ON THE FINE SPECTRA OF THE DIFFERENCE OPERATOR Δ OVER THE SEQUENCE SPACE lp, (1≤p<∞)

By A. Akhmedov, Feyzi Basar. 2006

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In the present paper, the norm of the bounded linear operator A acting on the sequence space £p has been found and the fine spectrum with respect to Goldberg's classification of the difference operator A over the sequence space t p has been determined, where 1 < p < oo. 1. Preliminaries, background and notation By B(X), we denote the set of all bounded linear operators on the Banach space X into itself. The space of all bounded linear functionals on a normed space X is denoted by X*. If X is a Banach space and T G B(X) then the adjoint T* of T is a bounded linear operator on the dual X* defined by (T*f){x) = f ( T x ) for all f X with domain V(T) C X. With T, we associate the operator Ta = T al, where a is a complex number and I is the identity operator on V(T). If Ta has an inverse, which is linear, we denote it by T", that is T~ = { T a i r \ and call it the resolvent operator of T. The name resolvent is appropriate, since T" helps to solve the equation Tax = y. Thus, x = T~y provided T~ exists. More important, the investigation of properties of T~ will be basic for an understanding of the The authors wish to express their thanks to TUBITAK-BAYG for supplying the financial support by PC-B Programme during the preparation of the present work in winter 2004 and their common project. 2000 Mathematics Subject Classification: Primary: 47A10, Secondary: 47B37.

Subfields: Applied Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/d43e48fae1d18d4f471bce418037f42534a397b8

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