Hausdorff Matrices as Bounded Operators over /
نویسنده
چکیده
A necessary and sufficient condition is obtained for an arbitrary Hausdorff matrix to belong to B(l). It is then shown that every conservative quasi-Hausdorff matrix is of type M. Let (H, u) denote the Hausdorff method with generating sequence ¡x = {/L,}, / = {{xn} \2„\xn\ < oo}, B(l) the algebra of bounded linear operators on /. A necessary and sufficient condition is obtained for an arbitrary Hausdorff method to belong to B(l). It is then shown that every conservative quasi-Hausdorff matrix is of type M. An infinite matrix is called triangular if it has only zeros above the main diagonal. Let B(c) denote the algebra of bounded linear operators in c, the space of convergent sequences. Lemma. Let A be a triangular matrix satisfying: (1) A GB(l), (2) t* = 2^_0|ûnfc| is monotone increasing in n, (3) lim„ *n Muts, where t„ = 2*_0a„/f Then A G B(c). Proof. Condition (1) is equivalent to sup¿ ^=k\ank\ < oo, which implies 2^_fc|anA| < M for every N > k, where M is independent of N and k. Summing k over [0, N] yields 2*_o2"_*|a,,*| < M(N + 1). Interchanging the order of summation gives 2 t*/(N+l) oo, a contradiction. Therefore ||^4||c is finite. Condition (1) implies A has zero column limits and (3) assures the existence of the limit of the row sums. Therefore A G B(c). Received by the editors March 21, 1979. AMS (MOS) subject classifications (1970). Primary 40G05; Secondary 40C05.
منابع مشابه
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This paper is concerned with the problem of finding a lower bound for certain matrix operators such as Hausdorff and Hilbert matrices on sequence spaces lp(w) and Lorentz sequence spaces d(w,p), which is recently considered in [7,8], similar to [13] considered by J. Pecaric, I. Peric and R. Roki. Also, this study is an extension of some works which are studied before in [1,2,7,8].
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تاریخ انتشار 2010