Compact Differences of Weighted Composition Operators on Weighted Banach Spaces of Analytic Functions

Let be the weighted Banach space of analytic functions with a topology generated by weighted sup-norm. In the present article, we investigate the analytic mappings and which characterize the compactness of differences of two weighted composition operators on the space . As a consequence we characterize the compactness of differences of composition operators on weighted Bloch spaces.      

Passive Systems with a Normal Main Operator and Quasi-selfadjoint Systems

Passive systems with and as an input and output space and as a state space are considered in the case that the main operator on the state space is normal. Basic properties are given and a general unitary similarity result involving some spectral theoretic conditions on the main operator is established. A passive system with is said to be quasi-selfadjoint if ran . The subclass of the Schur class is the class formed by all transfer functions of quasi-selfadjoint passive systems. The subclass is characterized and minimal passive quasi-selfadjoint realizations are studied. The connection between the transfer function belonging to the subclass and the Q-function of T is given. Received: December 16, 2007., Accepted: March 4, 2008.     

On the Structure of the C *-Algebra Generated by Toeplitz Operators with Piece-wise Continuous Symbols

We study the C ^*-algebra generated by Toeplitz operators with piece-wise continuous symbols acting on the Bergman space on the unit disk in . We describe explicitly each operator from this algebra and characterize Toeplitz operators which belong to the algebra. To the memory of G. S. Litvinchuk     

The Extremal Truncated Moment Problem

For a degree 2n real d-dimensional multisequence to have a representing measure μ, it is necessary for the associated moment matrix to be positive semidefinite and for the algebraic variety associated to β, , to satisfy rank card as well as the following consistency condition: if a polynomial vanishes on , then . We prove that for the extremal case , positivity of and consistency are sufficient for the existence of a (unique, rank -atomic) representing measure. We also show that in the preceding result, consistency cannot always be replaced by recursiveness of . The first-named author’s research was partially supported by NSF Research Grants DMS-0099357 and DMS-0400741. The second-named author’s research was partially supported by NSF Research Grant DMS-0201430 and DMS-0457138.     

On the spectrum of lamplighter groups and percolation clusters

Let be a finitely generated group and X its Cayley graph with respect to a finite, symmetric generating set S. Furthermore, let be a finite group and the lamplighter group (wreath product) over with group of “lamps” . We show that the spectral measure (Plancherel measure) of any symmetric “switch–walk–switch” random walk on coincides with the expected spectral measure (integrated density of states) of the random walk with absorbing boundary on the cluster of the group identity for Bernoulli site percolation on X with parameter . The return probabilities of the lamplighter random walk coincide with the expected (annealed) return probabilities on the percolation cluster. In particular, if the clusters of percolation with parameter are almost surely finite then the spectrum of the lamplighter group is pure point. This generalizes results of Grigorchuk and Żuk, resp. Dicks and Schick regarding the case when is infinite cyclic. Analogous results relate bond percolation with another lamplighter random walk. In general, the integrated density of states of site (or bond) percolation with arbitrary parameter is always related with the Plancherel measure of a convolution operator by a signed measure on , where or another suitable group. M. Neuhauser’s research supported by the Marie-Curie Excellence Grant MEXT-CT-2004-517154. The research of W. Woess was partially supported by Austrian Science Fund (FWF) P18703-N18.     

On the Circumference of 2-Connected $$\mathcal{P}_{3}$$-Dominated Graphs

Let G be a connected graph. For at distance 2, we define , and , if then . G is quasi-claw-free if it satisfies , and G is P ₃-dominated() if it satisfies , for every pair (x, y) of vertices at distance 2. Certainly contains as a subclass. In this paper, we prove that the circumference of a 2-connected P ₃-dominated graph G on n vertices is at least min or , moreover if then G is hamiltonian or , where is a class of 2-connected nonhamiltonian graphs.     

Semicrossed products of simple C*-algebras

Let and be C*-dynamical systems and assume that is a separable simple C*-algebra and that α and β are *-automorphisms. Then the semicrossed products and are isometrically isomorphic if and only if the dynamical systems and are outer conjugate. K. R. Davidson was partially supported by an NSERC grant. E. G. Katsoulis was partially supported by a summer grant from ECU     

$${\mathcal{C}}_{0}$$ ( X)-algebras,stability and strongly self-absorbing $${\mathcal{C}}^{*}$$ -algebras

We study permanence properties of the classes of stable and so-called -stable -algebras, respectively. More precisely, we show that a (X)-algebra A is stable if all its fibres are, provided that the underlying compact metrizable space X has finite covering dimension or that the Cuntz semigroup of A is almost unperforated (a condition which is automatically satisfied for -algebras absorbing the Jiang–Su algebra tensorially). Furthermore, we prove that if is a K ₁-injective strongly self-absorbing -algebra, then A absorbs tensorially if and only if all its fibres do, again provided that X is finite-dimensional. This latter statement generalizes results of Blanchard and Kirchberg. We also show that the condition on the dimension of X cannot be dropped. Along the way, we obtain a useful characterization of when a -algebra with weakly unperforated Cuntz semigroup is stable, which allows us to show that stability passes to extensions of -absorbing -algebras. Research supported by: Deutsche Forschungsgemeinschaft (through the SFB 478), by the EU-Network Quantum Spaces - Noncommutative Geometry (Contract No. HPRN-CT-2002-00280), and by the Center for Advanced Studies in Mathematics at Ben-Gurion University     

Stable Ranks of Banach Algebras of Operator-Valued Analytic Functions

Let E be a separable infinite-dimensional Hilbert space, and let denote the algebra of all functions that are holomorphic. If is a subalgebra of , then using an algebraic result of Corach and Larotonda, we derive that under some conditions, the Bass stable rank of is infinite. In particular, we deduce that the Bass (and hence topological stable ranks) of the Hardy algebra , the disk algebra and the Wiener algebra are all infinite. Submitted: October 10, 2007., Revised: January 11, 2008., Accepted: January 12, 2007.     

C*-Algebras of Integral Operators with Piecewise Slowly Oscillating Coefficients and Shifts Acting Freely

We establish a symbol calculus for the C*-subalgebra of generated by the operators of multiplication by slowly oscillating and piecewise continuous functions and the operators where is the Cauchy singular integral operator and The C*-algebra is invariant under the transformations

On the Intertwining of \partial{\mathcal{D}}-Isometries

Let be a strictly pseudoconvex bounded domain in with C ² boundary . If a subnormal m-tuple T of Hilbert space operators has the spectral measure of its minimal normal extension N supported on , then T is referred to as a -isometry. Using some non-trivial approximation theorems in the theory of several complex variables, we establish a commutant lifting theorem for those -isometries whose (joint) Taylor spectra are contained in a special superdomain Ω of . Further, we provide a function-theoretic characterization of those subnormal tuples whose Taylor spectra are contained in Ω and that are quasisimilar to a certain (fixed) -isometry T (of which the multiplication tuple on the Hardy space of the unit ball in is a rather special example). Submitted: September 9, 2007. Revised: October 10, 2007. Accepted: October 24, 2007.     

Additive maps preserving the minimum and surjectivity moduli of operators

Let be a complex Hilbert space and let be the algebra of all bounded linear operators on . We characterize additive maps from onto itself preserving different spectral quantities such as the minimum modulus, the surjectivity modulus, and the maximum modulus of operators. Received: 15 July 2008     

Measures on Graphs and Groupoid Measures

The main purpose of this paper is to introduce several measures determined by a given finite directed graph. To construct σ-algebras for those measures, we consider several algebraic structures induced by G; (i) the free semigroupoid of the shadowed graph (ii) the graph groupoid of G, (iii) the disgram set and (iv) the reduced diagram set . The graph measures determined by (i) is the energy measure measuing how much energy we spent when we have some movements on G. The graph measures determined by (iii) is the diagram measure measuring how long we moved consequently from the starting positions (which are vertices) of some movements on G. The graph measures and determined by (ii) and (iv) are the (graph) groupoid measure and the (quotient-)groupoid measure, respectively. We show that above graph measurings are invariants on shadowed graphs of finite directed graphs. Also, we will consider the reduced diagram measure theory on graphs. In the final chapter, we will show that if two finite directed graphs G ₁ and G ₂ are graph-isomorphic, then the von Neumann algebras L ^∞(μ ₁) and L ^∞(μ ₂) are *-isomorphic, where μ ₁ and μ ₂ are the same kind of our graph measures of G ₁ and G ₂, respectively. Received: December 7, 2006. Revised: August 3, 2007. Accepted: August 18, 2007.     

Parametrization of Contractive Block Operator Matrices and Passive Discrete-Time Systems

Passive linear systems τ = have their transfer function in the Schur class S . Using a parametrization of contractive block operators the transfer function is connected to the Sz.-Nagy–Foiaş characteristic function of the contraction A. This gives a new aspect and some explicit formulas for studying the interplay between the system τ and the functions and . The method leads to some new results for linear passive discrete-time systems. Also new proofs for some known facts in the theory of these systems are obtained. Dedicated to Eduard Tsekanovskiĭ on the occasion of his seventieth birthday This work was supported by the Research Institute for Technology at the University of Vaasa. The first author was also supported by the Academy of Finland (projects 212146, 117617) and the Dutch Organization for Scientific Research N.W.O. (B 61-553). Received: December 22, 2006. Revised: February 6, 2007.     

A spectral characterization of the uniform continuity of strongly continuous groups

Let X be a Banach space and a strongly continuous group of linear operators on X. Set and where is the unit circle and denotes the spectrum of T(t). The main result of this paper is: is uniformly continuous if and only if is non-meager. Similar characterizations in terms of the approximate point spectrum and essential spectra are also derived. Received: 14 June 2006, Revised: 27 September 2007     

Quasilinearization and curvature of Aleksandrov spaces

We present a new distance characterization of Aleksandrov spaces of non-positive curvature. By introducing a quasilinearization for abstract metric spaces we draw an analogy between characterization of Aleksandrov spaces and inner product spaces; the quasi-inner product is defined by means of the quadrilateral cosine—a metric substitute for the angular measure between two directions at different points. Our main result states that a geodesically connected metric space is an Aleksandrov domain (also known as a CAT(0) space) if and only if the quadrilateral cosine does not exceed one for every two pairs of distinct points in . We also observe that a geodesically connected metric space is an domain if and only if, for every quadruple of points in , the quadrilateral inequality (known as Euler’s inequality in ) holds. As a corollary of our main result we give necessary and sufficient conditions for a semimetric space to be an domain. Our results provide a complete solution to the Curvature Problem posed by Gromov in the context of metric spaces of non-positive curvature.      

The growth of a C 0-semigroup characterised by its cogenerator

We characterise contractivity, boundedness and polynomial growth for a C ₀-semigroup in terms of its cogenerator V (or the Cayley transform of the generator) or its resolvent. In particular, we extend results of Gomilko and Brenner, Thomée and show that polynomial growth of a semigroup implies polynomial growth of its cogenerator. As is shown by an example, the result is optimal. For analytic semigroups we show that the converse holds, i.e., polynomial growth of the cogenerators implies polynomial growth of the semigroup. In addition, we show by simple examples in (), , that our results on the characterization of contractivity are sharp. These examples also show that the famous Foiaş-Sz.-Nagy theorem on cogenerators of contractive C ₀-semigroups on Hilbert spaces fails in () for .      

On Similarity of Multiplication Operator on Weighted Bergman Space

Let D be the unit disk and be the weighted Bergman space. In this paper, we prove that the multiplication operator is similar to M _z on . The author was supported in part by NSF Grant (10571041, L2007B05).     

Beurling-type Representation of Invariant Subspaces in Reproducing Kernel Hilbert Spaces

By Beurling’s theorem, the orthogonal projection onto an invariant subspace M of the Hardy space on the unit disk can be represented as where Φ is an inner multiplier of . This concept can be carried over to arbitrary Nevanlinna-Pick spaces but fails in more general settings. This paper introduces the notion of Beurling decomposable subspaces. An invariant subspace M of a reproducing kernel Hilbert space will be called Beurling decomposable if there exist (operator-valued) multipliers such that and . We characterize the finite-codimensional and the finite-rank Beurling decomposable subspaces by means of their core function and core operator. As an application, we show that in many analytic Hilbert modules , every finite-codimensional submodule M can be written as with suitable polynomials p _i .      

Tolokonnikov’s Lemma for Real H ∞ and the Real Disc Algebra

We prove Tolokonnikov’s Lemma and the inner-outer factorization for the real Hardy space , the space of bounded holomorphic (possibly operator-valued) functions on the unit disc all of whose matrix-entries (with respect to fixed orthonormal bases) are functions having real Fourier coefficients, or equivalently, each matrix entry f satisfies for all z ∈ . Tolokonnikov’s Lemma for means that if f is left-invertible, then f can be completed to an isomorphism; that is, there exists an F, invertible in , such that F = [ f f _c ] for some f _c in . In control theory, Tolokonnikov’s Lemma implies that if a function has a right coprime factorization over , then it has a doubly coprime factorization in . We prove the lemma for the real disc algebra as well. In particular, and are Hermite rings. The work of the first author was supported by Magnus Ehrnrooth Foundation. Received: December 5, 2006. Revised: February 4, 2007.