Operator-valued Kernels Research Articles

QUANTUM INTEGRAL EQUATIONS OF VOLTERRA TYPE WITH GENERALIZED OPERATOR-VALUED KERNELS

Quantum integral equation of Volterra type with generalized operator-valued kernel is introduced. Existence and uniqueness of solutions are established, explicit expression of the solution is given, the continuity, continuous dependence on free terms and other properties of the solution are proved.

Noncommutative joint dilations and free product operator algebras

Let An (n = 2, 3, . . . , or n = ∞) be the noncommutative disc algebra, and On (resp. Tn) be the Cuntz (resp. Toeplitz) algebra on n generators. Minimal joint isometric dilations for families of contractive sequences of operators on a Hilbert space are obtained and used to extend the von Neumann inequality and the commutant lifting theorem to our noncommutative setting. We show that the universal algebra generated by k contractive sequences of operators and the identity is the amalgamated free product operator algebra ∗CAni for some positive integers n1, n2, . . . , nk ≥ 1, and characterize the completely bounded representations of ∗CAni . It is also shown that ∗CAni is completely isometrically imbedded in the “biggest” free product C∗-algebra ∗CTni (resp. ∗COni), and that all these algebras are completely isometrically isomorphic to some universal free operator algebras, providing in this way some factorization theorems. We show that the free product disc algebra ∗CAni is not amenable and the set of all its characters is homeomorphic to (C1)1 × · · · × (Ck)1. An extension of the Naimark dilation theorem to free semigroups is considered. This is used to construct a large class of positive definite operator-valued kernels on the unital free semigroup on n generators and to study the class Cρ (ρ > 0) of ρ-contractive sequences of operators. The dilation theorems are also used to extend the operatorial trigonometric moment problem to the free product C∗-algebras ∗CTni and ∗COni .

Operator-valued positive definite kernels on tubes

In this paper we generalize the concept of an infinite positive measure on a σ-algebra to a vector valued setting, where we consider measures with values in the compactification of a convex coneC which can be described as the set of monoid homomorphisms of the dual coneC* into [0, ∞]. Applying these concepts to measures on the dual of a vector space leads to generalizations of Bochner's Theorem to operator valued positive definite functions on locally compact abelian groups and likewise to generalizations of Nussbaum's Theorem on positive definite functions on cones. In the latter case we use the Laplace transform to realize the corresponding Hilbert spaces by holomorphic functions on tube domains.

Weakly singular integral equations with operator-valued kernels and an application to radiation transfer problems

The standard problem of radiation transfer in a bounded regionG ⊂ ℝ n can be reformulated as a weakly singular integral equation with an unknown functionu: G→C(Sn−1) and a kernelK: ((G × G ∖ }x=y}, which is continuously differentiable with respect to the operator strong convergence topology. We take these observations into the basis of an abstract treatment of weakly singular integral equations with ℒ(E)-valued kernels, whereE is a Banach space. Our purpose is to characterize the smoothness of the solution by proving that it belongs to special weighted spaces of smooth functions. On the way, realizing the proof techniques, we establish the compactness of the integral operator or its square inL p (G,E),BC(G,E), and other spaces of interest in numerical analysis as well as in weighted spaces of smooth functions. The smoothness results are specified for the standard problem of radiation transfer as well as for the corresponding eigenvalue problem.

An operator-valued Poisson kernel

Versions of the Cauchy and Poisson formulas in the unit ball of C n are defined, by means of appropriate operator-valued kernels. Connections with some problems of dilation theory are then made.

Singular Integral Operators with Operator-Valued Kernels on Spaces of Homogeneous Type

We give a generalized form of the Hormander-Schwartz-Triebel theorem for singular integral operators of potential type as well as a sequentiaI version. Applications include fractionary maximal inequalities of Hardy- Littlewood and Fefferman-Stein type.

A lifting theorem for subordinated invariant kernels

A lifting property for subordinated invariant kernels is proved and extended to operator-valued kernels. The connections with the classical Nagy-Foias lifting theorem and with Naimark's dilation theorem are indicated, as well as possible applications to scattering structures and quantum field theory.

On a nonlinear Volterra integral equation in a Banach space

The equation u( t) + ∝ 0 t k( t − s) g( s) ds ϵ f( t), t ⩾ 0, is studied in a real Banach space with uniformly convex dual. Conditions, sufficient for the existence of a unique solution, are given for the operatorvalued kernel k, the nonlinear m-accretive operators g( t) and the function f. The case when k is realvalued, g( t) ≡ g and X a reflexive Banach space is also considered. These results extend earlier results by Barbu, Londen and MacCamy.

Dilations as Propagators of Hilbertian Varieties

In this paper W-to-$W^ * $ operator-valued positive definite kernels $K( \cdot \cdot )$ are defined on $\Lambda \times \Lambda $, where W is a Banach space and $\Lambda $ an arbitrary set, and Hibertian varieties $X( \cdot )$ with such covariance kernels are studied with the aid of the Kolmogorov–Aronszajn–Pedrick kernel theorem. The general notion of a propagator of $X( \cdot )$ is introduced in terms of the action on $\Lambda $ of a semi-group $\Gamma $, and necessary and sufficient conditions are established for its existence. We show that these conditions simplify substantially when the semi-group $\Gamma $ is involutory, especially so when $\Lambda = \Gamma $. For $\Gamma = \Lambda $ equal to a unitized Banach algebra with isometric involution, these conditions subsume those given by Stinespring for $C^ * $-algebras.If for Hilbert spaces W and semi-groups $\Lambda $, dilations are redefined in terms of isometrics rather than projections, then the dilation $\tilde R( \cdot )$ of a given W-to-W operato...

Abstract Interpolation and Operator-Valued Kernels

The complex interpolation method is used to prove a sharp inequality of Hausdorff-Young type for integral operators with operator-valued kernels.

Gentle Perturbations of with Application to

The theory of gentle perturbations was introduced by Friedrichs [3] as a tool to study the perturbation theory of the absolutely continuous spectrum of a self-adjoint operator H0 and developed in an abstract form by Rejto [7; 8]. Two examples of gentle structures are well knowTn. In the first of these, the gentle operators have Hölder continuous complex or operator-valued kernels, and in the second, the kernels are Fourier transforms of L1 functions [4].The gentle structure has traditionally been verified in the case when H0 is in its spectral representation, that is, when H0 is the simple differentiation operator. This is not the natural setting for the second example mentioned above where one should consider the simple differentiation operator in a suitable L2-space and perturbations with L1 kernels.