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In this paper we prove a result of the Trotter–Kato type in the weak topology. Let { A e } e > 0 be a family of quasi m-accretive linear operators on a Hilbert space X and let us denote by J λ e the resolvent of A e . Under certain conditions, the result states that if for any x ∈ X and k = 1 , 2 , … , the sequence ( J λ e ) k x converges weakly to ( J λ ) k x as e → 0 , where J λ is the resolvent of a linear quasi m-accretive operator A on X, then the sequence of the semigroups generated by − A e tends weakly to the semigroup generated by −A, uniformly with respect to t on compact intervals. The result is different from other results of the same type (see e.g., Yosida (1980) [9, p. 269] ) and gives an answer to an open problem put in Eisner and Sereny (2010) [3] . It is finally applied to compare the asymptotic behavior of a singular perturbation problem associated to a first order hyperbolic problem with diffusion.

We show that the unit ball of l p (Γ) in its weak topology is a continuous image of σ 1 (Γ) N , and we deduce some combinatorial properties of its lattice of open sets which are not shared by the balls of other equivalent norms when r is uncountable.

(2006). Curiosities concerning Weak Topology in Hilbert Space. The American Mathematical Monthly: Vol. 113, No. 5, pp. 447-452.

In this paper we introduce the invex programming problem in Hilbert space. The requisite theory has been established to characterize the solution of such class of problems.

A lower hemi-continuous correspondence with open and convex values in Rn must have open lower sections. This well- known fact has been used to establish the existence of continuous selections, maximal elements, and fixed points of correspondences in various economic applications. Since there is an increasing number of economic models that use correspondences in an infinite-dimensional setting, it is important to know whether or not the above fact remains valid in such applications. The aim of this paper is to show that the above fact no longer holds when Rn is replaced with an infinite-dimensional space. This is accomplished by using the standard orthonormal base in a Hilbert space H to construct two correspondences with values in H equipped with the weak topology. The first correspondence is lower hemi-continuous with open and convex values but does not have open lower sections. The second is a lower hemi-continuous correspondence that fails to have an open graph despite having open and convex upper and lower sections. These counter-examples demonstrate that in an infinite-dimensional setting, it is no longer possible to rely on the geometric properties of a lower hemi-continuous map (the convexity of its sections) to establish the topological properties (open lower sections, open graph) needed in many economic applications.

We study the computational difficulty of the problem of finding fixed points of nonexpansive mappings in uniformly convex Banach spaces. We show that the fixed point sets of computable nonexpansive self-maps of a nonempty, computably weakly closed, convex and bounded subset of a computable real Hilbert space are precisely the nonempty, co-r.e. weakly closed, convex subsets of the domain. A uniform version of this result allows us to determine the Weihrauch degree of the Browder-Goehde-Kirk theorem in computable real Hilbert space: it is equivalent to a closed choice principle, which receives as input a closed, convex and bounded set via negative information in the weak topology and outputs a point in the set, represented in the strong topology. While in finite dimensional uniformly convex Banach spaces, computable nonexpansive mappings always have computable fixed points, on the unit ball in infinite-dimensional separable Hilbert space the Browder-Goehde-Kirk theorem becomes Weihrauch-equivalent to the limit operator, and on the Hilbert cube it is equivalent to Weak Koenig's Lemma. In particular, computable nonexpansive mappings may not have any computable fixed points in infinite dimension. We also study the computational difficulty of the problem of finding rates of convergence for a large class of fixed point iterations, which generalise both Halpern- and Mann-iterations, and prove that the problem of finding rates of convergence already on the unit interval is equivalent to the limit operator., 44 pages

In topological dynamics a theory of recurrence properties via (Fursten- berg) families was established in the recent years. In the current paper we aim to establish a corresponding theory of ergodicity via families in measurable dynamical systems (MDS). For a family F (of subsets of Z+) and a MDS (X,B,µ,T), several notions of ergodicity re- lated to F are introduced, and characterized via the weak topology in the induced Hilbert space L 2 (µ).

We study extension operators between spaces $\sigma_n(2^X)$ of subsets of $X$ of cardinality at most $n$. As an application, we show that if $B_H$ is the unit ball of a nonseparable Hilbert space $H$, equipped with the weak topology, then, for any $0

A characterisation of a type of weak-operator continuous linear functional on certain linear subsets of B(H), where H is a Hilbert space, is derived within Bishop-style constructive mathematics.

In this paper, we develop a new approach to obtain the compactness of the fluctuation processes for Boltzmann dynamics. Our method is applicable to Kac's model, already studied by Uchiyama, but it covers many other cases. A novelty worth mentioning is the use of the weak topology of a Hilbert space.

In the framework of white noise analysis a Gel'fand triple ( L )⊂(L 2 )⊂( L ) ∗ has been defined (e.g. Kubo and Yokoi, 1989), the space of smooth test functionals ( L ) and the space of Hida distributions ( L )∗ play some important roles. It has been shown (e.g. Yokoi, 1990) that a positive Hida distribution Φ is given by a positive measure νΦ on the space of real tempered distributions L ∗. Thus the space (L2)ΦL2( L ∗; B , νΦ) can be defined, where B is the Borel σ-algebra on L ∗ generated by the weak topology. The present article is concerned with a special choice of pre-Dirichlet forms with domain ( L ) on (L2)Φ which is a generalization of the energy form (Hida, Potthoff and Streit, 1988) and of the type e(F)= Φ; ∑ j,k H j,k ·∂ j F·∂ k F , for each Fϵ( L ) and where (Hj,k; j, kϵ N 0) is a double sequence of test functionals satisfying some natural conditions. Some closability results are given in the last section under mild conditions.

We give an alternate description of algebras in the class of ultraweakly closed subspaces of q(X) via the preannihilator. We then apply this result to show that proper ultraweakly closed algebras of bounded operators on an infinite-dimensional Hilbert space X have infinite codimension. We also use this alternate description of algebras to say something the structure of rank-one operators in unicellular algebras. We begin with some basic definitions and notation. For ', an infinitedimensional Hilbert space, let 5F(X) denote the set of all bounded linear operators on X and let Y(X) denote the set of all trace-class operators on X'. Then Y('), equipped with the trace norm, is a Banach space whose dual is 5( ). The duality is given by the linear functional on 7(X) x (X) defined by (t, x) tr(tx) for t EzY(X), x E (X), where tr denotes the trace. Thus we get a w*-topology on 5(X), which is also known as the ultraweak topology. The weak-operator topology (which we shall refer to as the weak topology) is actually weaker than the ultraweak topology, so all the results stated in this paper for ultraweakly closed subspaces are true for weakly closed subspaces. An excellent exposition of the role of this duality theory in invariant subspace theory is [ 1]. We shall follow the notation of [ 1], which the reader can consult for more background. As usual, we can use the above duality to define preannihilators. For ? an ultraweakly closed subspace of (X), the preannihilator is X1 = {t EY7(X) I tr(tm) = 0 for all m Ecz}. The codimension of A'(codim(.')) is the vector space dimension of {f(X)/ X'}. If we identify all infinite cardinals, then this is also equal to the vector space dimension of X1 . Given x, y E X, the operator x 0 y is the rank-one operator defined by x X y(z) = (z, y)x for z E X. Received by the editors July 3, 1989. 1980 Mathematics Subject Classification (1985 Revision). Primary 47D25. ? 1990 American Mathematical Society 0002-9939/90 $1.00 + $.25 per page

This paper introduces a general implicit iterative method for finding zeros of a maximal monotone operator in a Hilbert space which unifies three previously studied strategies: relaxation, inertial type extrapolation and projection step. The first two strategies are intended to speed up the convergence of the standard proximal point algorithm, while the third permits one to perform inexact proximal iterations with fixed relative error tolerance. The paper establishes the global convergence of the method for the weak topology under appropriate assumptions on the algorithm parameters.

A theorem of Dixmier states that each bounded linear functional $f$ on the algebra of bounded linear operators on a separable Hilbert space is a direct sum of a trace functional $g$ and a singular functional $h$, vanishing on the compact operators, such that $\Vert f \Vert= \Vert g \Vert +\Vert h \Vert$. We use elementary methods to construct, via the state space of a $C^\ast$-algebra, a Banach space of $C^\ast$ matrices that contains a closed subspace on which a version of Dixmier's theorem is proved. When the $C^\ast$-algebra is taken to be the complex numbers our approach gives elementary and transparent proofs of Dixmier's theorem and the trace formula $\rm{tr}(AB) = \rm{tr}(BA)$, without using the operator theoretical machineries used in the known proofs.

For every weakly statistically convergent sequence (x(n)) with increasing norms in a Hilbert space we prove that sup(n) parallel to x(n)parallel to/root n < infinity This estimate is sharp We study analogous problem for sonic other types of weak filter convergence. in particular for the Erdos-Ulam filters, analytical P-filters and F-sigma filters We present also a refinement of the recent Aron-Garcia-Maestre result on weakly dense sequences that tend to infinity in norm (C) 2010 Elsevier Inc. All rights reserved.

Within the context of infinite-dimensional representations of the rotation group the Dirac monopole problem is studied in details. Irreducible infinite-dimensional representations, being realized in the indefinite metric Hilbert space, are given by linear unbounded operators in infinite-dimensional topological spaces, supplied with a weak topology and associated weak convergence. We argue that an arbitrary magnetic charge is allowed, and the Dirac quantization condition can be replaced by a generalized quantization rule yielding a new quantum number, the so-called topological spin, which is related to the weight of the Dirac string., Comment: JHEP style. Extended version of hep-th/0403146. Revised version, title and some notations are changed. References and Appendix B are added

In this paper, I first prove an integral representation theorem: Every quasi-integralon a Stone lattice can be represented by a unique upper-continuous capacity. I then apply this representation theorem to study the topological structure of the space of all upper-continuous capacities on a compact space, and to prove the existence of an upper-continuous capacity on the product space of infinitely many compact Hausdorff spaces with a collection of consistent finite marginals.

The group of measure preserving transformations of the unit interval equipped with the weak topology is an absolute retract, hence is homeomorphic to a separable Hilbert space.

In [2] and [3], Strauss studied this problem when q has sufficiently rapid decay at infinity. In this paper we will derive some information for the case q -1. The basic parts used will be the energy estimates of Strauss derived in [2] and an elementary lemma about Hilbert spaces. While the results in this paper refer to the existence of wave operators in the weak topology, they open new methods of attacking the problem in the strong topology.

The approximate subdifferential introduced by Mordukhovich has attracted much attention in recent works on nonsmooth optimization. Potential advantages over other concepts of subdifferentiability might be related to its nonconvexity. This is motivation to study some topological properties more in detail. As the main result, it is shown that any weakly compact subset of any Hilbert space may be obtained as the Kuratowski–Painleve limit of approximate subdifferentials from a one-parametric family of Lipschitzian functions. Sharper characterizations are possible for strongly compact subsets. As a consequence, in any Hilbert space the approximate subdifferential of a suitable Lipschitzian function may be homeomorphic (both in the strong and weak topology) to the Cantor set. Further results relate the approximate subdifferential to specific topological types, to the one-dimensional case (which is extraordinary in some sense), and to the value function of a C 1-optimization problem.

Let A be a real nest algebra of B ( H ), where H is a real and separable Hilbert space. We show that the following conditions are equivalent for a weak topology continuous linear map ϕ : A → B ( H ) : (1) ϕ is a *-left preserving kernel-into-range mapping, i.e., ϕ ( T )(ker( T )) ⊆ ran( T *) for any T ∈ A . (2) ϕ is a generalized *-left inner derivations, i.e., ϕ ( T ) = T * A + BT for some A , B ∈ B ( H ). (3) ϕ is a generalized Jordan *-left derivations, i.e., ϕ ( T 2 ) = T * ϕ ( T ) + ϕ ( T ) T − T * ϕ ( I ) T for any T ∈ A . (4) ϕ is a *-left 1-preserving kernel-into-range mapping, i.e., ϕ ( T )(ker( T )) ⊆ ran( T *) for any rank one operator T ∈ A .

Given a locally compact Abelian group and the space integrable functions on said group; with the space of multipliers on the functions defined as the space of bounded operators on the functions which commute with translations by elements of the group; it is proved that the space of multipliers is the conjugate space of a Banach space of continuous functions on the group. As a direct consequence, it is shown that the space of multipliers is the closure in the weak (strong) operator topology of the span of the translation operators. The results are applied to the study of translates of multipliers and to the relations between multipliers and lacunary sets of the character group of the compact group. (Author)

We study transition semigroups and Kolmogorov equations corresponding to stochastic semilinear equations on a Hilbert space H . It is shown that the transition semigroup is strongly continuous and locally equicontinuous in the space of polynomially increasing continuous functions on H when endowed with the so-called mixed topology. As a result we characterize cores of certain second order differential operators in such spaces and show that they have unique extensions to generators of strongly continuous semigroups.

=forte R, and f(t+ T)=f(t) fora.e.t eR), we consider the equation (E) u’(t)+3t(u(t)) f(t), t e J, where J is an interval in R, u’(t)=(d/dt)u(t) and * isthesubdifferential of. For related studies on (E) we refer to [2, 4, 7, 8, 11, 12, 13]. In [3], Baillon and Haraux treated he time-independent case of i.e.’, and proved that any solution on J=[t0,) is asymptotically Tperiodic in the weak topology ofH and the difference of any two T-periodic solutions is a constant vector on R. Subsequently, Haraux [5] and Ishii [6] discussed the equation from the same viewpoint as in [3], when and f is almost periodic on R. In this paper we shall show by a simple example in 3-dimensional space that the equation with the time-dependent *

Representation theory for finite dimensional systems has been the subject of a great deal of discussion in recent years. In the case of linear systems defined over fields an account of the theory can be found in the books of BROCKETT and KALMAN-FALB-ARBIB and in the case of systems defined over rings in the book of EILENBERG, the papers of ROUCHALEAU-WYMAN-KALMAN, ROUCHALEAU-WYMAN and the thesis of JOHNSTON. For a lucid survey of results on systems defined over commutative rings, see SONTAG.

There have been two main contributions to the theory of operator algebras on Hilbert space. In a series of five memoirs, Murray and von Neumann have made important strides toward the structure theory of the weakly closed case. In work begun by the Russian school, a study has been made of the more general case where merely uniform closure is assumed. In terminology suggested by Segal, we call these W*-algebras and C*-algebras respectively. A notable advantage of the C*-case is the existence of an elegant system of intrinsic postulates due to Gelfand and Neumark [2]; so one can, and does, study C*-algebras in an abstract fashion that pays no attention to any particular representation. A corresponding characterization of TV*-algebras is not known, but nevertheless several substitutes have been suggested. In [6] von Neumann postulated from the start a second topology behaving like the weak topology. Steen [8] assumed completeness relative to a topology induced by positive functionals. In the present paper the entire burden will be thrown upon a more algebraic, and in some sense more elementary assumption; briefly put, our postulate is the assumption of least upper bounds in the partially ordered set of projections-the precise axioms are given in ?2. We call the algebras in question ATV*algebras (the "A" suggesting "abstract"). This work is in essence a continuation of the study that was begun by Rickart

Variational inequality problems over fixed point sets of nonexpansive mappings include many practical problems in engineering and applied mathematics, and a number of iterative methods have been presented to solve them. In this paper, we discuss a variational inequality problem for a monotone, hemicontinuous operator over the fixed point set of a strongly nonexpansive mapping on a real Hilbert space. We then present an iterative algorithm, which uses the strongly nonexpansive mapping at each iteration, for solving the problem. We show that the algorithm potentially converges in the fixed point set faster than algorithms using firmly nonexpansive mappings. We also prove that, under certain assumptions, the algorithm with slowly diminishing step-size sequences converges to a solution to the problem in the sense of the weak topology of a Hilbert space. Numerical results demonstrate that the algorithm converges to a solution to a concrete variational inequality problem faster than the previous algorithm. MSC:47H06, 47J20, 47J25.