Compactly bounded convolutions of measures

In this paper we extend classical results concerning generalized convolution structures on measure spaces. Given a locally compact Hausdorff space , we show that a compactly bounded convolution of point masses that is continuous in the topology of weak convergence with respect to can be extended to a general convolution of measures which is separately continuous in the topology of weak convergence with respect to .

     

Sequential Fourier-Feynman transform, convolution and first variation

Cameron and Storvick introduced the concept of a sequential Fourier-Feynman transform and established the existence of this transform for functionals in a Banach algebra of bounded functionals on classical Wiener space. In this paper we investigate various relationships between the sequential Fourier-Feynman transform and the convolution product for functionals which need not be bounded or continuous. Also we study the relationships involving this transform and the first variation.

     

Classes of convolution type operators on unions of bounded intervals

An invertibility theory for classes of convolution type operators on a union of bounded intervals whose kernels have Fourier transforms which are related to solutions of corona problems is established and the corresponding formulas for the inverse operators are given. A generalization of the portuguese transformation for matrix functions is obtained and is used to establish the invertibility theory for one of the above mentioned classes of operators. The same transformation allows, also, to establish the equivalence between convolution type operators on an union of disjoint intervals and convolution type operators on a bounded interval.     

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.

     

A counterexample to ``An extension of the Vitali-Hahn-Saks theorem' and a compactness result

We give a counterexample to ``An extension of the Vitali-Hahn-Saks theorem' and from that highlight the sharp frame within which any attempt to change the version of such an extension should be possible. Lastly a sequential compactness criterion for Radon measures absolutely continuous with respect to a prescribed Radon measure defined on a locally compact separable metric space (taking into account the ideas of Hernandez-Lerma and Lasserre) is proved. The results deal with Radon measures but yield obvious corollaries on real (or vector-valued) Radon measures and so on functions with bounded variation on open subsets of .

     

Convolution operators on the weighted Herz-type Hardy spaces

A molecular characterization of the weighted Herz-type Hardy spaces and is given, by which the boundedness of the Hilbert transform and the Riesz transforms are proved on these space for 0<p1. These results are obtained by first deriving that the convolution operator Tf=k*f is bounded on the weighted Herz-type Hardy spaces.     

The index of a critical point for densely defined operators of type in Banach spaces

The purpose of this paper is to demonstrate that it is possible to define and compute the index of an isolated critical point for densely defined operators of type acting from a real, reflexive and separable Banach space into This index is defined via a degree theory for such operators which has been recently developed by the authors. The calculation of the index is achieved by the introduction of a special linearization of the nonlinear operator at the critical point. This linearization is a new tool even for continuous everywhere defined operators which are not necessarily Fréchet differentiable. Various cases of operators are considered: unbounded nonlinear operators with unbounded linearization, bounded nonlinear operators with bounded linearization, and operators in Hilbert spaces. Examples and counterexamples are given in 2,$"> illustrating the main results. The associated bifurcation problem for a pair of operators is also considered. The main results of the paper are substantial extensions and improvements of the classical results of Leray and Schauder (for continuous operators of Leray-Schauder type) as well as the results of Skrypnik (for bounded demicontinuous mappings of type Applications to nonlinear Dirichlet problems have appeared elsewhere.

     

On injective or dense-range operators leaving a given chain of subspaces invariant

In this paper we prove the existence of dense-range or one-to-one compact operators on a separable Banach space leaving a given finite chain of subspaces invariant. We use this result to prove that a semigroup of bounded operators is reducible if and only if there exists an appropriate one-to-one compact operator such that the collection of compact operators is reducible.

     

Convex Separable Minimization Subject to Bounded Variables

A minimization problem with convex and separable objective function subject to a separable convex inequality constraint and bounded variables is considered. A necessary and sufficient condition is proved for a feasible solution to be an optimal solution to this problem. Convex minimization problems subject to linear equality/linear inequality constraint, and bounds on the variables are also considered. A necessary and sufficient condition and a sufficient condition, respectively, are proved for a feasible solution to be an optimal solution to these two problems. Algorithms of polynomial complexity for solving the three problems are suggested and their convergence is proved. Some important forms of convex functions and computational results are given in the Appendix.     

A counterexample to a conjecture of S. E. Morris

We give a counterexample to a conjecture of S. E. Morris by showing that there is a compact plane set such that has no nonzero, bounded point derivations but such that is not weakly amenable. We also give an example of a separable uniform algebra such that every maximal ideal of has a bounded approximate identity but such that is not weakly amenable.

     

Carrier and nerve theorems in the extension theory

We show that a regular cover of a general topological space provides structure similar to a triangulation. In this general setting we define analogues of simplicial maps and prove their existence and uniqueness up to homotopy. As an application we give simple proofs of sharpened versions of nerve theorems of K. Borsuk and A. Weil, which state that the nerve of a regular cover is homotopy equivalent to the underlying space.

Next we prove a nerve theorem for a class of spaces with uniformly bounded extension dimension. In particular we prove that the canonical map from a separable metric -dimensional space into the nerve of its weakly regular open cover induces isomorphisms on homotopy groups of dimensions less than .

     

The Kitai criterion and backward shifts

It is proved that for any separable infinite dimensional Banach space , there is a bounded linear operator on such that satisfies the Kitai criterion. The proof is based on a quasisimilarity argument and on showing that satisfies the Kitai criterion for certain backward weighted shifts .

     

The convolution equation of Choquet and Deny on [IN]-groups

Let be a probability measure on a locally compact groupG. A real Borel functionf onG is called -harmonic if it satisfies the convolution equation *f=f. Given that isnonsingular with its translates, we show that the bounded -harmonic functions are constant on a class of groups including the almost connected [IN]-groups. If is nondegenerate and absolutely continuous, we solve the more general equation *= for positive measure on those groups which are metrizable and separable.Supported by Hong Kong RGC Earmarked Grant and CUHK Direct Grant     

Resolutions of the identity in Fréchet spaces

It is a classical result that every Bade -complete Boolean algebra of (selfadjoint) projections in a separable Hilbert space coincides with the projections forming the resolution of the identity of some bounded selfadjoint operator. This result is extended to the setting of separable Fréchet spaces. Namely, every Bade -complete Boolean algebra of projections in such a space coincides with the resolution of the identity of some (continuous) scalar-type spectral operator having spectrum a compact subset of.     

On the derivatives of the Berezin transform

Improving upon a recent result of L. Coburn and J. Xia, we show that for any bounded linear operator on the Segal-Bargmann space, the Berezin transform of is a function whose partial derivatives of all orders are bounded. Similarly, if is a bounded operator on any one of the usual weighted Bergman spaces on a bounded symmetric domain, then the appropriately defined ``invariant derivatives' of any order of the Berezin transform of are bounded. Further generalizations are also discussed.

     

A universal property of reflexive hereditarily indecomposable Banach spaces

It is shown that every separable Banach space universal for the class of reflexive Hereditarily Indecomposable space contains isomorphically and hence it is universal for all separable spaces. This result shows the large variety of reflexive H.I. spaces.

     

Fourier multipliers of classical modulation spaces

Based on the observation that translation invariant operators on modulation spaces are convolution operators we use techniques concerning pointwise multipliers for generalized Wiener amalgam spaces in order to give a complete characterization of the Fourier multipliers of modulation spaces. We deduce various applications, among them certain convolution relations between modulation spaces, as well as a short proof for a generalization of the main result of a recent paper by Bènyi et al., see [À. Bènyi, L. Grafakos, K. Gröchenig, K.A. Okoudjou, A class of Fourier multipliers for modulation spaces, Appl. Comput. Harmon. Anal. 19 (1) (2005) 131–139]. Finally, we show that any function with ([d/2]+1)-times bounded derivatives is a Fourier multiplier for all modulation spaces with p(1,∞) and q[1,∞].     

Generalization of the determinants for trace-potent linear operators

SupposeA is a bounded linear operator on a separable Hilbert space withA ^m of trace class for some positive integerm. A generalized determinant for the operatorI–A is defined, its properties studied and this determinant is then used to exhibit an inversion formula forI–A.     

Geometric characterization of RN-operators

Let X and Y be Banach spaces andtl (x, y). An operator T: X Y is called an RN-operator if it transforms every X-valued. measure ¯m of bounded variation into a Y-valued measure having a derivative with respect to the variation of the measure ¯m. The notions of T-dentability and Ts-dentability of bounded sets in Banach spaces are introduced and in their terms are given conditions equivalent to the condition that T is an RN-operator (Theorem 1). It is also proved that the adjoint operator is an RN-operator if and only if for every separable subspace X_o of X the set (T|X_o)*(Y*) is separable (Theorem 2).Translated from Matematicheskie Zametki, Vol. 22, No. 2, pp. 189–202, August, 1977.