IJPAM: Volume 83, No. 5 (2013)

SOME CLASS OF OPERATOR IDEALS

P.D. Srivastava1, Amit Maji2
1,2Department of Mathematics
Indian Institute of Technology Kharagpur
Kharagpur, West Bengal, 721302, INDIA

Proceedings of
NCRTMSA - 2012


Abstract. In 1975, Rhoades [9] generalized the classes of, operators of lp type and operators of Cesaro type by introducing an arbitrary infinite matrix A = (ank) using approximation numbers of a bounded linear operator. In the same paper Rhoades has proved that for each fixed matrix A satisfying the condition |an, 2k-1| + |an, 2k| ≤ M|a_nk| on the matrix A = (ank) for each n, k = 1, 2, ... and each p, 0 < p ≤ ∞ the set A-p type operators is a linear space and raised an open question whether this condition is necessary to be a linear space? In this paper we have answered the question in negation. Further, we have introduced and studied the class A(s)-p of s-type cesp operators using s-number sequence and Cesaro sequence spaces. We have also shown that the class A^(s)-p forms a quasi-Banach operator ideal. Moreover, the inclusion relations among the operator ideals are established.

Received: January 10, 2013

AMS Subject Classification: 47B06, 47L20

Key Words and Phrases: $s$-numbers, approximation numbers, operator ideals, sequence space

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DOI: 10.12732/ijpam.v83i5.16 How to cite this paper?
Source:
International Journal of Pure and Applied Mathematics
ISSN printed version: 1311-8080
ISSN on-line version: 1314-3395
Year: 2013
Volume: 83
Issue: 5