Manjul Gupta

Generalized Orlicz-Lorentz sequence spaces and corresponding operator ideals

In this paper we introduce generalized or vector-valued Orlicz-Lorentz sequence spaces lp,q,M(X) on Banach space X with the help of an Orlicz function M and for different positive indices p and q. We study their structural properties and investigate cross and topological duals of these spaces. Moreover these spaces are generalizations of vector-valued Orlicz sequence spaces lM(X) for p = q and also Lorentz sequence spaces for M(x) = xq for q ≥ 1. Lastly we prove that the operator ideals defined with the help of scalar valued sequence spaces lp,q,M and additive s-numbers are quasi-Banach operator ideals for p < q and Banach operator ideals for p ≥ q. The results of this paper are more general than the work of earlier mathematicians, say A. Pietsch, M. Kato, L. R. Acharya, etc.

TOPOLOGICAL PROPERTIESAND MATRIXTRANSFORMATIONS OF CERTAIN ORDERED GENERALIZED SEQUENCE SPACES

In this note, we carry out investigations related to the mixed impact of ordering and topological structure of a locally convex solid Riesz space (X, ) and a scalar valued sequence space ,, on the vector valued sequence space x (X) which is formed and topologized with the help of x and X, and vice versa. Besides,we also characterize o-matrix trans- formations from c(X), = (X) to themselves, cs(X) to c(X) and derive neces- sary conditions for a matrix of linear operators to transform .1 (X) into a simple ordered vector valued sequence space ^(X).

Weak unconditional Cauchy series

A notion corresponding to a given sequence space λ more general than the weakly unconditionally Cauchy series in a locally convex space is introduced. We investigate some of its applications which in particular extend a few known results.

The Compact Approximation Property for Weighted Spaces of Holomorphic Mappings

In this paper, we examine the compact approximation property for the weighted spaces of holomorphic functions. We show that a Banach space E has the compact approximation property if and only if the predual \(\mathcal {G}_v(U)\) of the space \(H_v(U)\) consisting of all holomorphic mappings \(f:U\rightarrow \mathbb {C}\) (complex plane) with \(\sup \limits _{x\in U}v(x)\Vert f(x)\Vert <\infty \) has the compact approximation property, where v is a radial weight defined on a balanced open subset U of E such that \(H_v(U)\) contains all the polynomials. We have also studied the compact approximation property for the weighted (LB)-space VH(E) of holomorphic mappings and its predual VG(E) for a countable decreasing family V of radial rapidly decreasing weights on E.

A Note on Generalized Approximation Property

We introduce a notion of generalized approximation property, which we refer to as --AP possessed by a Banach space , corresponding to an arbitrary Banach sequence space and a convex subset of , the class of bounded linear operators on . This property includes approximation property studied by Grothendieck, -approximation property considered by Sinha and Karn and Delgado et al., and also approximation property studied by Lissitsin et al. We characterize a Banach space having --AP with the help of -compact operators, -nuclear operators, and quasi--nuclear operators. A particular case for () has also been characterized.

λ-similar bases

Corresponding to an arbitrary sequence space λ, a sequence {xn} in a locally convex space (l.c.s.) (X,T) is said to be λ-similar to a sequence {yn} in another l.c.s. (Y,S) if for an arbitrary sequence {αn} of scalars, {αn p(xn)} ϵ λ for all p ϵ DT⇔{αn q(yn)} ϵ λ, for all q ϵ DS, where DT and DS are respectively the family of all T and S continuous seminorms generating T and S.

Unconditional bases and the order structure

The paper deals with an analytic characterization of unconditional bases, a characterization of the unconditional character of a Schauder base for the associated cone and a number of counterexamples supporting various conditions appearing in the theorems.

Dominating sequences and functional equations

The notion of domination of sequences has been used by a number of mathematicians in the past with slight variations of terminology. Indeed, the domination between pair of sequences was considered by Banach ([4], p. 1638; [5], p. 11) around the year 1925. Later, Arsove ([2], [3]) used these notions in the s tudy of Schauder basis in Fr~chet spaces and constructed some examples of such sequences (for domination of infinite matrices, one is referred to [9], [10]). However, these concepts gained independent importance in the work of Singer [23] who also employed these notions to reformulate certain results on sequence spaces. He dealt with all the problems in Banach spaces. In this note, we give various equivalent conditions for strict domination in terms of functional equations in topological vector spaces, characterize a base through the notions of domination and strict domination, and apply them to obtain a result on weakly ~-unconditionally Cauchy ser~es -a concept introduced in [12] (cf. also Chapter 3 of [13]).