On the geometry in projective limits of Hilbert spaces

In this paper we consider the concept of orthogonality with respect to infinitely many inner products. We describe geometric properties related to this concept of orthogonality in certain Köthe sequence spaces (power series spaces), spaces of holomorphic functions in one and several variables and spaces of infinitely differentiable functions. The methods are required from a mixture of functional analysis (theory of bases), theory of functions of one complex variable, Fourier analysis and interpolation theory.     

Criteria for biholomorphic convex mappings on the unit ball in Hilbert spaces

In this paper, we give a necessary and sufficient condition that a locally biholomorphic mapping f on the unit ball B in a complex Hilbert space X is a biholomorphic convex mapping, which improves some results of Hamada and Kohr and solves the problem which is posed by Graham and Kohr. From this, we derive some sufficient conditions for biholomorphic convex mapping. We also introduce a linear operator in purpose to construct some concrete examples of biholomorphic convex mappings on B in Hilbert spaces. Moreover, we give some examples of biholomorphic convex mappings on B in Hilbert spaces.     

A note on the star order in Hilbert spaces

We study the star order on the algebra L(?) of bounded operators on a Hilbert space ?. We present a new interpretation of this order which allows to generalize to this setting many known results for matrices: functional calculus, semi-lattice properties, shorted operators and orthogonal decompositions. We also show several properties for general Hilbert spaces regarding the star order and its relationship with the functional calculus and the polar decomposition, which were unknown even in the finite-dimensional setting. We also study the existence of strong limits of star-monotone sequences and nets.     

Dunford–Pettis properties, Hilbert spaces and projective tensor products

We study complete continuity properties of operators onto ℓ₂ and prove several results in the Dunford–Pettis theory of JB^*-triples and their projective tensor products, culminating in characterisations of the alternative Dunford–Pettis property for where E and F are JB^*-triples.     

On p-adic vector measure spaces

For a separating algebra R of subsets of a set X, E a complete Hausdorff non-Archimedean locally convex space and m:R→E a bounded finitely additive measure, we study some of the properties of the integrals with respect to m of scalar-valued functions on X. The concepts of convergence in measure, with respect to m, and of m-measurable functions are introduced and several results concerning these notions are given.     

Estimates for norms of resolvents of~operators on tensor products of Hilbert spaces

A class of linear operators on tensor products of Hilbert spaces is considered. That class contains integro-differential operators arising in various applications. Estimates for the norm of the resolvent of considered operators are derived. By virtue of the obtained estimates, the spectrum of perturbed operators is investigated. These results are new even in the finite-dimensional case. Applications to integro-differential operators are also discussed. This revised version was published online in August 2006 with corrections to the Cover Date.     

On the proximal point method for equilibrium problems in Hilbert spaces

We analyse a proximal point method for equilibrium problems in Hilbert spaces, improving upon previously known convergence results. We prove global weak convergence of the generated sequence to a solution of the problem, assuming existence of solutions and rather mild monotonicity properties of the bifunction which defines the equilibrium problem, and we establish existence of solutions of the proximal subproblems. We also present a new reformulation of equilibrium problems as variational inequalities ones.     

Convergence theorems of fixed points for Lipschitz pseudo-contractions in Hilbert spaces

Let C be a closed convex subset of a real Hilbert space H and assume that T is a κ-strict pseudo-contraction on C. Consider Mann's iteration algorithm given by

     

Strong convergence of projection-like methods in Hilbert spaces

Many mathematical and applied problems can be reduced to finding a common point of a system of convex sets. The aim of this paper is twofold: first, to present a unified framework for the study of all the projection-like methods, both parallel and serial (chaotic, mostremote set, cyclic order, barycentric, extrapolated, etc.); second, to establish strong convergence results for quite general sets of constraints (generalized Slater, generalized uniformly convex, made of affine varieties, complementary, etc.). This is done by introducing the concept of regular family. We proceed as follows: first, we present definitions, assumptions, theorems, and conclusions; thereafter, we prove them.     

Generalized system for relaxed cocoercive mixed variational inequalities in Hilbert spaces

The approximate solvability of a generalized system for relaxed cocoercive mixed variational inequality is studied by using the resolvent operator technique. The results presented in this paper are more general and include many previously known results as special cases.     

On the normability of generalized Šerstnev PN spaces

Probabilistic Normed spaces have been redefined by C. Alsina, B. Schweizer and A. Sklar. But even today, whether the generalized Šerstnev PN space is normable or not is still an open question. In this paper, through applying Kolmogorov's theorem, we will give several sufficient conditions, under which many generalized Šerstnev PN spaces are normable. As the application of our results, we will give two examples which are normable generalized Šerstnev PN space, but not Šerstnev space.     

A proximal-Newton method for unconstrained convex optimization in Hilbert spaces

We propose and study the iteration-complexity of a proximal-Newton method for finding approximate solutions of the problem of minimizing a twice continuously differentiable convex function on a (possibly infinite dimensional) Hilbert space. We prove global convergence rates for obtaining approximate solutions in terms of function/gradient values. Our main results follow from an iteration-complexity study of an (large-step) inexact proximal point method for solving convex minimization problems.     

On the Hilbert function of curvilinear zero-dimensional subschemes of projective spaces

Here we show the existence of strong restrictions for the Hilbert function of zerodimensional curvilinear subschemes of P ⁿ with one point as support and with high regularity index.     

Norm estimates for functions of two operators on tensor products of Hilbert spaces

Analytic operator valued functions of two operators on tensor products of Hilbert spaces are considered. A precise norm estimate is established. Applications to operator differential equations are also discussed. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Sequential completeness and regularity of inductive limits of webbed spaces

Any inductive limit of bornivorously webbed spaces is sequentially complete iff it is regular.     

Schatten classes for Toeplitz operators with Hilbert space windows on modulation spaces

We prove that if ω, ω₁, ω₂, v₁, v₂ are appropriate, , j=1,2, and ωa∈Lp, then the Toeplitz operator Tph₁,h₂(a) from to belongs to the Schatten-von Neumann class of order p. From this property we prove convolution properties between weighted Lebesgue spaces and Schatten-von Neumann classes of symbols in pseudo-differential calculus.     

Compact coverings for Baire locally convex spaces

Very recently Tkachuk has proved that for a completely regular Hausdorff space X the space Cp(X) of continuous real-valued functions on X with the pointwise topology is metrizable, complete and separable iff Cp(X) is Baire (i.e. of the second Baire category) and is covered by a family of compact sets such that Kα⊂Kβ if α?β. Our general result, which extends some results of De Wilde, Sunyach and Valdivia, states that a locally convex space E is separable metrizable and complete iff E is Baire and is covered by an ordered family of relatively countably compact sets. Consequently every Baire locally convex space which is quasi-Suslin is separable metrizable and complete.     

关于Hilbert空间单位球面上等距延拓的注记

本文指出,文[1]中有关在单位球面S1[H]上的1-Lipschitz假设条件减弱为“局部1-Lipschitz”条件时,那里的等距定理仍然成立.