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Measure of noncompactness of matrix operators on some difference sequence spaces of weighted means. (English) Zbl 1247.47009

Summary: For a sequence x=(x k ), we denote the difference sequence by Δx=(x k -x k-1 ). Let u=(u k ) k=0 and v=(v k ) k=0 be sequences of real numbers such that u k 0, v k 0 for all k. The difference sequence spaces of weighted means λ(u,v,Δ) are defined as λ(u,v,Δ)={x=(x k ):W(x)λ}, where λ is either of c,c 0 , and the matrix W=(w nk ) is defined by

w nk =u n (v k -v k+1 )ifk<n,u n v n ifk=n,0ifk>n·

In this paper, we establish some identities or estimates for the operator norms and the Hausdorff measures of noncompactness of certain matrix operators on λ(u,v,Δ). Further, we characterize some classes of compact operators on these spaces by using the Hausdorff measure of noncompactness.

MSC:
47B37Operators on special spaces (weighted shifts, operators on sequence spaces, etc.)
47B07Operators defined by compactness properties
47H08Measures of noncompactness and condensing mappings, K-set contractions, etc.
40C05Matrix methods in summability
46B45Banach sequence spaces
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