Invariant means and some matrix transformations (Q5917977)

Let \(\sigma\) be an injection of the set of positive integers into itself having no finite orbits, and \(T\) be the operator defined on \({\mathcal L}_\infty\), the Banach space of bounded sequences, by \(Tx(n)= x(\sigma n)\). Let a linear functional \({\mathcal L}\) be called a \(\sigma\)-mean if \({\mathcal L} (x) ={\mathcal L} (Tx)\) for all \(x\) in \({\mathcal L}_\infty\). Let \(V_\sigma\) be the set of \(\sigma\)-convergence sequences, i.e. the set of bounded sequences all of whose invariant means are equal. In this papers the authors proved a theorem which proves that \(V_\sigma\) is a Banach space under the norm \(\|x \|_\sigma =\sup_{mn} |t_{mn} (x)|\), where \(t_{mn}(x) ={1\over m+1} (x_n+Tx_n+ \cdots+ T^mx_n)\). Also they proved a theorem which characterizes the matrices of type \((l_1,V_\sigma)\) where \(l_1=\{x: \sum^\infty_{k=0} |x_k |< \infty\}\). Besides, they have defined \({\mathcal L}_\sigma\), the space of absolutely \(\sigma\)-convergence, and \(m_\sigma\), the space of absolutely \(\sigma\)-boundedness, and investigated some properties of their new sequence spaces derived from the concept of invariant means. These spaces are further extended to \({\mathcal L}_\sigma(p)\) and \(m_\sigma(p)\) for a sequence \(p=(p_m)\) of positive reals with \(\sup_mp_m<\infty\).