Narrow Results

Let T be a bounded linear operator on an infinite dimensional, separable Banach space X. We consider a class of supercyclic operators, whose point spectrum of their adjoints are nonempty, and prove that under certain conditions, the orbit of every supercyclic vector for such an operator is unbounded. This result has some nice applications: (1) We obtain some conditions equivalent to the supercyclicity of extreme points of the closed unit ball of on a separable infinite dimensional Hilbert space; this helps us to characterize all supercyclic operators in this class. (2) The adjoint of composition operators on certain weighted Hardy spaces are never supercyclic. Next, we turn our attention to the commutant and show that if T is an operator in the mentioned class, then every operator in the commutant of T is not hypercyclic.

The set of anchor points is an important subset of the set of extreme efficient points of the production possibility set (PPS) in data envelopment analysis (DEA). In this paper, some new results, under variable returns to scale (VRS) assumption, are given which characterize these points from different points of view. First, we prove two necessary and sufficient conditions leading to geometrical characterizations. Afterwards, four sufficient conditions are given with respect to the level values of the supporting hyperplanes, infeasibility of the super efficiency models, maximum/minimum data, and zero data. Some of the given conditions can be checked without solving any mathematical programming problems. Also, some of them can be tested by solving standard DEA models. The investigated conditions are useful in designing pre-processors and heuristics.

In this paper, we will study the extremal structure of the unit ball of U(M,X), the space of uniformly continuous and bounded functions, from a not necessarily compact metric space M into a normed space X. Concretely, if X is uniformly convex and dim X ≥ 2, where dim X denotes the dimension of X as a real vector space, it is proved that every element y in U(M,X), with ‖y‖ < 1, is a convex combination of a finite number of extreme points of the unit ball. As a result, the unit ball of U(M,X) coincides with the closed-convex hull of its extreme points.

Let A be an f-ring with identity u and B be an archimedean f-ring. For every idempotent element w in B, let denote the set of all positive group homomorphisms ℌ : A → B with ℌ(u) = w. We prove that is a ring homomorphism if and only if ℌ is an extreme point of . As a consequence, we derive a characterization of ring homomorphisms in in terms of a Gelfand-type transform. Moreover, we show that ring homomorphisms in are, up to multiplicative constants, all the basic elements of the ℓ-group of all bounded group homomorphisms from A into ℝ.

By making use of Wanas operator, we aim to introduce and investigate a certain family of univalent holomorphic functions with negative coefficients defined on complex Hilbert space. We present some important geometric properties of this family such as coefficient estimates, convexity, distortion and growth, radii of starlikeness and convexity. We also discuss the extreme points for functions belonging to this family.