On some Lambda-Pascal sequence spaces and compact operators. (English) Zbl 1502.46015

Yaying, Taja; Başar, Feyzi

Rocky Mt. J. Math. 52, No. 3, 1089-1103 (2022).

Summary: We introduce Lambda-Pascal sequence spaces \(\ell_q (G), c_0 (G), c(G)\) and \(\ell_\infty (G)\) generated by the matrix \(G\) which is obtained by the product of Pascal matrix and \(\Lambda\)-matrix. It is proved that the Lambda-Pascal sequence spaces \(\ell_q (G), c_0 (G), c(G)\) and \(\ell_\infty (G)\) are BK-spaces and linearly isomorphic to \(\ell_q, c_0, c\) and \(\ell_\infty\), respectively. We construct Schauder bases and obtain \(\alpha\)-, \(\beta\)- and \(\gamma\)-duals of the new spaces. We state and prove characterization theorems related to matrix transformation from the space \(\ell_q (G)\) to the spaces \(\ell_\infty, c\) and \(c_0\). Finally, we determine necessary and sufficient conditions for a matrix operator to be compact from the space \(c_0 (G)\) to any one of the spaces \(\ell_\infty, c, c_0\) or \(\ell_1\).

MSC:

46B45	Banach sequence spaces
46A45	Sequence spaces (including Köthe sequence spaces)
40C05	Matrix methods for summability
47B07	Linear operators defined by compactness properties
47B37	Linear operators on special spaces (weighted shifts, operators on sequence spaces, etc.)

Keywords:

sequence space; Pascal matrix; Schauder basis; Köthe duals; matrix mappings; compact operator

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