The Vector-Valued Big <Emphasis Type="Italic">q</Emphasis>-Jacobi Transform

Big q-Jacobi functions are eigenfunctions of a second-order q-difference operator L. We study L as an unbounded self-adjoint operator on an L ²-space of functions on ℝ with a discrete measure. We describe explicitly the spectral decomposition of L using an integral transform ℱ with two different big q-Jacobi functions as a kernel, and we construct the inverse of ℱ.      

Characteristic properties of strong integral and strong B-integral self-adjoint operators

A proof that a strong integral (strong B-integral) self-adjoint operator in L₂ (a, b) is a Hilbert-Schmidt operator (a kernel operator).Translated from Matematicheskie Zametki, Vol. 8, No. 5, pp. 653–661, November, 1970.     

Matrix-valued Schrödinger operators over local fields

We consider quantum systems that have as their configuration spaces finite dimensional vector spaces over local fields. The quantum Hilbert space is taken to be a space with complex coefficients and we include in our model particles with internal symmetry. The Hamiltonian operator is a pseudo-differential operator that is initially only formally defined. For a wide class of potentials we prove that this Hamiltonian is well-defined as an unbounded self-adjoint operator. The free part of the operator gives rise to ameasure on the Skorokhod space of paths,D[0,∞), and with respect to this measure there is a path integral representation for the semigroup associated to the Hamiltonian. We prove this Feynman-Kac formula in the local field setting as a consequence of the Hille-Yosida theory of semi-groups. The text was submitted by the authors in English.     

On generalized resolvents and spectral functions of differential operators of even order in a space of vector-valued functions

We consider a self-adjoint differential expression of even order in a space of vector-valued functions. We prove that an arbitrary generalized resolvent of the minimal operator L₀ induced by this differential expression is an integral operator and we derive a formula for all of the spectral functions (orthogonal and nonorthogonal) of the operator L₀.     

Optimality of function spaces for kernel integral operators

We explore boundedness properties of kernel integral operators acting on rearrangement-invariant (r.i.) spaces. In particular, for a given r.i. space X we characterize its optimal range partner, that is, the smallest r.i. space Y such that the operator is bounded from X to Y. We apply the general results to Lorentz spaces to illustrate their strength.     

Dissipative q-Dirac operator with general boundary conditions

In this paper, we consider the symmetric q-Dirac operator. We describe dissipative, accumulative, self-adjoint and the other extensions of such operators with general boundary conditions. We construct a self-adjoint dilation of dissipative operator. Hence, we determine the scattering matrix of dilation. Later, we construct a functional model of this operator and define its characteristic function. Finally, we prove that all root vectors of this operator are complete.     

On the Norm of a Self-Adjoint Operator and a New Bilinear Integral Inequality

In this paper, the expression of the norm of a self-adjoint integral operator T ： L^2（0, ∞） → L^2 （0, ∞） is obtained. As applications, a new bilinear integral inequality with a best constant factor is established and some particular cases are considered.     

Symplectic leaves in real banach Lie–Poisson spaces

We present several large classes of real Banach Lie–Poisson spaces whose characteristic distributions are integrable, the integral manifolds being symplectic leaves just as in finite dimensions. We also investigate when these leaves are embedded submanifolds or when they have K?hler structures. Our results apply to the real Banach Lie–Poisson spaces provided by the self-adjoint parts of preduals of arbitrary W*-algebras, as well as of certain operator ideals. Received: April 2004 Accepted: September 2004     

Analysis of the M/M/1 Queue with Processor Sharing via Spectral Theory

We show in this paper that the computation of the distribution of the sojourn time of an arbitrary customer in a M/M/1 with the processor sharing discipline (abbreviated to M/M/1 PS queue) can be formulated as a spectral problem for a self-adjoint operator. This approach allows us to improve the existing results for this queue in two directions. First, the orthogonal structure underlying the M/M/1 PS queue is revealed. Second, an integral representation of the distribution of the sojourn time of a customer entering the system while there are n customers in service is obtained.     

Galerkin method for a nonstationary equation with a monotone operator

In the present paper, we consider the Galerkin method for a quasilinear differentialoperator equation with a leading self-adjoint operator A(t) and a subordinate monotone operator K. For the projection subspaces we take linear spans of eigenelements of an operator similar to the leading operator A(t). We obtain new estimates for the Galerkin method and consider applications to an initial-boundary value problem for a parabolic equation of higher order.     

Vector-valued singular integral operators with rough kernels

In this paper, we establish a weak-type (1,1) boundedness criterion for vector-valued singular integral operators with rough kernels. As applications, we obtain weak-type (1,1) bounds for the convolution singular integral operator taking value in the Banach space Y with a rough kernel, the maximal operator taking vector value in Y with a rough kernel and several square functions with rough kernels. Here, $Y={[H,X]}_{\theta }$ is a complex interpolation space between a Hilbert space H and a UMD space X.     

Error estimates for projection-difference methods for differential equations with differentiable operators

We study the projection-difference methods for approximate solving the Cauchy problem for operator-differential equations with a leading self-adjoint operator A(t) and a subordinate linear operator K(t), whose definition domain is independent of t. Operators A(t) and K(t) are assumed to be sufficiently smooth. We obtain estimates for the rate of convergence of approximate solutions to the exact solution as well as those for fractional degrees of an operator similar to A(0).     

Similarity of a Triangular Operator to a Diagonal Operator

We give sufficient conditions under which the non-self-adjoint operator A = G + iV ^1/2 JV ^1/2 (with a well-defined imaginary part) is similar to a self-adjoint one. We also give sufficient conditions (these conditions become necessary in the dissipative case) under which the triangular operator is similar to a self-adjoint one. Bibliography: 34 titles.     

Similarity of Some Singular Operators to Self-Adjoint Ones

The singular differential operator is studied. It is proved that if the second moment of p is finite and L has no nonreal eigenvalues, then L is similar to a self-adjoint operator. The proof is based on an integral resolvent criterion of similarity applied to a wide class of functions p(x). Bibliography: 20 titles.     

On uniform positivity of a symmetrizer

We exhibit ah effective method of constructing a positive definite symmetrizer of a linear operator that is spectrally equivalent to a self-adjoint operator bundle whose operator coefficients satisfy certain conditions. It is proved that these conditions are sufficient for positive definiteness of the symmetrizer.Translated fromMatematicheskie Melody i Fiziko-Mekhanicheskie Polya, Issue 34, 1991, pp. 12–15.     

Scattering of elastic waves in a perturbed isotropic half space with a free boundary. The limiting absorption principle

In this article we consider the self-adjoint operator governing the propagation of elastic waves in a perturbed isotropic half space with a free boundary condition. We prove the limiting absorption principle in appropriate Hilbert spaces for this operator. We also prove decreasing properties for the eigenfunctions associated with strictly positive eigenvalues of this operator. The proofs are based on the limiting absorption principle for the self-adjoint operator governing the propagation of elastic waves in a homogeneous isotropic half space with a free boundary and on the so called division theorem for it. Both perturbations of R ₊² ={(x₁, x₂) ? R ²; x₂ > 0} and R ₊² = {(x₁, x₂, x₃) ? R ³; x₃ > 0} are considered.     

-Operator frames for a Banach space

In this paper, (p,Y)-Bessel operator sequences, operator frames and (p,Y)-Riesz bases for a Banach space X are introduced and discussed as generalizations of the usual concepts for a Hilbert space and of the g-frames. It is proved that the set of all (p,Y)-Bessel operator sequences for a Banach space X is a Banach space and isometrically isomorphic to the operator space B(X,ℓ^p(Y)). Some necessary and sufficient conditions for a sequence of operators to be a (p,Y)-Bessel operator sequence are given. Also, a characterization of an independent (p,Y)-operator frame for X is obtained. Lastly, it is shown that an independent (p,Y)-operator frame for X is just a (p,Y)-Riesz basis for X and has a unique dual (q,Y^*)-operator frame for X^*.     

The controlled separable projection property for Banach spaces

Let X, Y be two Banach spaces. We say that Y is a quasi-quotient of X if there is a continuous operator R: X → Y such that its range, R(X), is dense in Y.     

Criteria for the single-valued metric generalized inverses of multi-valued linear operators in banach spaces

Let X, Y be Banach spaces and M be a linear subspace in X × Y = {{x, y}|x ∈ X, y ∈ Y }. We may view M as a multi-valued linear operator from X to Y by taking M (x) = {y|{x, y} ∈ M }. In this paper, we give several criteria for a single-valued operator from Y to X to be the metric generalized inverse of the multi-valued linear operator M . The principal tool in this paper is also the generalized orthogonal decomposition theorem in Banach spaces.     

Banach spaces of operators that are complemented in their biduals

Let [A, a] be a normed operator ideal. We say that [A, a] is boundedly weak*-closed if the following property holds: for all Banach spaces X and Y, if T: X → Y** is an operator such that there exists a bounded net (T _i ) _i∈I in A(X, Y) satisfying lim _i 〈y*, T _i x y*〉 for every x ∈ X and y* ∈ Y*, then T belongs to A(X, Y**). Our main result proves that, when [A, a] is a normed operator ideal with that property, A(X, Y) is complemented in its bidual if and only if there exists a continuous projection from Y** onto Y, regardless of the Banach space X. We also have proved that maximal normed operator ideals are boundedly weak*-closed but, in general, both concepts are different.