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Generalized difference sequence spaces on seminormed space defined by Orlicz functions. (English) Zbl 1174.40003

Let \(w(X)\), \(\ell _{\infty }(X)\), \(c(X)\) and \(c_{0}(X)\) be the classes of all, bounded, convergent and to null convergent \(X\)-valued sequences, where \(X\) is a seminormed space. Let \(M\) be an Orlicz function and \(X\) be a seminormed space with the seminorm\(q\). The authors consider the sequence space \[ \ell _{M}(\Delta ^{m},p,q,s)=\Biggl \{x\in w(X):\sum \limits _{k=1}^{\infty }k^{-s}\biggl [M\Bigl (q\Bigl (\frac {\Delta ^{m}x_{k}}{\rho }\Bigr )\Bigr )\biggr ]^{p_{k}}\!<\infty ,\;s\geq 0, \rho >0\Biggr \}, \] where \(\Delta ^{m}\) is the operator of the \(m\)th order difference for a nonnegative fixed integer \(m\) and \(p=(p_{k})\) is a sequence of strictly positive real numbers. The different algebraic and topological properties, like solidness, symmetricity, monotonicity and convergence freeness, are studied in the paper.

MSC:

40C05 Matrix methods for summability
46A45 Sequence spaces (including Köthe sequence spaces)
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