On hypersphericity of manifolds with finite asymptotic dimension

We prove the following embedding theorems in the coarse geometry:



The Corollary is used in the proof of the following.

Theorem B together with a theorem of Gromov-Lawson implies the result, previously proven by G. Yu (1998), which states that an aspherical manifold whose fundamental group has a finite asymptotic dimension cannot carry a metric of positive scalar curvature.

We also prove that if a uniformly contractible manifold of bounded geometry is large scale uniformly embeddable into a Hilbert space, then is stably integrally hyperspherical.

     

A representation theorem for Schauder bases in Hilbert space

A sequence of vectors in a separable Hilbert space is said to be a Schauder basis for if every element has a unique norm-convergent expansion

If, in addition, there exist positive constants and such that

then we call a Riesz basis. In the first half of this paper, we show that every Schauder basis for can be obtained from an orthonormal basis by means of a (possibly unbounded) one-to-one positive self adjoint operator. In the second half, we use this result to extend and clarify a remarkable theorem due to Duffin and Eachus characterizing the class of Riesz bases in Hilbert space.

     

For which reproducing kernel Hilbert spaces is Pick's theorem true?

Pick's theorem tells us that there exists a function inH , which is bounded by 1 and takes given values at given points, if and only if a certain matrix is positive.H is the space of multipliers ofH ², and this theorem has a natural generalisation whenH is replaced by the space of multipliers of a general reproducing kernel Hilbert spaceH(K) (whereK is the reproducing kernel). J. Agler has shown that this generalised theorem is true whenH(K) is a certain Sobolev space or the Dirichlet space, so it is natural to ask for which reproducing kernel Hilbert spaces this generalised theorem is true. This paper widens Agler's approach to cover reproducing kernel Hilbert spaces in general, replacing Agler's use of the deep theory of co-analytic models by a relatively elementary, and more general, matrix argument. The resulting theorem gives sufficient (and usable) conditions on the kernelK, for the generalised Pick's theorem to be true forH(K), and these are then used to prove Pick's theorem for certain weighted Hardy and Sobolev spaces and for a functional Hilbert space introduced by Saitoh.     

Weyl's theorem for operator matrices

Weyl's theorem holds for an operator when the complement in the spectrum of the Weyl spectrum coincides with the isolated points of the spectrum which are eigenvalues of finite multiplicity. By comparison Browder's theorem holds for an operator when the complement in the spectrum of the Weyl spectrum coincides with Riesz points. Weyl's theorem and Browder's theorem are liable to fail for 2×2 operator matrices. In this paper we explore how Weyl's theorem and Browder's theorem survive for 2×2 operator matrices on the Hilbert space.Supported in part by BSRI-97-1420 and KOSEF 94-0701-02-01-3.     

A note on the almost sure approximation of weakly dependent random variables

A strong approximation theorem for Hilbert space valued martingales, due to Morrow and Philipp, is improved. This result is then used to strengthen and, at the same time, to extend a recent strong approximation theorem for weakly dependent ^d -valued random variables, due to Eberlein, to Hilbert space valued random variables.Dedicated to Professor E. Hlawka on the occasion of his seventieth birthday     

Spaces that admit hypercyclic operators with hypercyclic adjoints

A continuous linear operator is hypercyclic if there is an such that the orbit is dense. A result of H. Salas shows that any infinite-dimensional separable Hilbert space admits a hypercyclic operator whose adjoint is also hypercyclic. It is a natural question to ask for what other spaces does contain such an operator. We prove that for any infinite-dimensional Banach space with a shrinking symmetric basis, such as and any , there is an operator , where both and are hypercyclic.

     

A version of Burkholder's theorem for operator-weighted spaces

Let be an operator weight, i.e. a weight function taking values in the bounded linear operators on a Hilbert space . We prove that if the dyadic martingale transforms are uniformly bounded on for each dyadic grid in , then the Hilbert transform is bounded on as well, thus providing an analogue of Burkholder's theorem for operator-weighted -spaces. We also give a short new proof of Burkholder's theorem itself. Our proof is based on the decomposition of the Hilbert transform into ``dyadic shifts'.

     

A von Neumann type inequality for certain domains in

Strict contractions on a Hilbert space have a functional calculus with functions that are analytic in the unit disc of the complex plane; an estimate of the norm is then provided by von Neumann's inequality. We consider functions that satisfy related inequalities with respect to multioperators connected to certain domains in ; a representation formula and a Nevanlinna-Pick type theorem are obtained.

     

Characterization of the Hilbert ball by its automorphism group

Let be a bounded, convex domain in a separable Hilbert space. The authors prove a version of the theorem of Bun Wong, which asserts that if such a domain admits an automorphism orbit accumulating at a strongly pseudoconvex boundary point, then it is biholomorphic to the ball. Key ingredients in the proof are a new localization argument using holomorphic peaking functions and the use of new ``normal families' arguments in the construction of the limit biholomorphism.

     

The Arveson Extension Theorem and coanalytic models

We develop techniques which allow one to describe in simple terms the set of operators on Hilbert space of the form M^{* ()} |M, where M is multiplication by z on a Hilbert space of analytic functions satisfying certain technical assumptions, M^{* ()} is the direct sum of a countably infinite number of copies of M^*, andM is invariant for M^{* ()}. One of the main ingredients in our technique is the Arveson Extension Theorem and this paper illustrates the great power and tractability of that theorem in a concrete setting.Research partially supported by NSF grant MCS 81-02518     

An expansion theorem for an eigenvalue problem of an arbitrary multiply connected domain with an eigenparameter in a general type of boundary conditions

The object of this paper is to establish an expansion theorem for a regular right-definite eigenvalue problem with an eigenvalue parameter which is contained in the Schrödinger partial differential equation and in a general type of boundary conditions on the boundary of an arbitrary multiply connected bounded domain inR ⁿ (n2). We associate with this problem an essentially self-adjoint operator in a suitably defined Hilbert space and then we develop an associated eigenfunction expansion theorem.     

A generalization of Andô's theorem and Parrott's example

Andô's theorem states that any pair of commuting contractions on a Hilbert space can be dilated to a pair of commuting unitaries. Parrott presented an example showing that an analogous result does not hold for a triple of pairwise commuting contractions. We generalize both of these results as follows. Any -tuple of contractions that commute according to a graph without a cycle can be dilated to an -tuple of unitaries that commute according to that graph. Conversely, if the graph contains a cycle, we construct a counterexample.

     

On a Beurling-Arveson Type Theorem for Some Functional Hilbert Spaces and Related Questions

We prove in terms of Berezin symbols a Beurling-Arveson type theorem for some functional Hilbert space over the unit disc of the complex plane . We give also an affirmative answer to a question of Yang [15].      

Invarianz gewisser Operatorenklassen unter nichtkommutativen ergodischen Mitteln

Let be a group of *-automorphisms on the algebra of bounded linear operators on a complex Hilbert space H. Then the strongly closed convex hull of the orbit of any compact operator under consists of compact operators. The same is true if one replaces compact by nuclear, Hilbert-Schmidt or positive Fredholm. We further discuss these results in the framework of the noncommutative mean ergodic theorem of KOVACS and SZ#x00FC;CS and formulate an analogous theorem for the algebra of compact operators on a complex Hilbert space.

---

---

Gefördert von der Deutschen Forschungsgemeinschaft im Rahmen des Forschungsvorhabens Ko 506/1.     

Embedding Riemannian manifolds by their heat kernel

By embedding a class of closed Riemannian manifolds (satisfying some curvature assumptions and with diameter bounded from above) into the same Hilbert space, we interpret certain estimates on the heat kernel as giving a precompactness theorem on the class considered.This research has been supported in part by the E.C. Contract SC 1-0105-C G.A.D.G.E.T.     

Coarse embeddings of metric spaces into Banach spaces

There are several characterizations of coarse embeddability of locally finite metric spaces into a Hilbert space. In this note we give such characterizations for general metric spaces. By applying these results to the spaces , we get their coarse embeddability into a Hilbert space for . This together with a theorem by Banach and Mazur yields that coarse embeddability into and into are equivalent when . A theorem by G.Yu and the above allow us to extend to , , the range of spaces, coarse embeddings into which is guaranteed for a finitely generated group to satisfy the Novikov Conjecture.

     

An Extension of Lomonosov's Techniques to non-compact Operators

The aim of this work is to generalize Lomonosov's techniques in order to apply them to a wider class of not necessarily compact operators. We start by establishing a connection between the existence of invariant subspaces and density of what we define as the associated Lomonosov space in a certain function space. On a Hilbert space, approximation with Lomonosov functions results in an extended version of Burnside's Theorem. An application of this theorem to the algebra generated by an essentially self-adjoint operator yields the existence of vector states on the space of all polynomials restricted to the essential spectrum of . Finally, the invariant subspace problem for compact perturbations of self-adjoint operators acting on a real Hilbert space is translated into an extreme problem and the solution is obtained upon differentiating certain real-valued functions at their extreme.

     

Asymptotic behavior of bounded solutions to a nonhomogeneous second-order evolution equation of monotone type

By using previous results of Djafari Rouhani [B. Djafari Rouhani, Ergodic theorems for nonexpansive sequences in Hilbert spaces and related problems, Ph.D. Thesis, Yale university, 1981, part I, pp. 1-76; B. Djafari Rouhani, Asymptotic behaviour of quasi-autonomous dissipative systems in Hilbert spaces, J. Math. Anal. Appl. 147 (1990) 465-476; B. Djafari Rouhani, Asymptotic behaviour of almost nonexpansive sequences in a Hilbert space, J. Math. Anal. Appl. 151 (1990) 226-235] for dissipative systems, we study the asymptotic behavior of solutions to the following system of second-order nonhomogeneous evolution equations:

     

A rigid space homeomorphic to Hilbert space

A rigid space is a topological vector space whose endomorphisms are all simply scalar multiples of the identity map. This is in sharp contrast to the behavior of operators on , and so rigid spaces are, from the viewpoint of functional analysis, fundamentally different from Hilbert space. Nevertheless, we show in this paper that a rigid space can be constructed which is topologically homeomorphic to Hilbert space. We do this by demonstrating that the first complete rigid space can be modified slightly to be an AR-space (absolute retract), and thus by a theorem of Dobrowolski and Torunczyk is homeomorphic to .

     

Stochastic processes with sample paths in reproducing kernel Hilbert spaces

A theorem of M. F. Driscoll says that, under certain restrictions, the probability that a given Gaussian process has its sample paths almost surely in a given reproducing kernel Hilbert space (RKHS) is either or . Driscoll also found a necessary and sufficient condition for that probability to be .

Doing away with Driscoll's restrictions, R. Fortet generalized his condition and named it nuclear dominance. He stated a theorem claiming nuclear dominance to be necessary and sufficient for the existence of a process (not necessarily Gaussian) having its sample paths in a given RKHS. This theorem - specifically the necessity of the condition - turns out to be incorrect, as we will show via counterexamples. On the other hand, a weaker sufficient condition is available.

Using Fortet's tools along with some new ones, we correct Fortet's theorem and then find the generalization of Driscoll's result. The key idea is that of a random element in a RKHS whose values are sample paths of a stochastic process. As in Fortet's work, we make almost no assumptions about the reproducing kernels we use, and we demonstrate the extent to which one may dispense with the Gaussian assumption.