Spectral representation for generalized operator-valued Toeplitz kernels

We prove an integral representation for operator-valued Toeplitz kernels. The proof is based on the spectral theory of the corresponding differential operator constructed from this kernel and acting in a Hilbert space. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 57, No. 12, pp. 1698–1710, December, 2005.     

Solving integro-differential equations with Cauchy kernel

A simple method for solving the Fredholm singular integro-differential equations with Cauchy kernel is proposed based on a new reproducing kernel space. Using a transformation and modifying the traditional reproducing kernel method, the singular term is removed and the analytical representation of the exact solution is obtained in the form of series in the new reproducing kernel space. The advantage of the approach lies in the fact that, on the one hand, by improving the definition of traditional inner product, the representation of new reproducing kernel function becomes simple and requirement for image space of operator is weakened comparing with traditional reproducing kernel method; on the other hand, the approximate solution and its derivatives converge uniformly to the exact solution and its derivatives. Some examples are displayed to demonstrate the validity and applicability of the proposed method.     

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.     

Convergence of high-order deterministic particle methods for the convection-diffusion equation

A proof of high-order convergence of three deterministic particle methods for the convection-diffusion equation in two dimensions is presented. The methods are based on discretizations of an integro-differential equation in which an integral operator approximates the diffusion operator. The methods differ in the discretization of this operator. The conditions for convergence imposed on the kernel that defines the integral operator include moment conditions and a condition on the kernel's Fourier transform. Explicit formulae for kernels that satisfy these conditions to arbitrary order are presented. © 1997 John Wiley & Sons, Inc.     

Integral Kernel Operators on Fock Space – Generalizations and Applications to Quantum Dynamics

A general theory of operators on Boson Fock space is discussed in terms of the white noise distribution theory on Gaussian space (white noise calculus). An integral kernel operator is generalized from two aspects: (i) The use of an operator-valued distribution as an integral kernel leads us to the Fubini type theorem which allows an iterated integration in an integral kernel operator. As an application a white noise approach to quantum stochastic integrals is discussed and a quantum Hitsuda–Skorokhod integral is introduced. (ii) The use of pointwise derivatives of annihilation and creation operators assures the partial integration in an integral kernel operator. In particular, the particle flux density becomes a distribution with values in continuous operators on white noise functions and yields a representation of a Lie algebra of vector fields by means of such operators.     

N-group neutron transport theory: A criticality problem in slab geometry

The steady-state equation for N-group neutron transport in slab geometry is written as an integral equation. A spectral analysis is made of the integral operator and related to the criticality problem. The method depends on a representation for the resolvent kernel for a subcritical slab and on analytic continuation in a complex parameter to characterize eigenvalues in terms of singularities of the resolvent. The analytic continuation is based on a bifurcation analysis of some nonlinear matrix integral equations whose solutions provide a matrix Wiener-Hopf factorization of the Fourier transform of the kernel of the transport operator.     

On a Series Representation for Integral Kernels of Transmutation Operators for Perturbed Bessel Equations

A representation for the kernel of the transmutation operator relating a perturbed Bessel equation to the unperturbed one is obtained in the form of a functional series with coefficients calculated by a recurrent integration procedure. New properties of the transmutation kernel are established. A new representation of a regular solution of a perturbed Bessel equation is given, which admits a uniform error bound with respect to the spectral parameter for partial sums of the series. A numerical illustration of the application of the obtained result to solve Dirichlet spectral problems is presented.     

Neumann problem with the integro-differential operator in the boundary condition

The Neumann problem for a second-order parabolic equation with integro-differential operator in the boundary condition is considered. A well-posedness theorem is proved, in particular, the integral representation of the solution is obtained, estimates for the derivatives of the solution are established, and the kernel of the inverse operator of the problem is explicitly expressed.     

Eventual Positivity of Hermitian Polynomials and Integral Operators

Quillen proved that if a Hermitian bihomogeneous polynomial is strictly positive on the unit sphere, then repeated multiplication of the standard sesquilinear form to this polynomial eventually results in a sum of Hermitian squares. Catlin-D'Angelo and Varolin deduced this positivstellensatz of Quillen from the eventual positive-definiteness of an associated integral operator. Their arguments involve asymptotic expansions of the Bergman kernel. The goal of this article is to give an elementary proof of the positive-definiteness of this integral operator.     

Pseudodifferential Multi-Product Representation of the Solution Operator of a Parabolic Equation

By using a time slicing procedure, we represent the solution operator of a second-order parabolic pseudodifferential equation on ? ⁿ as an infinite product of zero-order pseudodifferential operators. A similar representation formula is proven for parabolic differential equations on a compact Riemannian manifold. Each operator in the multi-product is given by a simple explicit Ansatz. The proof is based on an effective use of the Weyl calculus and the Fefferman-Phong inequality.     

Tauberian-type theorem for (e)-convergent sequences

We prove a Tauberian-type theorem for (e)-convergent sequences, which were introduced by the author in Karaev (2010) [4]. Our proof is based on the Berezin symbols technique of operator theory in the reproducing kernel Hilbert space.     

On extensions of positive definite integral Fredholm operators

The main result of this paper states that a positive definite Fredholm integral operator acting on L²([0,1]) can be modified on a Lebesque measurable set D\mit\Delta in [0,1]² such that the resulting operator is positive definite and its resolvent kernel is zero on D\mit\Delta . This answers a question raised in [3]. The proof is based on extension results for positive definite operator matrices and their connection to generalized determinants.     

Theta function transformation formulas and the Weil representation

A short proof is given that the theta functional is invariant under the Weil representation, and the explicit determination of the eighth root of unity which arises is also shown. Namely, the action of the Weil representation on the theta functional is described as the limit of integration against a specialization of the symplectic theta function as the symplectic variable approaches a cusp. The invariance is then a consequence of the automorphic nature of this theta function, coupled with the fact that in the limit it acts as the reproducing kernel for a certain lattice. Using results of Stark and Styer, this also allows one to determine the root of unity involved.     

The Fredholm Determinant Representation for the Partition Function of the Six-Vertex Model

The six-vertex model with domain-wall boundary conditions is considered. The Fredholm determinant representation for the partition function of the model is obtained. The kernel of the corresponding integral operator depends on Laguerre polynomials. Bibliography: 13 titles.     

INTEGRAL REPRESENTATION OF NONLINEAR OPERATORS

A continuous linear functional on some function space can be represented by an integral which in its usual form is linear. In this paper, we give an integral representation of a nonlinear operator on the space C=C([0,1],X) of continuous functions on [0,1] with values in a Banach space X. This is done by means of a nonlinear integral using a kernel type function.     

Haar-like expansions and boundedness of a Riesz operator on a solvable Lie group

Consider the group of affine transformations of the line. Denote by X and Y the right-invariant vector fields corresponding to the s and t directions, respectively, and let We prove that the first-order Riesz operator is of weak type (1, 1) with respect to left Haar measure. This operator is therefore also bounded on . Our results provide answers, in a particular instance, to the open question of the boundedness of Riesz operators on Lie groups of exponential growth. The main parts of the proof concern the behaviour of the kernel of the operator at infinity, and exploit cancellation. A key technique is to use expansion with respect to scales of Haar-like functions. Received March 16, 1998; in final form June 22, 1998     

Fredholm Determinant Evaluations of the Ising Model Diagonal Correlations and their λ Generalization

The diagonal spin–spin correlations of the square lattice Ising model, originally expressed as Toeplitz determinants, are given by two distinct Fredholm determinants—one with an integral operator having an Appell function kernel and another with a summation operator having a Gauss hypergeometric function kernel. Either determinant allows for a Neumann expansion possessing a natural λ‐parameter generalization and we prove that both expansions are in fact equal, implying a continuous and a discrete representation of the form factors. Our proof employs an extension of the classic study by Geronimo and Case [ 1 ], applying scattering theory to orthogonal polynomial systems on the unit circle, to the bi‐orthogonal situation.     

A Fredholm determinant formula for Toeplitz determinants

We prove a formula expressing a generaln byn Toeplitz determinant as a Fredholm determinant of an operator 1 –K acting onl ₂ (n,n+1,...), where the kernelK admits an integral representation in terms of the symbol of the original Toeplitz matrix. The proof is based on the results of one of the authors, see [14], and a formula due to Gessel which expands any Toeplitz determinant into a series of Schur functions. We also consider 3 examples where the kernel involves the Gauss hypergeometric function and its degenerations.     

Some essentially self-adjoint one-electron dirac operators

A class of smooth functions is introduced and it is shown that, the one-electron Dirac operator corresponding to an element with atomic number less than 102, is essentially self-adjoint on this class of functions. The proof makes essential use of the fact that each of these operators admits a complete family of reducing subspaces. At the same time we use an explicit formula for the resolvent kernel of the part of these operators over such a reducing subspace. Appendix by F. H. Brownell of the University of Washington This work was supported by N.S.F. grant GP-21330.