The inverse problem for the characteristic functions of several commuting operators

In this paper we derive conditions for an operator valued function to be the characteristic function of several commuting operators in a Hilbert space. We use the connection of this problem to some problems in partial differential equation to get a solution for a class of operator valued functions.     

Unbounded functions and positive linear operators

The approximation of unbounded functions by positive linear operators under multiplier enlargement is investigated. It is shown that a very wide class of positive linear operators can be used to approximate functions with arbitrary growth on the real line. Estimates are given in terms of the usual quantities which appear in the Shisha-Mond theorem. Examples are provided.     

On infinite products of positive linear operators reproducing linear functions

Infinite products of positive linear operators (p.l.o.) reproducing linear functions are considered from a quantitative point of view. Refining and generalizing convergence theorems of Gwó?d?-?ukawska, Jachymski, Gavrea, Ivan and the present authors, it is shown that infinite products of certain positive linear operators, all taken from a finite set of mappings reproducing linear functions, weakly converge to the first Bernstein operator. A discussion of products of Meyer-König and Zeller operators is included.     

Positive linear operators generated by analytic functions

Let φ be a power series with positive Taylor coefficients {a _k } _k=0^∞ and non-zero radius of convergence r ≤ ∞. Let ξ _x , 0 ≤ x < r be a random variable whose values α _k , k = 0, 1, …, are independent of x and taken with probabilities a _k x ^k /φ(x), k = 0, 1, …. The positive linear operator (A _φ f)(x):= E[f(ξ _x )] is studied. It is proved that if E(ξ _x ) = x, E(ξ _x ²) = qx ² + bx + c, q, b, c ∈ R, q > 0, then A _φ reduces to the Szász-Mirakyan operator in the case q = 1, to the limit q-Bernstein operator in the case 0 < q < 1, and to a modification of the Lupaş operator in the case q > 1.     

The Krein string and characteristic functions of non-self-adjoint operators

The operator generated by the Krein string is investigated in the framework of the extension theory of symmetric operators. A simple proof of the complete non-self-adjointness of the operator is proposed. The scattering function of the string is obtained with the help of the Derkach-Malamud formula for characteristic functions of almost solvable extensions. Bibliography: 16 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 327, 2005, pp. 115–134.     

A class of linear operators in the space of abstract continuous functions

This paper describes a class of linear operators A which have the property that I+A = 1 + A , and which act in the space C of abstract continuous functions. In particular, to such operators, called rammers, are referred completely continuous operators and operators acting in C not only from C, but from some essentially broader space. As an application, it is shown that each linear operator B in C, whose norm is 1 and which is the identity of a subspace of finite deficiency, coincides with the identity on the whole space.Translated from Matematicheskie Zametki, Vol. 2, No. 6, 599–604, December, 1967.The author is grateful to P. P. Zabreiko for doscussion of this note.     

Tame linear extension operators for smooth Whitney functions

For a compact set K⊆Rd we present a rather easy construction of a linear extension operator E:E(K)→C^∞(Rd) for the space of Whitney jets E(K) which satisfies linear tame continuity estimates , where ‖⋅s_‖ denotes the s-th Whitney norm. The construction turns out to be possible if and only if the local Markov inequality LMI(s) introduced by Bos and Milman holds for every s>r on K. In particular, E(K) admits a tame linear extension operator if and only if the local Markov inequality LMI(s) holds on K for some s?1.     

On an extrapolation problem for characteristic functions

Lithuanian Mathematical Journal - Let f be the characteristic function of a probability measure μf on ?n, and let σ &gt; 0. We study the following extrapolation problem: under...     

On constructing new expansions of functions using linear operators

Let T,U be two linear operators mapped onto the function f such that U(T(f))=f, but T(U(f))≠f. In this paper, we first obtain the expansion of functions T(U(f)) and U(T(f)) in a general case. Then, we introduce four special examples of the derived expansions. First example is a combination of the Fourier trigonometric expansion with the Taylor expansion and the second example is a mixed combination of orthogonal polynomial expansions with respect to the defined linear operators T and U. In the third example, we apply the basic expansion U(T(f))=f(x) to explicitly compute some inverse integral transforms, particularly the inverse Laplace transform. And in the last example, a mixed combination of Taylor expansions is presented. A separate section is also allocated to discuss the convergence of the basic expansions T(U(f)) and U(T(f)).     

Positive linear extension operators for spaces of affine functions

The following result is proved: letE be anF-space (that is, the space of all continuous affine functions defined on a compact universal cap van shing at zero) and letMχE be anM-ideal. Then, ifE/M is a π₁ with positive defining projections, then there is a positive linear operator ϱ:E/M→E of norm one such that ϱ lifts the canonical mapE→E/M. In the proof, which heavily depends on work of Ando, we study ensor products of certain convex cones with compact bases, and we calculate the norm of a positive linear operator defined on a finite dimensional space with range in aF-space. Various corollaries are deduced for split faces of compact convex sets and for morphisms ofC ^*-algebras.     

Bounded characteristic functions and models for noncontractive sequences of operators

The Sz.-Nagy-FoiaŞ functional model for completely non-unitary contractions is extended to completely non-coisometric sequences of bounded operatorsT = (T₁,...,T _d) (d finite or infinite) on a Hilbert space, with bounded characteristic functions. For this class of sequences, it is shown that the characteristic function θ_T is a complete unitary invariant. We obtain, as the main result, necessary and sufficient conditions for a bounded multi-analytic operator on Fock spaces to coincide with the characteristic function associated with a completely non-coisometric sequence of bounded operators on a Hilbert space. Research supported in part by a COBASE grant from the National Research Council. The first author was partially supported by a grant from Ministerul Educaţiei Şi Cercetarii. The second author was partially supported by a National Science Foundation grant.     

Approximation of analytical functions by sequences of k-positive linear operators

Approximation properties of sequences of k-positive operators, i.e. linear operators acting in the space of analytical functions and preserving the cone of functions with non-negative Taylor coefficients, are studied. Some general theorems which are valid in the space of functions that are analytical in a bounded simply connected domain are proved.