The eigenvalues of limits of radial Toeplitz operators

Let A² be the Bergman space on the unit disk. A bounded operatorS on A² is called radial if Szⁿ = _n zⁿ for all n 0, where _nis a bounded sequence of complex numbers. We characterize theeigenvalues of radial operators that belong to the Toeplitzalgebra.     

Invariant Subspaces of Bergman Spaces on Slit Domains

In this paper, we characterize the z-invariant subspaces thatlie between the Bergman spaces A^p(G) and A^p(G\K), where 1 <p < , G is a bounded region in C, and K is a closed subsetof a simple compact C¹ arc.     

On full groups of measure-preserving and ergodic transformations with uncountable cofinalities

The group of all measure-preserving permutations of the unitinterval and the full group of an ergodic transformation ofthe unit interval are shown to have uncountable cofinality andthe Bergman property. Here, a group G is said to have the Bergmanproperty if, for any generating subset E of G, some boundedpower of EE^–1{1} already covers G. This property arosein a recent interesting paper of Bergman, where it was derivedfor the infinite symmetric groups. We give a general sufficientcriterion for groups G to have the Bergman property. We showthat the criterion applies to a range of other groups, includingsufficiently transitive groups of measure-preserving, non-singular,or ergodic transformations of the reals; it also applies tolarge groups of homeomorphisms of the rationals, the irrationals,or the Cantor set.     

Local Rigidity of Infinite-Dimensional Teichmuller Spaces

This paper presents a rigidity theorem for infinite-dimensionalBergman spaces of hyperbolic Riemann surfaces, which statesthat the Bergman space A¹(M), for such a Riemann surface M,is isomorphic to the Banach space of summable sequence, l¹.This implies that whenever M and N are Riemann surfaces thatare not analytically finite, and in particular are not necessarilyhomeomorphic, then A¹(M) is isomorphic to A¹(N). It is knownfrom V. Markovic that if there is a linear isometry betweenA¹(M) and A¹(N), for two Riemann surfaces M and N of non-exceptionaltype, then this isometry is induced by a conformal mapping betweenM and N. As a corollary to this rigidity theorem presented here,taking the Banach duals of A¹(M) and l¹ shows that the spaceof holomorphic quadratic differentials on M, Q(M), is isomorphicto the Banach space of bounded sequences, l. As a consequenceof this theorem and the Bers embedding, the Teichmüllerspaces of such Riemann surfaces are locally bi-Lipschitz equivalent.     

On Hardy, BMO and Lipschitz spaces of invariantly harmonic functions in the unit ball

The invariantly harmonic functions in the unit ball Bⁿ in Cⁿare those annihilated by the Bergman Laplacian . The Poisson-Szegökernel P(z,) solves the Dirichlet problem for : if f C(Sⁿ),the Poisson-Szegö transform of f, where d is the normalized Lebesgue measure on Sⁿ,is the unique invariantly harmonic function u in Bⁿ, continuousup to the boundary, such that u=f on Sⁿ. The Poisson-Szegötransform establishes, loosely speaking, a one-to-one correspondencebetween function theory in Sⁿ and invariantly harmonic functiontheory in Bⁿ. When n 2, it is natural to consider on Sⁿ functionspaces related to its natural non-isotropic metric, for theseare the spaces arising from complex analysis. In the paper,different characterizations of such spaces of smooth functionsare given in terms of their invariantly harmonic extensions,using maximal functions and area integrals, as in the correspondingEuclidean theory. Particular attention is given to characterizationin terms of purely radial or purely tangential derivatives.The smoothness is measured in two different scales: that ofSobolev spaces and that of Lipschitz spaces, including BMO andBesov spaces. 1991 Mathematics Subject Classification: 32A35,32A37, 32M15, 42B25.     

The Dunford-Pettis Property for some Function Algebras in Several Complex Variables

The Dunford-Pettis property is shown to hold for the uniformalgebra A() and its dual for some standard domains , includingstrongly pseudoconvex bounded domains in Cⁿ, pseudoconvex boundeddomains of finite type in C², and bounded domains in C. Previouslythe result was known for the unit ball and unit polydisc inCⁿ. Techniques used involve Bourgain algebras, Hankel operators,properties of the Bergman kernel, quasi-metrics on the boundary,and -theory.     

Lorentz Spaces Of Vector-Valued Measures

Given a non-atomic, finite and complete measure space (,,µ)and a Banach space X, the modulus of continuity for a vectormeasure F is defined as the function _F(t) = sup_µ(E)t |F|(E)and the space V^p,q(X) of vector measures such that t^–1/p'_F(t) L^q((0,µ()],dt/t) is introduced. It is shown thatV^p,q(X) contains isometrically L^p,q(X) and that L^p,q(X) = V^p,q(X)if and only if X has the Radon–Nikodym property. It isalso proved that V^p,q(X) coincides with the space of cone absolutelysumming operators from L^p',q^' into X and the duality V^p,q(X^*)=(Lp^',q^'(X))^*where 1/p+1/p^'= 1/q+1/q^' = 1. Finally, V^p,q(X) is identifiedwith the interpolation space obtained by the real method (V¹(X),V(X))_1/p^',q. Spaces where the variation of F is replaced bythe semivariation are also considered.     

A Characterization of Fredholm Pseudo-Differential Operators

We give a necessary and sufficient condition on an ellipticsymbol of order m to ensure that the unique closed extensionin L^p(Rⁿ) for 1 < p < , of the pseudo-differential operatorT, initially defined on the Schwartz space, is a Fredholm operatorfrom L^p(Rⁿ) into L^p(Rⁿ) with domain H^{m, p}, where H^{m, p} is theL^p Sobolev space of order m.     

Plane with A{infty}-Weighted Metric not Bilipschitz Embeddable to Rn

A planar set G R² is constructed that is bilipschitz equivalentto (G,d^z), where (G, d) is not bilipschitz embeddable to anyuniformly convex Banach space. Here, Z (0, 1) and d^z denotesthe zth power of the metric d. This proves the existence ofa strong A weight in R², such that the corresponding deformedgeometry admits no bilipschitz mappings to any uniformly convexBanach space. Such a weight cannot be comparable to the Jacobianof a quasiconformal self-mapping of R². 2000 Mathematics SubjectClassification 54E40 (primary); 30C62, 30C65, 28A80 (secondary).     

A fourth-order cubic spline method for linear second-order two-point boundary-value problems

A cubic spline method for linear second-order two-point boundary-valueproblems is analysed. The method is a Petrov-Galerkin methodusing a cubic spline trial space, a piecewise-linear test space,and a simple quadrature rule for the integration, and may beconsidered a discrete version of the H¹-Galerkin method. Themethod is fully discrete, allows an arbitrary mesh, yields alinear system with bandwidth five, and under suitable conditionsis shown to have an 0(h^4– rate of convergence in the W_p¹norm for i = 0, 1, 2, 1p. The H¹-Galerkin method and orthogonalspline collocation with Hermite cubics are also discussed.     

The l{infty} direct sum of Lp (1 < p < {infty}) is primary

Using Szemeredi's theorem on arithmetic progressions, it isshown that, for 1 < p < , the infinite l direct sum (L^p L^p · · · )_l is a primary Banach space.     

On v-Distal Flows on 3-Manifolds

In [2], H. Furstenberg studied a distal action of a locallycompact group G on a compact metric space X, and establisheda structure theorem. As a consequence, he showed that if G isabelian, then a simply connected space X does not admit a minimaldistal G-action. In this paper we concern ourselves with a nonsingular flow = {^t} on a closed 3-manifold M. Recall that is called distalif for any distinct two points x, y M, the distance d(^tx, ^ty)is bounded away from 0. The distality depends strongly uponthe time parametrization. For example, there exists a time parametrizationof a linear irrational flow on T² which yields a nondistal flow[4, 6]. 1991 Mathematics Subject Classification 58F25, 57R30.     

Rank Distributions on Coarse Spaces and Ideal Structure of Roe Algebras

The notions of controlled truncations for operators in the Roealgebras C^* (X) of a coarse space (X, ) with uniformly locallyfinite coarse structure, and rank distributions on (X, ) areintroduced. It is shown that the controlled propagation operatorsin an ideal I of C^* (X) are exactly the controlled truncationsof elements in I. It follows that the lattice of the idealsof C^* (X) in which controlled propagation operators are denseis isomorphic to the lattice of all rank distributions on (X,). If X is a discrete metric space with Yu's property A, thenthe ideal structure of the Roe algebra C^* (X is completely determinedby the rank distributions on X. 2000 Mathematics Subject Classification46L80, 46L89.     

REAL HYPERSURFACES IN A EUCLIDEAN COMPLEX SPACE FORM

Let M be an orientable connected and compact real hypersurfaceof the complex space form C^{(n + 1)/2}. If the mean curvature and the function f = g(A, ) of hypersurface M satisfy the inequalityn²² (n² – 1) + f², where is the characteristic vectorfield, A is the shape operator and (n – 1) is the infimumof the Ricci curvatures of hypersurface M, then it is shownthat is a constant and M is the sphere Sⁿ(²) in C^{(n + 1)/2}.     

Interpolation by Polynomials in z and z-1 on an Annulus

A polynomial of degree n in z^–1 and n–1 in z isdefined by an interpolation projection from the space of functionsanalytic in the annulus r|z|R and continuous on its boundary.The points of interpolation are chosen to coincide with then roots of zⁿ=Rⁿeⁱⁿ (0<<2/n) and the n roots of zⁿ=rⁿ.The behaviour of the corresponding Lebesgue function is studied,and an estimate for the operator norm is obtained. The resultsof the present paper give a partial affirmative answer to twoconjectures suggested earlier by Mason on the basis of numericalcomputations.     

The Alternative Dunford-Pettis Property, Conjugations and Real Forms of C*-Algebras

Let be a conjugation, alias a conjugate linear isometry oforder 2, on a complex Banach space X and let X be the real formof X of -fixed points. In contrast to the Dunford–Pettisproperty, the alternative Dunford–Pettis property neednot lift from X to X. If X is a C^*-algebra it is shown thatX has the alternative Dunford–Pettis property if and onlyif X does and an analogous result is shown when X is the dualspace of a C^*-algebra. One consequence is that both Dunford–Pettisproperties coincide on all real forms of C^*-algebras.     

Distributed control for cooperative systems involving parabolic operators with an infinite number of variables

^** Email: serraghm{at}yahoo.com The optimal control for cooperative systems involving parabolicoperators with an infinite number of variables is considered.First the existence and uniqueness of the states are proved;then the necessary and sufficient condition for the controlto be optimal is obtained by a set of inequalities. The controlin our problems is of distributed type and is allowed to bein the Hilbert space (L²(0, T, L²()))ⁿ.     

The Hardy Operator and the Gap Between L{infty} and BMO

We study boundedness and compactness properties of the Hardyintegral operator from a weighted Banach function space X(v) into L and BMO. We give a new simplecharacterization of compactness of T from X(v) into BMO. Weconstruct examples of spaces X(v) such that T(X(v)) is (a) boundedin L but not compact in BMO; (b) compact in BMO but not boundedin L; (c) bounded in BMO but neither bounded in L nor compactin BMO; (d) bounded in L, compact in BMO and yet not compactin L. In order to obtain the last of the counterexamples weconstruct a new weighted Banach function space.     

UNIVERSAL SPACES OF TWO-CELL COMPLEXES AND THEIR EXPONENT BOUNDS

Let P²ⁿ⁺¹ be a two-cell complex which is formed by attachinga (2n + 1)-cell to a 2m-sphere by a suspension map. We constructa universal space U for P²ⁿ⁺¹ in the category of homotopy associative,homotopy commutative H-spaces. By universal, we mean that Uis homotopy associative, homotopy commutative and has the propertythat any map f: P²ⁿ⁺¹ Y to a homotopy associative, homotopycommutative H-space Y extends to a uniquely determined H-map: U Y. We then prove upper and lowerbounds of the H-homotopy exponent of U. In the case of a modp^r, Moore space U is the homotopy fibre S²ⁿ⁺¹{p^r} of the p^r-powermap on S²ⁿ⁺¹, and we reproduce Neisendorfer's result that S²ⁿ⁺¹{p^r}is homotopy associative, homotopy commutative and that the p^r-powermap on S²ⁿ⁺¹{p^r} is null homotopic.     

A qualocation method for parabolic partial differential equations

In this paper a qualocation method is analysed for parabolicpartial differential equations in one space dimension. Thismethod may be described as a discrete H¹-Galerkin method inwhich the discretization is achieved by approximating the integralsby a composite Gauss quadrature rule. An O (h^4-i) rate of convergencein the W^i.p norm for i = 0, 1 and 1 p is derived for a semidiscretescheme without any quasi-uniformity assumption on the finiteelement mesh. Further, an optimal error estimate in the H² normis also proved. Finally, the linearized backward Euler methodand extrapolated Crank-Nicolson scheme are examined and analysed.