Factorization of Hilbert port operators with poles on the imaginary axis

A factorization method is given to extract poles located on the imaginary axis for J-biexpansive meromorphic operator-valued functions acting on an infinite-dimensional Hilbert space. Decomposition of a real operator in terms of real factors, applicable to Hilbert ports, is also described, thus generalizing synthesis techniques originally developed for passive n-ports.     

On the representation of a class of contractive matrix-valued functions

By a decomposition of L₊²(Cⁿ) in two orthogonal subspaces we obtain a representation of a matrix-valued function of the class $\text{J}$Π, defined by Arov (Darlington realization of matrix-valued functions, Izv. Akad. Nauk. SSSR, Ser. Mat. Tom.37 (1973), No. 6); (Math. USSR Izvestija7 (1973), No. 6, 1295–1326). Real matrix-valued functions of this class play an important role in methods of synthesis of scattering matrices of linear passive n-ports.     

Real separated algebraic curves, quadrature domains, Ahlfors type functions and operator theory

The aim of this paper is to inter-relate several algebraic and analytic objects, such as real-type algebraic curves, quadrature domains, functions on them and rational matrix functions with special properties, and some objects from operator theory, such as vector Toeplitz operators and subnormal operators. Our tools come from operator theory, but some of our results have purely algebraic formulation. We make use of Xia's theory of subnormal operators and of the previous results by the author in this direction. We also correct (in Section 5) some inaccuracies in the works of [D.V. Yakubovich, Subnormal operators of finite type I. Xia's model and real algebraic curves in C², Rev. Mat. Iberoamericana 14 (1998) 95-115; D.V. Yakubovich, Subnormal operators of finite type II. Structure theorems, Rev. Mat. Iberoamericana 14 (1998) 623-681] by the author.     

Every completely polynomially bounded operator is similar to a contraction

It is proved that a bounded operator on a Hilbert space is similar to a contraction if and only if it is completely polynomially bounded. This gives a partial answer to Problem 6 of Halmos (Bull. Amer. Math. Soc.76 (1970). 877–933). The set of completely bounded maps between C¹-algebras is studied to obtain some structure, representation, and extension theorems for this class of maps. These allow a characterization of the completely bounded representations, on a Hilbert space, of any subalgebra of a C¹-algebra to be obtained. The result in the title follows by applying this characterization to the disk algebra.     

Applications of autoreproducing kernel grammian moduli to S (U,H)-valued stationary random functions

It is shown that the analytical characterizations of q-variate interpolable and minimal stationary processes obtained by H. Salehi (Ark. Mat., 7 (1967), 305–311; Ark. Mat., 8 (1968), 1–6; J. Math. Anal. Appl., 25 (1969), 653–662), and later by A. Weron (Studia Math., 49 (1974), 165–183), can be easily extended to Hilbert space valued stationary processes when using the two grammian moduli that respectively autoreproduce their correlation kernel and their spectral measure. Furthermore, for these processes, a Wold-Cramér concordance theorem is obtained that generalizes an earlier result established by H. Salehi and J. K. Scheidt (J. Multivar. Anal., 2 (1972), 307–331) and by A. Makagon and A. Weron (J. Multivar. Anal., 6 (1976), 123–137).     

On the spectral Stefan–Florin problem with classical boundary condition

The purpose of this work is to study the spectral properties of the problem of transmission arising after the linearization of two-phase problems of Stefan and Florin with classical boundary condition on a small time interval. With the help of the operator methods of mathematical physics, a boundary-value problem is reduced to the study of the spectrum of a weakly perturbed compact self-adjoint operator in a Hilbert space. On the basis of the theorems of M. V. Keldysh and V. B. Lidskii, we have established the basis property of the system of eigen- and associated elements by Abel–Lidskii in some Hilbert space. It is proved that the spectrum is discrete with the single limiting point at infinity. It is situated on the positive semiaxis or, except for a finite number of eigenvalues, in the aperture angle ε. The growth of the moduli of eigenvalues is estimated, and some asymptotic formulas are obtained.     

Factorizing operators on Banach function spaces through spaces of multiplication operators

In order to extend the theory of optimal domains for continuous operators on a Banach function space X(μ) over a finite measure μ, we consider operators T satisfying other type of inequalities than the one given by the continuity which occur in several well-known factorization theorems (for instance, Pisier Factorization Theorem through Lorentz spaces, pth-power factorable operators …). We prove that such a T factorizes through a space of multiplication operators which can be understood in a certain sense as the optimal domain for T. Our extended optimal domain technique does not need necessarily the equivalence between μ and the measure defined by the operator T and, by using δ-rings, μ is allowed to be infinite. Classical and new examples and applications of our results are also given, including some new results on the Hardy operator and a factorization theorem through Hilbert spaces.     

Elementary localization theorems for nonlinear eigenproblems

The minimax formula for linear eigenvalues of a linear operator is used to estimate the parameter values (λ) for which the self-adjoint operator L(λ) on Hilbert space to itself fails to have a bounded inverse. Such λ compose the “nonlinear spectrum” of L. The parameter spaces include regions in real or complex n-space. The localization theorems are used to demonstrate certain necessary conditions for stability of linear integro-partial-differential delay equations.     

Reproducing kernels of Sobolev spaces via a green kernel approach with differential operators and boundary operators

We introduce a vector differential operator P and a vector boundary operator B to derive a reproducing kernel along with its associated Hilbert space which is shown to be embedded in a classical Sobolev space. This reproducing kernel is a Green kernel of differential operator L:?=?P ^???T P with homogeneous or nonhomogeneous boundary conditions given by B, where we ensure that the distributional adjoint operator P ^??? of P is well-defined in the distributional sense. We represent the inner product of the reproducing-kernel Hilbert space in terms of the operators P and B. In addition, we find relationships for the eigenfunctions and eigenvalues of the reproducing kernel and the operators with homogeneous or nonhomogeneous boundary conditions. These eigenfunctions and eigenvalues are used to compute a series expansion of the reproducing kernel and an orthonormal basis of the reproducing-kernel Hilbert space. Our theoretical results provide perhaps a more intuitive way of understanding what kind of functions are well approximated by the reproducing kernel-based interpolant to a given multivariate data sample.     

Universal inequalities for lower order eigenvalues of self-adjoint operators and the poly-Laplacian

In this paper, we first establish an abstract inequality for lower order eigenvalues of a self-adjoint operator on a Hilbert space which generalizes and extends the recent results of Cheng et al. （Calc. Var. Partial Differential Equations, 38, 409-416 （2010））. Then, making use of it, we obtain some universal inequalities for lower order eigenvalues of the biharmonic operator on manifolds admitting some speciM functions. Moreover, we derive a universal inequality for lower order eigenvalues of the poly-Laplacian with any order on the Euclidean space.     

Definitizability of a Class of J-Selfadjoint Operators with Applications

We formulate an abstract result concerning the definitizability of J-selfadjoint operators which, roughly speaking, differ by at most finitely many dimensions from the orthogonal sum of a J-selfadjoint operator with finitely many negative squares and a semibounded selfadjoint operator in a Hilbert space. The general perturbation result is applied to a class of singular Sturm–Liouville operators with indefinite weight functions. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Meromorphic Factorization Revisited and Application to Some Groups of Matrix Functions

Some properties and applications of meromorphic factorization of matrix functions are studied. It is shown that a meromorphic factorization of a matrix function G allows one to characterize the kernel of the Toeplitz operator with symbol G without actually having to previously obtain a Wiener–Hopf factorization. A method to turn a meromorphic factorization into a Wiener–Hopf one which avoids having to factorize a rational matrix that appears, in general, when each meromorphic factor is treated separately, is also presented. The results are applied to some classes of matrix functions for which the existence of a canonical factorization is studied and the factors of a Wiener–Hopf factorization are explicitly determined. Submitted: April 15, 2007. Revised: October 26, 2007. Accepted: December 12, 2007.     

On the <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript> Helmholtz decomposition: A review of a result due to Solonnikov<Superscript>*</Superscript>

We consider the Helmholtz decomposition of the Lebesgue space L^p(Ω). We essentially reproduce a proof given by Solonnikov in [V.A. Solonnikov, Estimates of the solutions of the nonstationary Navier–Stokes system, Boundary Value Problems of Mathematical Physics and Related Questions in the Theory of Functions. Part 7, Zap. Nauchn. Semin. Leningr. Otd. Mat. Inst. Steklova, Vol. 38, Nauka, Leningrad, 1973, pp. 153–231 (in Russian)] and [V.A. Solonnikov, Estimates for solutions of nonstationary Navier–Stokes equations, J. Sov. Math., 8(4):467–529, 1977].     

A canonical factorization for meromorphic Herglotz functions on the unit disk and sum rules for Jacobi matrices

We prove a general canonical factorization for meromorphic Herglotz functions on the unit disk whose notable elements are that there is no restriction (other than interlacing) on the zeros and poles for their Blaschke product to converge and there is no singular inner function. We use this result to provide a significant simplification in the proof of Killip-Simon (Ann. Math. 158 (2003) 253) of their result characterizing the spectral measures of Jacobi matrices, J, with J−J₀ Hilbert-Schmidt. We prove a nonlocal version of Case and step-by-step sum rules.     

Operator algebras of functions

We present some general theorems about operator algebras that are algebras of functions on sets, including theories of local algebras, residually finite-dimensional operator algebras and algebras that can be represented as the scalar multipliers of a vector-valued reproducing kernel Hilbert space. We use these to further develop a quantized function theory for various domains that extends and unifies Agler's theory of commuting contractions and the Arveson-Drury-Popescu theory of commuting row contractions. We obtain analogous factorization theorems, prove that the algebras that we obtain are dual operator algebras and show that for many domains, supremums over all commuting tuples of operators satisfying certain inequalities are obtained over all commuting tuples of matrices.     

Extremals of functions on graphs with applications to graphs and hypergraphs

The method used in an article by T. S. Matzkin and E. G. Straus [Canad. J. Math.17 (1965), 533–540] is generalized by attaching nonnegative weights to t-tuples of vertices in a hypergraph subject to a suitable normalization condition. The edges of the hypergraph are given weights which are functions of the weights of its t-tuples and the graph is given the sum of the weights of its edges. The extremal values and the extremal points of these functions are determined. The results can be applied to various extremal problems on graphs and hypergraphs which are analogous to P. Turán's Theorem [Colloq. Math.3 (1954), 19–30: (Hungarian) Mat. Fiz. Lapok48 (1941), 436–452].     

On the generalized invertibility of Wiener-Hopf operators in Banach spaces

It is proved that a Wiener-Hopf operator T_p (A) on a Banach space PX is generalized invertible iff A has a cross factorization with respect toX and P. IfX is a separable Hilbert space, then a criterion for the weak factorization of A can be concluded.     

Multivariate Bernoulli splines and the periodic interpolation problem

Periodic spline interpolation in Euclidian spaceR d is studied using translates of multivariate Bernoulli splines introduced in [25]. The interpolating polynomial spline functions are characterized by a minimal norm property among all interpolants in a Hilbert space of Sobolev type. The results follow from a relation between multivariate Bernoulli splines and the reproducing kernel of this Hilbert space. They apply to scattered data interpolation as well as to interpolation on a uniform grid. For bivariate three-directional Bernoulli splines the approximation order of the interpolants on a refined uniform mesh is computed.     

UNIQUENESS PROBLEM FOR MEROMORPHIC MAPPINGS IN SEVERAL COMPLEX VARIABLES INTO P^N（C） WITH TRUNCATED MULTIPLICITIES

This paper proves some uniqueness theorems for meromorphic mappings in several complex variables into the complex projective space P N(C) with truncated multiplicities,and our results improve some earlier work.     

Regularization with differential operators. I. General theory

The method of regularization is used to obtain least squares solutions of the linear equation Kx = y, where K is a bounded linear operator from one Hilbert space into another and the regularizing operator L is a closed densely defined linear operator. Existence, uniqueness, and convergence analyses are developed. An application is given to the special case when K is a first kind integral operator and L is an nth order differential operator in the Hilbert space L²[a, b].