Showing papers on "Dual norm published in 2003"

Continuity of the Norm of a Composition Operator

Abstract: We explore the continuity of the map which, given an analytic selfmap of the disk, takes as its value the norm of the associated composition operator on the Hardy space \( H^{2} \). We also examine the continuity the functions which assign to a self-map of the disk the Hilbert-Schmidt norm or the essential norm of the associated composition operator and show these to be discontinuous. Additionally, we characterize when the norm of a composition operator is minimal.

Set-valued nonlinear variational inequalities for H -monotone mappings in nonreflexive Banach spaces

Abstract: Let H be a mapping from a normed space X to a normed space Y. In the monograph by Vainberg Variational method and method of monotone operators, Nauka, Moskva, 1972, a mapping T from a normed space X to the dual space Y* of a normed space Y is said to be H-monotone if 〈Tx - Ty, H(x - y)〉 ≥ 0, ∀x, y ∈ X, where H is a mapping from X to Y.In nonreflexive Banach spaces nonlinear variational inequalities for upper-semi-continuous H-monotone set-valued mappings T have not been investigated yet even though this is an interesting problem. The present paper is concerned with this difficult one.

On the density of positive proper efficient points in a normed space

Abstract: In the context of vector optimization and generalizing cones with bounded bases, we introduce and study quasi-Bishop-Phelps cones in a normed space X. A dual concept is also presented for the dual space X*. Given a convex subset A of a normed space X partially ordered by a closed convex cone S with a base, we show that, if A is weakly compact, then positive proper efficient points are sequentially weak dense in the set E(A, S) of efficient points of A; in particular, the connotation weak dense in the above can be replaced by the connotation norm dense if S is a quasi-Bishop-Phelps cone. Dually, for a convex subset of X* partially ordered by the dual cone S+, we establish some density results of positive weak* efficient elements of A in E(A, S+).

Two New Neuron Models Based on T/S Norms and Their Applications

Abstract: Based on T norm/ S norm, F 1 and F2 fuzzy neuron models are proposed for the first time in the paper. Their properties and applications are discussed here. F1 model holds good sensitivity and poor robustness. However F2 model holds good robustness and poor sensitivity. The paper presents a necessary and sufficient condition of that a generalized AND/OR operator based on F1 model is a weak T norm/ S norm cluster. The paper set forth the concept of weak bound triangular norm for the first time, further, finds out a particular F2 model which can realize weak bound triangular norm. Then the paper points out that F1 model is more suitable for industrial control systems and F2 model is more suitable for those computer application systems such as fuzzy expert system in the fields of medicine, law and strategic decision et al. Finally, a fuzzy neural network based on a particular F2 neuron is applied in fuzzy inference. The new inference method generalizes traditional Zadeh’s CRI and satisfies modus ponens and other inference principles when the weights are adjusted in right way.

Renorming of $C(K)$ spaces

Abstract: If K is a scattered Eberlein compact space, then C(K)* admits an equivalent dual norm that is uniformly rotund in every direction. The same is shown for the dual to the Johnson-Lindenstrauss space JL 2 .

Almost normed spaces of functions with polynomial asymptotic behaviour

Abstract: We consider a linear space of functions asymptotically approaching polynomials of degree not higher than a fixed one as the independent variable approaches infinity. Such a space cannot be normed if the functions in it possess a certain smoothness. For this reason the concept of almost normed space is introduced and the spaces in question, namely, spaces of functions asymptotically or strongly asymptotically approaching polynomials, are shown to be almost normed. The completeness of these spaces in the metric generated by their almost norm is also proved, the connection between the asymptotic approach and the strong asymptotic approach of functions to polynomials is studied, and a new (and shorter) proof of the criterion for the asymptotic approach of functions to polynomials is presented.

Reduction of Opial-type inequalities to norm inequalities

Abstract: Weighted Opial-type inequalities are shown to be equivalent to weighted norm inequalities for sublinear operators and for nearly positive operators. Examples involving the Hardy-Littlewood maximal function and the non-increasing rearrange- ment are presented. Opial-type inequalities are related to norm inequalities much as quadratic forms are related to bilinear forms. A linear operator T on Hilbert space gives rise to the bilinear form (f,g) 7! hTf,gi and the quadratic form f 7! hTf,fi. Duality shows that the norm of T and the norm of the bilinear form coincide and a standard polarization argument shows that this norm is equivalent to but not necessarily equal to the norm of the quadratic form, called the numerical radius of T. In this paper, far from the luxuries of Hilbert spaces and linear operators, we show that the equivalence of operator norm and numerical radius persists. The work is in response to Richard Brown's suggestion that Steven Bloom's result (2, The- orem 1) which gives the equivalence for positive operators should apply in greater generality. Opial-type inequalities have been much studied since Opial's original paper in 1960 and the papers (2), (3) and (4) include many references. After the main theorem showing equivalence of Opial-type and norm inequali- ties, an example involving the Hardy-Littlewood maximal function is included to illustrate that the equivalence cannot be taken in a pointwise sense. To show that the method can be readily applied to generate non-trivial inequal- ities from known norm inequalities we give a simple weight characterization of an Opial-type inequality for the non-increasing rearrangement.

A note on the convergence of Orlicz-Bochner spaces with the Luxemburg norm

Abstract: In this paper, the convergence of the Orlicz-Bochner function spaces with the Luxemburg norm was proved.

An algorithm for a minimum norm solution of a system of linear inequalities

Abstract: A common problem encountered in the studies of the least squares problems is that of finding the minimum E 2 norm solution of a system of linear equations. In this paper, we consider an algorithm for computing the vector of minimum norm solution of a given system of linear inequalities. We replace the E 2 norm by $\ell^p\comma \, 1\lt p \lt\infty$ . Duality theorems and characterizations of the solution are given. The feasibility of the method is proved and some numerical experimentations are included.