Some results of operator ideals on s-type $|A, p|$ operators

Article Sidebar

PDF
Published: Jun 30, 2014
DOI: https://doi.org/10.5556/j.tkjm.45.2014.1297
Keywords:
s-numbers Approximation numbers Kolmogorov numbers Operator ideals.

Main Article Content

AMIT MAJI
P. D. Srivastava

Abstract

Let $s=(s_{n})$ be a sequence of $s$-numbers in the sense of Pietsch and $A$ be an infinite matrix. This paper presents a generalized class $\mathscr{A}^{(s)}-p$ of $s$-type $|A, p|$ operators using $s$-number sequence which unifies many earlier well known classes. It is shown that the class $\mathscr{A}^{(s)}-p$ forms a quasi-Banach operator ideal under certain conditions on the matrix $A$. Moreover, the inclusion relations among the operator ideals as well as the inclusion relations among their duals are established. It is also proved that for the Ces$\grave{\rm{a}}$ro matrix of order $1$, the operator ideal formed by approximation numbers is small for $1<p< \infty$.

Article Details

How to Cite
MAJI, A., & Srivastava, P. D. (2014). Some results of operator ideals on s-type $|A, p|$ operators. Tamkang Journal of Mathematics, 45(2), 119–136. https://doi.org/10.5556/j.tkjm.45.2014.1297
Issue
Vol. 45 No. 2 (2014)
Section
Papers
Author Biographies

AMIT MAJI

Department ofMathematics, Indian Institute of Technology Kharagpur, Kharagpur 721 302, West Bengal, India.

P. D. Srivastava

Department ofMathematics, Indian Institute of Technology Kharagpur, Kharagpur 721 302, West Bengal, India.

References

B. Carl, A. Hinrichs, On $s$-numbers and Weyl inequalities of operators in Banach spaces, Bull. Lond. Math. Soc. 41 (2009), no. 2, 332-340.

Gh. Constantin, Operators of ces-p type, Rend.Acc.Naz.Lincei. 52 (8) (1972), 875-878.

C. V. Hutton, On the approximation numbers of an operator and its adjoint, Math. Ann. 210 (1974), 277-280.

A. Pietsch, Einige neue klassen von kompakten linearen Abbildungen, Rev. Math. Pures Appl. (Bucarest) 8 (1963), 427-447.

A. Pietsch, $s$-numbers of operators in Banach spaces, Studia Math. 51 (1974), 201-223.

A. Pietsch, Small ideals of operators, Studia Math. 51 (1974), 265-267.

A. Pietsch, Operator Ideals, VEB Deutscher Verlag der Wissenschaften, Berlin, 1978.

A. Pietsch, Weyl numbers and eigenvalues of operators in Banach spaces, Math. Ann. 247 (1980), no. 2, 149-168.

A. Pietsch, Eigenvalues and s-numbers, Cambridge University Press, New York, NY, USA, 1986.

J. R. Retherford, Applications of Banach ideals of operators, Bull. Amer. Math. Soc. 81 (1975), no. 6, 978-1012.

B. E. Rhoades, Operators of $A-p$ type, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. (8) 59 (1975), no. 3-4, 238-241 (1976).

B. E. Rhoades, Some sequence spaces which include the $l^{p}$ spaces, Tamkang J. Math. 10 (1979), no. 2, 263-267.

J. S. Shiue, On the Cesaro sequence spaces, Tamkang J. Math. 1 (1970), no. 1, 19-25.

C. S. Wang, On Norlund sequence spaces, Tamkang J. Math. 9 (1978), no. 2, 269-274.