On Compact Operators whose Norms are Eigenvalues

Abstract

The class of compact operators is fundamental in operato_Lllle~+Y.• Char-acterization of compact operators acting on different sp~ces has been fas-cinating to many Mathernaticians., Many properties such as bounded¬ness, completeness, compactness on finite-dimensional Banach spaces are . - consequences of the Heine-Borel theorem, which implies that the closed unit ball in a finite-dimensional Banach space is compact, which may --, not apply to infinite-dimensional spaces. Thus, the problem of compact-

ness of operators is still open. Lin, established that the norm of a linear compact operator •r is an eigenvalue for the operator if it satisfies the Daugavet equation. So the question whether every linear compact op¬erator can be approximated by linear compact operator whose norms are eigenvalues has not been fully investigated. The objectives of this study are to: investigate completeness, establish the necessary and suffi¬cient conditions for invertibility and investigate convergence of compact operators whose norms are eigenvalues, Investigation on completeness involved the use of Theorem of completeness and Gram-Schmidt orthog¬onalization. Open mapping Theorem and direct sum decomposition have been used to study invertibility. The study of convergence employed ten¬sor products and Riesz representation Theorem. In this study, we have established that if {Tn}nEN is an orthonormal sequence of compact oper¬ators whose norms are eigenvalues, then {Tn}nEN is complete if and only if (T, Tn) = 0, \:/ n E N implies T = 0. Moreover, the projective (re-spectively, injective) tensor product of Tn and Sn tends to the projective (respectively, injective) tensor product of T and S. The results may be useful in many disciplines of science and engineering.