Operator ideal of Norlund-type sequence spaces

Awad A Bakery

doi:10.1186/s13660-015-0774-5

Awad A Bakery

Open Access

https://doi.org/10.1186/s13660-015-0774-5

Copy DOI

Abstract

Let E be a Norlund sequence space which is invariant under the doubling operator $$D:x=(x_{0}, x_{1}, x_{2},\ldots )\mapsto y=(x_{0}, x_{0}, x_{1}, x_{1}, x_{2}, x_{2},\ldots ). $$ Using the approximation numbers $(\alpha_{n}(T))^{\infty}_{n=0}$ of operators from a Banach space X into a Banach space Y, we give the sufficient (not necessary) conditions on E such that the components $$U_{E}^{\mathrm {app}}(X, Y):= \bigl\{ T\in L(X, Y):\bigl( \alpha_{n}(T)\bigr)_{n=0}^{\infty}\in E \bigr\} $$ form an operator ideal, the finite rank operators are dense in the complete space of operators $U_{E}^{\mathrm {app}}(X, Y)$ which is a longstanding open problem. Finally we give an answer for Rhoades (Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl. 59(3-4):238-241, 1975) about the linearity of E-type spaces $(U_{E}^{\mathrm {app}}(X, Y))$ , and we conclude under a few conditions that every compact operator would be approximated by finite rank operators. Our results agree with those in (J. Inequal. Appl., 2013, doi: 10.1186/1029-242x-2013-186 ) for the space $\operatorname {ces}((p_{n}))$ , where $(p_{n})$ is a sequence of positive reals.

Highlights

1 Introduction As an aftereffect of the enormous applications in geometry of Banach spaces, spectral hypothesis, hypothesis of eigenvalue dispersions and fixed point hypothesis and so on, the hypothesis of operator ideal goals possesses an uncommon essentialness in useful examination
A large portion of the administrator goals in the family of Banach spaces or normed spaces in straight practical examination are characterized by diverse scalar grouping spaces
Let L be the class of all bounded linear operators between any arbitrary Banach spaces

Summary

Introduction

As an aftereffect of the enormous applications in geometry of Banach spaces, spectral hypothesis, hypothesis of eigenvalue dispersions and fixed point hypothesis and so on, the hypothesis of operator ideal goals possesses an uncommon essentialness in useful examination. Appl., 2013, doi:10.1186/1029-242x-2013-186) for the space ces((pn)), where (pn) is a sequence of positive reals. Bakery [ , ] took some different mean of Cesaro type spaces involving Lacunary sequence Ces(θ , p) defined in [ ] to form an operator ideal with its approximation numbers.

Results

Conclusion