## Classification of Ideals in Banach Spaces

Wanjala Patrick Makila

Department of Mathematics, Masinde Muliro University of Science and Technology, P.O.Box 190-50100, Kakamega, Kenya.

Ojiema Michael Onyango *

Department of Mathematics, Masinde Muliro University of Science and Technology, P.O.Box 190-50100, Kakamega, Kenya.

Simiyu Achiles Nyongesa

Department of Mathematics, Masinde Muliro University of Science and Technology, P.O.Box 190-50100, Kakamega, Kenya.

*Author to whom correspondence should be addressed.

### Abstract

Let an operator T belong to an operator ideal J, then for any operators A and B which can be composed with T as BTA then BTA \(\in\) J. Indeed, J contains the class of finite rank Banach Space operators. Now given *L(X, Y )*. Then *J(X, Y )* \(\subseteq\) *L(X, Y )* such that *J(X Y )* = *{T : X \(\gets\) Y : T \(\subseteq\) }.* Thus an operator ideal is a subclass J of L containing every identity operator acting on a one-dimensional Banach space such that: *S + T \(\in\) J(X, Y )* where *S, T \(\in\) J(X, Y )*. If W,Z,X, Y \(\in\) \(\mathbb{K}\),*A \(\in\) L(W,X),B* *\(\in\) L(Y,Z)* then *BTA \(\in\) J(W,Z)* whenever *T \(\in\) J(X; Y) *These properties compare very well with the algebraic notion of ideals in Banach Algebras within whose classes lie compact operators, weakly compact operators, finitely strictly regular operators, completely continuous operators, strictly singular operators among others. Thus, the aim of this paper is to characterize the various classes of ideals in Banach spaces. Special attention is given to the characteristics involving the ideal properties, the metric approximation properties, the hereditary properties in relation to the ideal extensions in the Hahn-Banach space, projection and embedments in the biduals of the Banach Space.

Keywords: Operator ideals, Banach space ideals

**How to Cite**

*Asian Research Journal of Mathematics*,

*19*(11), 123–141. https://doi.org/10.9734/arjom/2023/v19i11760

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### References

Srivastava PD, Amit Maji. Some Classes of Operator Ideal. International Journal of Pure and Applied

Mathematics. 2013;83(5):731-740.

Rhoades BE. Operators of A-p type, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. 1975;8(59):238-

Alfsen R, E ros K. Structures in real Banach spaces. Ann.of Math. 1972;96:98-128.

Smith RR, Ward JD. M-ideal structure in Banach algebras. J. Functional Analysis. 1978;27(3):337-349.

Godefroy G, Kalton N, Saphar P. Unconditional ideals in Banach spaces. Studia Mathematica.

;1(104):13-59.

Godun BV. Equivalent Norms on nonre exive Banach Spaces. In Doklady Akademii Nauk. 1982;265:20-23. Russian Academy of Sciences.

Rao TSSRK. On Intersection of Ideals in Banach spaces Indian Statistical Institute. Bangalore Centre.

;1-7.

Rao TSSRK. On almost isometric ideals in Banach spaces. Monatsh Math. 2016;181:169-176.

Abrahamsen TA, Lima A, Lima V. Unconditional ideals of nite rank operators. Czechoslovak Mathematical

Journal. 2008;58(4):1257-1278.

Abrahamsen TA, Nygaard O. On (lambda) -strict ideals in Banach spaces. Bulletin of the Australian Mathematical Society. 2011;83(2):231-240.

Casazza PG, Jarchow H. Self-induced compactness in Banach spaces. Proc. Roy. Soc. 1973;126:355-362.

Godefroy G, Kalton NJ, Saphar PD. Duality in spaces of operators and smooth norms on Banach spaces.

Illinois J. Math.m. 1988;32:672-695.

Haro JCC, Lai HC. Multipliers in continuous vector-valued function spaces. Bull. Austral. Math. Soc.

;46:199{204.

John K. u-ideals of factorable operators. Czech.Math.J. 1999;49:616-667.

Johnson J. Remarks on Banach spaces of compact operators. Journal of functional analysis. 1979;32:302-311.

Lima V, Lima A, Nygaard O. On the compact approximation property. Studia math. 2004;160:185-200.

Harmad P, Werner D, Werner W. M-ideals in Banach spaces and Banach algebras. springer-Verlag; 1993.

Asvald L. The metric approximation property, norm-one projections and intersection properties of ball.

Israel.J.Math. 1993;84:451-475.

Chong Man Cho, Eun Joo Lee. Hereditary properties of certain ideals of compact operators. Bull. Korean

Math .Soc. 2004;3(41):457-464

Abrahamsen AT, Olav N. on (lambda) -strict ideals in Banach spaces. Bulletin of the Australian Mathematical

society. 2011;83:231-240.

John K. On results of J. Johnson. Czech. Math. J. 1973;45(120):235-240.