Teh Bloch space of M-harmonic functions on bounded symmetric domains

New characterizations of Bloch functions on a bounded symmetric domain in ⁿ are given in terms of the intrinsic metrics. Various criteria for an M-harmonic function to be Bloch or little Bloch are obtained. These criteria establish the links between the M-harmonic Bloch and little Bloch functionsf and other notions including the Laplace-Betrami operator. We also study some basic properties of M-harmonic Bloch and little Bloch functions, extending previously known results for holomorphic functions. The space of M-harmonic functions contains the Poisson integrals of Borel measures and all pluriharmonic functions.     

Weakly compact operators onH ∞

We prove that a weakly compact operator fromH or any of its even duals into an arbitrary Banach space is uniformly convexifying. By using this, we establish three dicothomies: (1) every operator defined onH or any of its even duals either fixes a copy ofl or factors through a Banach space having the Banach-Saks property; (2) every quotient ofH or any of its even duals either contains a copy ofl or is super-reflexive; (3) every subspace ofL ¹/H ₀ ¹ or any of its even duals either contains a complemented copy ofl ¹ or is super-reflexive.     

Fredholm composition operators on spaces of holomorphic functions

Composition operators on vector spaces of holomorphic functions are considered. Necessary conditions that range of the operator is of a finite codimension are given. As a corollary of the result it is shown that a composition operatorC on a certain Banach space of holomorphic functions on a strictly pseudoconvex domain withC ² boundary or a polydisc or a compact bordered Riemann surface or a bounded domainD such that intD = D is invertible if and only if it is a Fredholm operator if and only if is a holomorphic automorphism.     

Some results about the spectrum of commutative Banach algebras under the weak topology and applications

LetA be a commutative Banach algebra with a nonempty spectrum A. By weak we denote the relative weak topology induced on A by (A ^*,A ^**). In this note we study some properties of the topological space (A, weak) and present some applications of the results obtained and tools used to amenability, weakly compact homomorphisms, weakly compact subsets of the spectrum of the uniform algebras and to a characterization of the synthesizable ideals of the algebraA.     

The property (DN) and the exponential representation of holomorphic functions

It is shown that a nuclear Fréchet spaceE has the property (DN) if and only if every holomorphic function onE ^*, the strongly dual space ofE, with values in the strongly dual space of a Fréchet spaceF having the property ( ) can be represented in the exponential form. Moreover, it is shown that the space of holomorphic functions onC , the space of all complex number sequences, has a linearly absolutely exponential representation system. But the space of holomorphic functions onE ^* does not have such a system whenE is a nuclear Fréchet space that does not have the property (DN).Supported by the State Program for Fundamental Researches in Natural Sciences     

Uniqueness in the characteristic Cauchy problem of the Klein–Gordon equation and tame restrictions of generalized functions

We show that every tempered distribution, which is a solution of the (homogenous) Klein–Gordon equation, admits a “tame” restriction to the characteristic (hyper)surface {x ⁰ + x ⁿ = 0} in (1 + n)-dimensional Minkowski space and is uniquely determined by this restriction. The restriction belongs to the space which we have introduced in (Ullrich in J. Math. Phys. 45, 2004). Moreover, we show that every element of appears as the “tame” restriction of a solution of the (homogeneous) Klein–Gordon equation.     

A note on weighted Banach spaces of holomorphic functions

For every open subset G of and for every continuous, strictly positive weight v on G, the Banach space of all the holomorphic functions f on G such that vanishes at infinity on G, endowed with the natural weighted sup-norm, is isomorphic to a closed subspace of the Banach space c₀; hence it is reflexive if and only if it is finite dimensional.Received: 30 September 2002     

Two-sided hyperbolic SVD

In this paper, we propose the two-sided hyperbolic SVD (2HSVD) for square matrices, i.e., A=UΣV^[∗], where U and V^[∗] are J-unitary (J=diag(±1)) and Σ is a real diagonal matrix of “double-hyperbolic” singular values. We show that, with some natural conditions, such decomposition exists without the use of hyperexchange matrices. In other words, U and V^[∗] are really J-unitary with regard to J and not some matrix which is permutationally similar to matrix J. We provide full characterization of 2HSVD and completely relate it to the semidefinite J-polar decomposition.     

Real analytic curves in Fréchet spaces and their duals

The following results are presented: 1) a characterization through the Liouville property of those Stein manifoldsU such that every germ of holomorphic functions on xU can be developed locally as a vector-valued Taylor series in the first variable with values inH(U); 2) ifT is a surjective convolution operator on the space of scalar-valued real analytic functions, one can find a solutionu of the equationT u=f which depends holomorphically on the parameterz wheneverf depends in the same manner. These results are obtained as an application of a thorough study of vector-valued real analytic maps by means of the modern functional analytic tools. In particular, we give a tensor product representation and a characterization of those Fréchet spaces or LB-spacesE for whichE-valued real analytic functions defined via composition with functionals and via suitably convergent Taylor series are the same.     

Overcompleteness of Sequences of Reproducing Kernels in Model Spaces

We give necessary conditions and sufficient conditions for sequences of reproducing kernels (k_Θ(·, λ_n))_{n ≥ 1} to be overcomplete in a given model space K_Θ^p where Θ is an inner function in H^∞, p ∈ (1, ∞), and where (λ_n)_{n ≥ 1} is an infinite sequence of pairwise distinct points of Under certain conditions on Θ we obtain an exact characterization of overcompleteness. As a consequence we are able to describe the overcomplete exponential systems in L² (0, a).     

Carleson Measures for Spaces of Dirichlet Type

If 0 < p < ∞ and α > − 1, the space consists of those functions f which are analytic in the unit disc and have the property that f ′ belongs to the weighted Bergman space A_α^p. In 1999, Z. Wu obtained a characterization of the Carleson measures for the spaces for certain values of p and α. In particular, he proved that, for 0 < p ≤ 2, the Carleson measures for the space are precisely the classical Carleson measures. Wu also conjectured that this result remains true for 2 < p < ∞. In this paper we prove that this conjecture is false. Indeed, we prove that if 2 < p < ∞, then there exists g analytic in such that the measure μ_g,p on defined by dμ_g,p (z) = (1 − |z|²)^{p - 1}| g ′ (z)|^p dx dy is not a Carleson measure for but is a classical Carleson measure. We obtain also some sufficient conditions for multipliers of the spaces      

Complement preserving operators

An operatorTVV on a real inner product space is called complement preserving if, wheneverU is aT-invariant subspace ofV the orthogonal complementU is alsoT-invariant. In this note we obtain some results on such operators.     

Harmonic analysis in infinite dimension

We study the Hermite transform onL ²() where is a Gaussian measure on a Lusin locally convex spaceE. We are then lead to a Hilbert space () of analytic functions onE which is also a natural range for the Laplace transform. LetB be a convenient Hilbert-Schmidt operator on the Cameron-Martin spaceH of . There exists a natural sequence Cap _n of capacities onE associated toB. This implies the Kondratev-Yokoi theorem about positive linear forms on the Hida test-functions space.     

SOME ASPECTS OF THE LIFTING PROPERTY OF THE SPACE £∞ (μ, X)

Abstract

Using a lifting of £_∞ (μ, X) ([5],[6]), we construct a lifting ρ x of the seminormed vector space £_∞ (μ, X) of measurable, essentially bounded X-valued functions. We show that in a certain sense such a lifting always exists. If μ is Lebesgue measure on (0, 1) we show that ρ x exists as map from £_∞ ((O, 1), X) → £_∞,((0, l), X) if and only if X is reflexive. In general the lifted function takes its values in X **. Therefore we investigate the question, when f ε £_∞ (μ, X) is strictly liftable in the sense that the lifted function is a map with values even in X.

As an application we introduce the space £_∞ ^strong (μ, L (X, Y**)), a subspace of the space of strongly measurable, essentially bounded L (X, Y, **)-valued functions, and the associated quotient space £_∞ ^strong (μ, L (X,Y**)). We show that this space is a Banach space because there is a kind of a Dunford-Pettis Theorem for a subspace of L (X, £_∞(μ Y**)). Finally we investigate the measurability property of functions in £_∞(μ Y**)) und see that there exists a connection to the Radon-Nikodym property of the space L (X, Y).     

Multiplicative ∗-lie triple higher derivations of standard operator algebras

Abstract

Let be a standard operator algebra on an infinite dimensional complex Hilbert space containing identity operator I. In this paper it is shown that if is closed under the adjoint operation, then every multiplicative ?-Lie triple derivation is a linear ?-derivation. Moreover, if there exists an operator S ∈ such that S + S^? = 0 then d(U) = U S ? SU for all U ∈ , that is, d is inner. Furthermore, it is also shown that any multiplicative ?-Lie triple higher derivation D = {δ_n}_n∈? of is automatically a linear inner higher derivation on with d(U)^? = d(U^?).     

On the corona theorem for almost periodic functions

LetAP ⁺ (R ⁿ ) denote the Banach algebra of all continuous almost periodic functions onR ⁿ whose Bohr-Fourier spectrum is contained in an additive semi-group [0, ) ⁿ . We show that the maximal ideal space ofAP ⁺ (R ⁿ ) may have a nonempty corona and we characterize all for which the corona is empty. Analogous results are established for algebras of almost periodic functions with absolutely convergent Fourier series.