S-Numbers of Weighted Shift Operators on P-Summable Formal Entire Functions of M-Variables

Nashat Faried,Z.A Hassanain,H Abd El Ghaffar,A Lokman

doi:10.3844/jmssp.2019.79.85

Abstract

The idea of multiplying a formal Taylor power series by z to make a right shift operator on the space of all square summable sequences of real numbers was due to A.L. Shield. In this work, we consider Taylor power series in m-variables and we give upper and lower estimations of s-numbers for multiplication of m- right weighted shift operators. This allowed us to estimate upper bounds for s-numbers of infinite series of m-right weighted shift operators and give some applications.

Highlights

For any bounded linear operator T from a Banach space E into a Banach space F there are associated some decreasing sequences of non negative numbers called snumbers satisfying certain conditions
The s-numbers of the unilateral forward weighted shift operator RJ have the following upper and lower estimations: ( ) ( ) ( ) sup sup inf β1 i1 + j1 ( ) ( ) ∏m ri ≤r +1cardξi =ri I∈ξ β1 i1 βm im + jm ⋯
The authors declare that there is no conflict of interests regarding the puplication of this article

Summary

Introduction

For any bounded linear operator T from a Banach space E into a Banach space F there are associated some decreasing sequences of non negative numbers called snumbers satisfying certain conditions. Hilbert Schmidt operators are those operators whose sequences of approximation numbers are square summable (Pietsch, 1980). Shields (1974) gave representation for weighted shift operators as formal power series in unilateral shifts and formal Laurent series in bilateral shifts. He suggested to express functions belonging to the space H2(β) by f (z). { } and f (n) is a sequence of real numbers. In this case, to see that ||zk|| = β(k) he considered the following:. Sn(UTV) ≤ ||U|| sn(T) ||V|| for V∈L(E0,E), T∈L(E, F). Let {τi} be a bounded family of real numbers. A z z ⋯ z ⋯ z i1 i2 i1 ,i2 ,⋯ij ,⋯im 1 2 ij −1 im ik =0 k =1,2,⋯,m respectively

Main Results

For the infinite series of unilateral weighted shift

From proposition