The spectra of weighted mean operators on bν0

B. E. Rhoades
1992 Journal of the Australian Mathematical Society  
In a series of papers, the author has previously investigated the spectra and fine spectra for weighted mean matrices, considered as bounded operators over various sequence spaces. This paper examines the spectra of weighted mean matrices as operators over bv 0 , the space of null sequences of bounded variation. 1991 Mathematics subject classification (Amer. Math. Soc.) 45 F 05, 47 A 10. Let x be a sequence, c Q the spaces of null sequences. Then bv 0 :-bv n c 0 , where bv = {x\L k \x k -x k _
more » ... {x\L k \x k -x k _ x \ < oo} . From [6, formula 119] for example, we have a matrix A: bv 0 -* bv Q if and only if A has null columns and (1) Y. U nk-a n-\,k < OO. A weighted mean matrix is a lower triangular matrix A -(a nk ) with a nk = Pkl p n ' w h e r e P 0 >0, P n >0 for n>0 and P n := £ L o^ • B(bv 0 ) will denote the set of bounded linear operators on bv 0 , and a (A) will denote the spectrum of A for A e B(bv 0 ). The results of this paper are similar to those obtained in [1], but the proofs are different because of the bv 0 norm. THEOREM 1. Let A be a weighted mean matrix with P n -> oo. Then
doi:10.1017/s1446788700034388 fatcat:aay74r5isnglrbjiaut3kmwymi