On locally definitizable matrix functions

We study analytic properties of special classes of matrix functions (locally definitizable and locally Nevanlinna functions) by methods of operator theory. The aim of this paper is to prove that if G(λ) is a locally definitizable or locally generalized matrix Nevanlinna function, then ?(G(λ))^?1 belongs to the same class.     

Stochastic generalized porous media and fast diffusion equations

We present a generalization of Krylov-Rozovskii's result on the existence and uniqueness of solutions to monotone stochastic differential equations. As an application, the stochastic generalized porous media and fast diffusion equations are studied for σ-finite reference measures, where the drift term is given by a negative definite operator acting on a time-dependent function, which belongs to a large class of functions comparable with the so-called N-functions in the theory of Orlicz spaces.     

Products of generalized Nevanlinna functions with symmetric rational functions

New classes of generalized Nevanlinna functions, which under multiplication with an arbitrary fixed symmetric rational function remain generalized Nevanlinna functions, are introduced. Characterizations for these classes of functions are established by connecting the canonical factorizations of the product function and the original generalized Nevanlinna function in a constructive manner. Also, a detailed functional analytic treatment of these classes of functions is carried out by investigating the connection between the realizations of the product function and the original function. The operator theoretic treatment of these realizations is based on the notions of rigged spaces, boundary triplets, and associated Weyl functions.     

Analytic characterizations of Jordan chains by pole cancellation functions of higher order

In this paper the analytic characterization of generalized poles of operator valued generalized Nevanlinna functions (including the length of Jordan chains of the representing relation) is completed. In particular, given a Jordan chain of the representing relation of length ?, we show that there exists a pole cancellation function of order at least ?, and, moreover, the construction shows that it is of surprisingly simple form.     

Eigenvalues and Pole Functions of Hamiltonian Systems with Eigenvalue Depending Boundary Conditions

This paper consists of two chapters. The first chapter concerns matrix functions belonging to the generalized Nevanlinna class N_k^{m × m}. We present results about the operator representation of such functions. These representations are then used to obtain information about the (generalized) poles of generalized Nevanlinna functions. The second chapter may be viewed as a continuation of our paper [DLS3] and treats Hamiltonian systems of differential equations with boundary conditions depending on the eigenvalue parameter. In particular we study the eigenvalues, both isolated and embedded eigenvalues.     

Linear Fractional Transformations of Nevanlinna Functions Associated with a Nonnegative Operator

In the present paper a subclass of scalar Nevanlinna functions is studied, which coincides with the class of Weyl functions associated to a nonnegative symmetric operator of defect one in a Hilbert space. This class consists of all Nevanlinna functions that are holomorphic on (?∞, 0) and all those Nevanlinna functions that have one negative pole a and are injective on ${(-\infty, a)\,\cup\, (a, 0)}$ . These functions are characterized via integral representations and special attention is paid to linear fractional transformations which arise in extension and spectral problems of symmetric and selfadjoint operators.     

A Hypergeometric Function Transform and Matrix-Valued Orthogonal Polynomials

The spectral decomposition for an explicit second-order differential operator T is determined. The spectrum consists of a continuous part with multiplicity two, a continuous part with multiplicity one, and a finite discrete part with multiplicity one. The spectral analysis gives rise to a generalized Fourier transform with an explicit hypergeometric function as a kernel. Using Jacobi polynomials, the operator T can also be realized as a five-diagonal operator, leading to orthogonality relations for 2×2-matrix-valued polynomials. These matrix-valued polynomials can be considered as matrix-valued generalizations of Wilson polynomials.     

The properties of orthogonal matrix-valued wavelet packets in higher dimensions

In the paper matrix-valued multiresolution analysis and matrix-valued wavelet packets of spaceL ²(R ⁿ ,C ^{s x s}) are introduced. A procedure for constructing a class of matrix-valued wavelet packets in higher dimensions is proposed. The properties for the matrix-valued multivariate wavelet packets are investigated by using integral transform, algebra theory and operator theory. Finally, a new orthonormal basis ofL ²(R ⁿ ,C ^{s x s}) is derived from the orthogonal multivariate matrix-valued wavelet packets.     

A characterization of generalized poles of generalized Nevanlinna functions

Generalized poles of a generalized Nevanlinna function Q ∈ ??_κ (??) are defined in terms of the operator representation of Q . In this paper those generalized poles that are not of positive type and their degrees of non‐positivity are characterized analytically by means of pole cancellation functions. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Examples of discontinuous maximal monotone linear operators and the solution to a recent problem posed by B.F. Svaiter

In this paper, we give two explicit examples of unbounded linear maximal monotone operators. The first unbounded linear maximal monotone operator S on ?² is skew. We show its domain is a proper subset of the domain of its adjoint S^∗, and −S^∗ is not maximal monotone. This gives a negative answer to a recent question posed by Svaiter. The second unbounded linear maximal monotone operator is the inverse Volterra operator T on L²[0,1]. We compare the domain of T with the domain of its adjoint T^∗ and show that the skew part of T admits two distinct linear maximal monotone skew extensions. These unbounded linear maximal monotone operators show that the constraint qualification for the maximality of the sum of maximal monotone operators cannot be significantly weakened, and they are simpler than the example given by Phelps-Simons. Interesting consequences on Fitzpatrick functions for sums of two maximal monotone operators are also given.     

Boundary Relations, Unitary Colligations, and Functional Models

Recently a new notion, the so-called boundary relation, has been introduced involving an analytic object, the so-called Weyl family. Weyl families and boundary relations establish a link between the class of Nevanlinna families and unitary relations acting from one Kreĭn space, a basic (state) space, to another Kreĭn space, a parameter space where the Nevanlinna family or Weyl family is acting. Nevanlinna families are a natural generalization of the class of operator-valued Nevanlinna functions and they are closely connected with the class of operator-valued Schur functions. This paper establishes the connection between boundary relations and their Weyl families on the one hand, and unitary colligations and their transfer functions on the other hand. From this connection there are various advances which will benefit the investigations on both sides, including operator theoretic as well as analytic aspects. As one of the main consequences a functional model for Nevanlinna families is obtained from a variant of the functional model of L. de Branges and J. Rovnyak for Schur functions. Here the model space is a reproducing kernel Hilbert space in which multiplication by the independent variable defines a closed simple symmetric operator. This operator gives rise to a boundary relation such that the given Nevanlinna family is realized as the corresponding Weyl family. Received: January 21, 2008., Revised: March 31, 2008.     

Weyl-Titchmarsh functions of vector-valued Sturm-Liouville operators on the unit interval

The matrix-valued Weyl-Titchmarsh functions M(λ) of vector-valued Sturm-Liouville operators on the unit interval with the Dirichlet boundary conditions are considered. The collection of the eigenvalues (i.e., poles of M(λ)) and the residues of M(λ) is called the spectral data of the operator. The complete characterization of spectral data (or, equivalently, N×N Weyl-Titchmarsh functions) corresponding to N×N self-adjoint square-integrable matrix-valued potentials is given, if all N eigenvalues of the averaged potential are distinct.     

COLLECTIVELY COMPACT COMPOSITION OPERATOR SEQUENCES BETWEEN BERGMAN AND HARDY SPACES

This paper studies the collective compactness of composition operator se-quences between the Bergman and Hardy spaces. Some sufficient and necessary conditionsinvolving the generalized Nevanlinna counting functions for composition operator sequencesto be collectively compact between weighted Bergman spaces are given.     

一类非紧算子正不动点的存在唯一性

本文利用锥理论和选代技巧,研究了一类非紧混合单调算子正不动点的存在唯一性,改进和推广了混合单调算子、增算子与减算子的某些相应结果．     

Generalized invex monotonicity

In this paper the generalized invex monotone functions are defined as an extension of monotone functions. A series of sufficient and necessary conditions are also given that relate the generalized invexity of the function θ with the generalized invex monotonicity of its gradient function ∇θ. This new class of functions will be important in order to characterize the solutions of the Variational-like Inequality Problem and Mathematical Programming Problem.     

Pseudo-localization of singular integrals and noncommutative Calderón-Zygmund theory

The weak type (1,1) boundedness of singular integrals acting on matrix-valued functions has remained open since the 1980s, mainly because the methods provided by the vector-valued theory are not strong enough. In fact, we can also consider the action of generalized Calderón-Zygmund operators on functions taking values in any other von Neumann algebra. Our main tools for its solution are two. First, the lack of some classical inequalities in the noncommutative setting forces to have a deeper knowledge of how fast a singular integral decreases—L₂ sense—outside of the support of the function on which it acts. This gives rise to a pseudo-localization principle which is of independent interest, even in the classical theory. Second, we construct a noncommutative form of Calderón-Zygmund decomposition by means of the recent theory of noncommutative martingales. This is a corner stone in the theory. As application, we obtain the sharp asymptotic behavior of the constants for the strong Lp inequalities, also unknown up to now. Our methods settle some basics for a systematic study of a noncommutative Calderón-Zygmund theory.     

Generalized Nevanlinna functions and multivalency

A new characterization of generalized Nevanlinna functions in terms of multivalency is established. By analyzing multivalent functions additional novel characterizations of (generalized) Nevanlinna functions are obtained. As a particular consequence of these investigations a function-theoretic proof of the factorization property of generalized Nevanlinna functions is obtained.     

The converse of the Loewner–Heinz inequality via perspective

The Loewner–Heinz inequality is not only the most essential one in operator theory, but also a fundamental tool for treating operator inequalities. The aim of this paper is to investigate the converse of the Loewner–Heinz inequality in the view point of perspective and generalized perspective of operator monotone and multiplicative functions. Indeed, we give perspective inequalities equivalent to the Loewner–Heinz inequality.     

Generalized Monotone Operators on Dense Sets

In the present work we show that the local generalized monotonicity of a lower semicontinuous set-valued operator on some certain type of dense sets ensures the global generalized monotonicity of that operator. We achieve this goal gradually by showing at first that the lower semicontinuous set-valued functions of one real variable, which are locally generalized monotone on a dense subsets of their domain are globally generalized monotone. Then, these results are extended to the case of set-valued operators on arbitrary Banach spaces. We close this work with a section on the global generalized convexity of a real valued function, which is obtained out of its local counterpart on some dense sets.