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.     

A Theorem of Beurling-sLax Type for Hilbert Spaces of Functions Analytic in the Unit Ball

Schur multipliers on the unit ball are operator-valued functions for which the N-variable Schwarz-Pick kernel is nonnegative. In this paper, the coefficient spaces are assumed to be Pontryagin spaces having the same negative index. The associated reproducing kernel Hilbert spaces are characterized in terms of generalized difference-quotient transformations. The connection between invariant subspaces and factorization is established.     

Factorizations along commutative subspace lattices

A theorem of D. R. Larson on the factorization of positive-definite operators along complete, countable nests in a Hilbert space is generalized to the case of commutative subspace lattices: We characterize those selfadjoint, positive-definite operators which can be factored A^* A where A and A^–1 leave invariant the subspaces in a given countable, complete, commutative subspace lattice. Applications to inner-outer factorization theory and factorization of a positive-definite operator with respect an uncountable commutative subspace lattice are given. These results have applications to function theory on the polydisk, nonanticipative representations of Gaussian random fields, and multiparameter systems theory.     

Extension to maximal semidefinite invariant subspaces for hyponormal matrices in indefinite inner products

It is proved that under certain essential additional hypotheses, a nonpositive invariant subspace of a hyponormal matrix admits an extension to a maximal nonpositive subspace which is invariant for both the matrix and its adjoint. Nonpositivity of subspaces and the hyponormal property of the matrix are understood in the sense of a nondegenerate inner product in a finite dimensional complex vector space. The obtained theorem combines and extends several previously known results. A Pontryagin space formulation, with essentially the same proof, is offered as well.     

分担四个或五个小函数的亚纯函数

本文证明了两个非常数亚约函数在具有五个ＩＭ分担小函数的前提下附加一定条件,则这两个函数必恒等,并研究了两个具有四个ＣＭ分担小函数的非常数亚纯函数之间的关系,推广了Ｎｅｖａｎｌｉｎｎａ四值定理．     

Approximate Factorization in Generalized Hardy Spaces

In this paper we establish a connection between the approximate factorization property appearing in the theory of dual algebras and the spectral inclusion property for a class of Toeplitz operators on Hilbert spaces of vector valued square integrable functions. As an application, it follows that a wide range of dual algebras of subnormal Toeplitz operators on various Hardy spaces associated to function algebras have property (A ₁(1)). It is also proved that the dual algebra generated by a spherical isometry (with a possibly infinite number of components) has the same property. One particular application is given to the existence of unimodular functions sitting in cyclic invariant subspaces of weak* Dirichlet algebras. Moreover, by this method we provide a unified approach to several Toeplitz spectral inclusion theorems. Research partially supported by grant CNCSIS GR202/2006 (cod 813).     

Pontryagin spaces of entire functions II

We continue the study of a generalization of L. de Branges's theory of Hilbert spaces of entire functions to the Pontryagin space setting. In this-second-part we investigate isometric embeddings of spaces of entire functions into spacesL ² () understood in a distributional sense and consider Weyl coefficients of matrix chains. The main task is to give a proof of an indefinite version of the inverse spectral theorem for Nevanlinna functions. Our methods use the theory developed by L. de Branges and the theory of extensions of symmetric operators of M.G.Krein.     

Pontryagin-de Branges-Rovnyak spaces of slice hyperholomorphic functions

We study reproducing kernel Hilbert and Pontryagin spaces of slice hyperholomorphic functions. These are analogs of the Hilbert spaces of analytic functions introduced by de Branges and Rovnyak. In the first part of the paper, we focus on the case of Hilbert spaces and introduce, in particular, a version of the Hardy space. Then we define Blaschke factors and Blaschke products and consider an interpolation problem. In the second part of the paper, we turn to the case of Pontryagin spaces. We first prove some results from the theory of Pontryagin spaces in the quaternionic setting and, in particular, a theorem of Shmulyan on densely defined contractive linear relations. We then study realizations of generalized Schur functions and of generalized Carathéodory functions.     

The Krein formula in almost Pontryagin spaces. A proof via orthogonal coupling

A new proof is provided for the Krein formula for generalized resolvents in the context of symmetric operators or relations with defect numbers $(1,1)$ in an almost Pontryagin space. The new proof is geometric and uses the orthogonal coupling of the almost Pontryagin spaces induced by the $Q$-function and the parameter function in the Krein formula.     

Multiple point interpolation in Nevanlinna classes

A Nevanlinna-Pick type interpolation problem for generalized Nevanlinna functions is considered. We prescribe the values of the function and its derivatives up to a certain order at finitely many points of the upper half plane. An operator theoretic approach is used to parametrize the solutions of this interpolation problem by means of selfadjoint extensions of a certain symmetry.     

A cross section from invariant subspaces to inner functions

Beurling's well known theorem connects the study of invariant subspaces to that of inner functions over the unit disc. In this paper, we will further explore this connection and, as a corollary of the result, show a one to one correspondence between the components of the invariant subspace lattice and the components of the space of inner functions.     

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.     

Generalized resolvent matrices and spaces of analytic functions

We introduce triplet spaces for symmetric relations with defect index (1, 1) in a Pontryagin space. Representations of Pontryagin spaces by spaces of vector-valued analytic functions are investigated. These concepts are used to study 2×2-matrix valued analytic functions which satisfy a certain kernel condition.     

A dual method of constructing hereditarily indecomposable Banach spaces

A new method of defining hereditarily indecomposable Banach spaces is presented. This method provides a unified approach for constructing reflexive HI spaces and also HI spaces with no reflexive subspace. All the spaces presented here satisfy the property that the composition of any two strictly singular operators is a compact one. This yields the first known example of a Banach space with no reflexive subspace such that every operator has a non-trivial closed invariant subspace.     

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.     

Characteristic functions and their factorizations

A new definition of the characteristic function is introduced for contractions on Hilbert spaces. The relationship with other definitions is established. A factorization formula corresponding to an invariant subspace is obtained. Bibliography: 3 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 247, 1997, pp. 71–78. Translated by V. V. Kapustin.     

Random dynamics of transcendental functions

This work concerns random dynamics of hyperbolic entire and meromorphic functions of finite order whose derivative satisfies some growth condition at ∞. This class contains most of the classical families of transcendental functions and goes much beyond. Based on uniform versions of Nevanlinna’s value distribution theory, we first build a thermodynamical formalism which, in particular, produces unique geometric and fiberwise invariant Gibbs states. Moreover, spectral gap property for the associated transfer operator along with exponential decay of correlations and a central limit theorem are shown. This part relies on our construction of new positive invariant cones that are adapted to the setting of unbounded phase spaces. This setting rules out the use of Hilbert’s metric along with the usual contraction principle. However, these cones allow us to apply a contraction argument stemming from Bowen’s initial approach.     

Shifted Hermite-Biehler Functions and Their Applications

We investigate a particular subclass of so-called symmetric indefinite Hermite-Biehler functions and give a characterization of functions of this class in terms of the location of their zeros. For the proof we employ the theory of de Branges Pontryagin spaces of entire functions. We apply our results to obtain information on the eigenvalues of some boundary value problems.     

Compact composition operators on weighted Hilbert spaces of analytic functions

We characterize the compactness of composition operators acting on a large family of Hilbert spaces of analytic functions which lie between Bergman and Dirichlet spaces. Our characterization is given in terms of generalized Nevanlinna counting functions.