping, II

SynopsisThe results of part I (see [5]) are applied to pairs of formally symmetric differential expressions, to Hermitian differential systems and to a reduced operator moment problem.

Moment methods in two point Padé approximation

In the separable Hilbert space ( H, 〈·, ·〉) the following “operator moment problem” is solved: given a complex sequence ( c k ) k ϵ Z generated by a meromorphic function f, find T ϵ B( H) and u 0 ϵ H such that 〈 T k u 0, u 0〉 = c k ( k ϵ Z). If the sequence ( c k ) k ϵ Z is “normal,” an adapted form of Vorobyev's method of moments yields a sequence of two point Padé approximants to f. A sufficient condition for convergence of this sequence of approximants is given.

Moment methods in Padé approximation

The complex sequence ( c k ) k = 0 ∞ is assumed to be normal. In the separable Hilbert space ( H, 〈·, ·〉) we solve the “operator moment problem”: Find A ϵ B( H) such that 〈 A k u 0, u 0〉 = c k ( k = 0, 1,…), ( u 0 ϵ H). A compact 2 ( c k ) k = 0 ∞ has meromorphic generating function ƒ(z) = ∑ k = 0 ∞ c kz k . Padé-type approximants to ƒ, sometimes the ordinary Padé approximants, are obtained as solutions of certain approximate Fredholm equations generated by projections onto finite dimensional subspaces.