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.     

Characterization of Amalgamated Free Blocks of a Graph von Neumann Algebra

We identify two noncommutative structures naturally associated with countable directed graphs. They are formulated in the language of operators on Hilbert spaces. If G is a countable directed graphs with its vertex set V(G) and its edge set E(G), then we associate partial isometries to the edges in E(G) and projections to the vertices in V(G). We construct a corresponding von Neumann algebra as a groupoid crossed product algebra of an arbitrary fixed von Neumann algebra M and the graph groupoid induced by G, via a graph-representation (or a groupoid action) α. Graph groupoids are well-determined (categorial) groupoids. The graph groupoid of G has its binary operation, called admissibility. This has concrete local parts , for all e ∈ E(G). We characterize of , induced by the local parts of , for all e ∈ E(G). We then characterize all amalgamated free blocks of . They are chracterized by well-known von Neumann algebras: the classical group crossed product algebras , and certain subalgebras (M) of operator-valued matricial algebra . This shows that graph von Neumann algebras identify the key properties of graph groupoids. Received: December 20, 2006. Revised: March 07, 2007. Accepted: March 13, 2007.     

Hermitean Clifford-Hermite Polynomials

Orthogonal Clifford analysis in flat m–dimensional Euclidean space focusses on monogenic functions, i.e. null solutions of the rotation invariant vector valued Dirac operator , where ( ) forms an orthogonal basis for the quadratic space underlying the construction of the Clifford algebra . When allowing for complex constants and taking the dimension to be even: m = 2n, the same set of generators produces the complex Clifford algebra , which we equip with a Hermitean Clifford conjugation and a Hermitean inner product. Hermitean Clifford analysis then focusses on the simultaneous null solutions of two mutually conjugate Hermitean Dirac operators, naturally arising in the present context and being invariant under the action of a realization of the unitary group U (n). In this so–called Hermitean setting Clifford–Hermite polynomials are constructed, starting from a Rodrigues formula involving both Dirac operators mentioned. Due to the specific features of the Hermitean setting, four different types of polynomials are obtained, two types of even degree and two types of odd degree. We investigate their properties: recurrence relations, structure, explicit form and orthogonality w.r.t. a deliberately chosen weight function. They also give rise to the definition of the Hermitean Clifford–Hermite functions, and may be used to develop a Hermitean continuous wavelet transform, see [4].     

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.     

On H. Weyl and J. Steiner Polynomials

The paper deals with root location problems for two classes of univariate polynomials both of geometric origin. The first class discussed, the class of Steiner polynomial, consists of polynomials, each associated with a compact convex set . A polynomial of this class describes the volume of the set V + tB ⁿ as a function of t, where t is a positive number and B ⁿ denotes the unit ball in . The second class, the class of Weyl polynomials, consists of polynomials, each associated with a Riemannian manifold , where is isometrically embedded with positive codimension in . A Weyl polynomial describes the volume of a tubular neighborhood of its associated as a function of the tube’s radius. These polynomials are calculated explicitly in a number of natural examples such as balls, cubes, squeezed cylinders. Furthermore, we examine how the above mentioned polynomials are related to one another and how they depend on the standard embedding of into for m > n. We find that in some cases the real part of any Steiner polynomial root will be negative. In certain other cases, a Steiner polynomial will have only real negative roots. In all of this cases, it can be shown that all of a Weyl polynomial’s roots are simple and, furthermore, that they lie on the imaginary axis. At the same time, in certain cases the above pattern does not hold.

Erasmus Darwin, the nephew of the great scientist Charles Darwin, believed that sometimes one should perform the most unusual experiments. They usually yield no results but when they do . . . . So once he played trumpet in front of tulips for the whole day. The experiment yielded no results.

Submitted: March 5, 2007., Revised: February 1, 2008., Accepted: February 2, 2008.     

Lie and Jordan Ideals in Reflexive Algebras

A CDCSL algebra is a reflexive operator algebra with completely distributive and commutative subspace lattice. In this paper, we show, for a weakly closed linear subspace of a CDCSL algebra , that is a Lie ideal if and only if for all invertibles A in , and that is a Jordan ideal if and only if it is an associative ideal.     

The Isotonic Cauchy Transform

Starting with an integral representation for the class of continuously differentiable solutions of the system

The Reduced Minimum Modulus in <Emphasis>C</Emphasis><Superscript>*</Superscript>-Algebras

We define the reduced minimum modulus of a nonzero element a in a unital C ^*-algebra by . We prove that . Applying this result to and its closed two side ideal , we get that dist , and for any if RR = 0, where and is the quotient homomorphism and . These results generalize corresponding results in Hilbert spaces.     

A note on weak convergence in L^{1}_{\rm loc}({\mathbb{R}})

Let be a sequence of Borel measurable functions satisfying, for a function the inequalities

Dicritical holomorphic flows on Stein manifolds

We study holomorphic flows on Stein manifolds. We prove that a holomorphic flow with isolated singularities and a dicritical singularity of the form on a Stein manifold with , is globally analytically linearizable; in particular M is biholomorphic to . A complete stability result for periodic orbits is also obtained. Bruno Scárdua: Partially supported by ICTP-Trieste-Italy. Received: 27 September 2006     

Partial Primitives for Clifford Algebra-Valued Functions

In this paper we generalize the concept of primitivation of monogenic functions taking values in a Clifford algebra, which is on its own a generalization to higher dimension of the primitivation problem for holomorphic functions in the complex plane. This problem can be stated as follows: given a monogenic function on , i.e. a solution for the generalized Cauchy-Riemann operator D on , construct a monogenic function such that . In view of the fact that, for monogenic functions g, this can be written as g = f, a straightforward generalization consists in replacing the scalar generator of translations in the x ₀-direction by a generator of another transformation group. In this paper we consider translations in more dimensions.     

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.     

The Resolvent Problem and $${{H^{\infty}}}$$ -calculus of the Stokes Operator in Unbounded Cylinders with Several Exits to Infinity

It is proved that the Stokes operator in L^q -space on an infinite cylindrical domain of , , with several exits to infinity generates a bounded and exponentially decaying analytic semigroup and admits a bounded -calculus. For the resolvent estimates, the Stokes resolvent system with a prescribed divergence in an infinite straight cylinder with bounded cross-section is studied in L ^q where and is an arbitrary Muckenhoupt weight. The proofs use cut-off techniques and the theory of Schauder decomposition of UMD spaces based on -boundedness of operator families and on square function estimates involving Muckenhoupt weights.     

An Operator Corona Theorem for Some Subspaces of <Emphasis Type="Italic">H</Emphasis><Superscript>∞</Superscript>

Let E, E_* be separable Hilbert spaces. If S is an open subset of , then denotes the space of all functions that are holomorphic in , and bounded and continuous on . In this article we prove the following results:

The Schur–Potapov Algorithm for Sequences of Complex p × q Matrices. II

This is the second and final part of a paper which appeared in a preceding issue of this journal. Herein the methods developed in the earlier sections of this paper are used first to develop a number of applications. A central theme of this paper is to study the interplay between functions from and their sequence of SP-parameters. In particular, we describe how certain summability properties of the SP-parameters are expressed in terms of the associated functions from . As a byproduct of our investigations on the interplay between the SP-algorithm for p × q Schur functions and the SP-algorithm for sequences of complex p × q matrices we present a new approach to the nondegenerate matricial Schur problem. Our method complies with the basic strategy in Schur’s classical paper [138] because it does not make use of any tools outside of the theory of Schur functions and Schur sequences. A closer look at the behaviour of distinguished subclasses of with respect to the SP-algorithm enables us to handle the corresponding versions of the matricial Schur problem restricted to these subclasses. Submitted: August 17, 2006. Accepted: September 13, 2006.     

The Inverse Problem for Perturbed Harmonic Oscillator on the Half-Line with a Dirichlet Boundary Condition

We consider the perturbed harmonic oscillator in , where is a real-valued potential. We prove that the mapping spectral data = {eigenvalues of T _D } {norming constants} is one-to-one and onto. The complete characterization of the set of spectral data which corresponds to is given. Dedicated to Vladimir Buslaev on the occasion of his 70th birthday Submitted: September 27, 2006. Accepted: January 9, 2007.     

Schur Complements in Krein Spaces

The aim of this work is to generalize the notions of Schur complements and shorted operators to Krein spaces. Given a (bounded) J-selfadjoint operator A (with the unique factorization property) acting on a Krein space and a suitable closed subspace of , the Schur complement of A to is defined. The basic properties of are developed and different characterizations are given, most of them resembling those of the shorted of (bounded) positive operators on a Hilbert space. To the memory of Professor Mischa Cotlar     

Polynomial automorphisms and hypercyclic operators on spaces of analytic functions

We consider hypercyclic composition operators on which can be obtained from the translation operator using polynomial automorphisms of . In particular we show that if C _S is a hypercyclic operator for an affine automorphism S on , then for some polynomial automorphism Θ and vectors a and b, where I is the identity operator. Finally, we prove the hypercyclicity of “symmetric translations” on a space of symmetric analytic functions on ℓ₁. Received: 8 June 2006 Revised: 26 September 2006     

Equidistribution of Rational Matrices in their Conjugacy Classes

Let G be a connected simply connected almost -simple algebraic group with non-compact and a cocompact congruence subgroup. For any homogeneous manifold of finite volume, and a , we show that the Hecke orbit T _a (x ₀ H) is equidistributed on as , provided H is a non-compact commutative reductive subgroup of G. As a corollary, we generalize the equidistribution result of Hecke points ([COU], [EO1]) to homogeneous spaces G/H. As a concrete application, we describe the equidistribution result in the rational matrices with a given characteristic polynomial. The second author partially supported by DMS 0333397. Received: May 2005 Revision: March 2006 Accepted: June 2006     

Infimal convolution and Muckenhoupt A p (.) condition in variable L p spaces

We study the Hardy-Littlewood maximal operator M on . Under the assumptions that the exponent p satisfies and is constant outside some large ball, we prove that if and only if . Received: 2 June 2006 Revised: 28 November 2006