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The Hausdorff measure of noncompactness of operators on the matrix domains of triangles in the spaces of strongly C 1 summable and bounded sequences. (English) Zbl 1244.47046

Let ω be the space of all complex sequences x=x k k=1 , A=a nk n,k=1 be an infinite matrix of complex numbers and X a subset of ω. The set

X A =xω:Ax= k=1 a nk x k X

is called the matrix domain of A in X. Given any sets X and Y in ω, let X,Y denote the class of all matrices A such that XY A . Let T=t nk n,k=1 be a triangle, that is, t nk =0 for k>n and t nn 0 n=1,2,, let eω be the sequence with e k =1 for all k and 1p<. The sets of strongly C 1 -summable and bounded sequences

w 0 p =xω:lim n 1 n k=1 n x k p =0,
w p =xω:x-ξ·ew 0 p forsomecomplexnumberξ

and

w p =xω:sup n1 n k=1 n x k p <

were defined and studied by I. J. Maddox [“On Kuttner’s theorem”, J. Lond. Math. Soc. 43, 285–290 (1968; Zbl 0155.38802)].

In the paper under review, the authors apply the Hausdorff measure of noncompactness to characterize the classes of compact operators given by infinite matrices AX,Y, where X is one of spaces w 0 p T ,w p T or w p T and Y is the space c 0 of null sequences or the space c of convergent sequences. Moreover, they give sufficient conditions for the compactness of operators AX,Y when X is again one of spaces w 0 p T ,w p T or w p T and the final Y is the space l of bounded sequences.

MSC:
47H08Measures of noncompactness and condensing mappings, K-set contractions, etc.
40C05Matrix methods in summability
40J05Summability in abstract structures
References:
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