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\(\sigma\)-core theorems for real bounded sequences. (English) Zbl 0951.40002

Let \(\sigma:\mathbb{N}\to \mathbb{N}\) be a one-to-one mapping such that \(\sigma^q(n)\neq n\) for all \(q, n= 1,2,\dots\). A continuous linear functional \(\varphi\) on \(m\), the Banach space of all real bounded sequences, is said to be a \(\sigma\)-mean if (a) \(x_n\geq 0\) for all \(n\in\mathbb{N}\) implies \(\varphi(x)\geq 0\), (b) \(\varphi(e)= 1\), where \(e:= (1,1,\dots)\) and (c) \(\varphi((x_{\sigma(n)}))= \varphi(x)\) for all \(x\in m\). Let \(V_\sigma\) be the set of all bounded sequences all of whose \(\sigma\)-means are equal and let \(\sigma\)-\(\lim x\) denote the common value of all \(\sigma\)-means at \(x\in V_\sigma\). Then [see R. A. Raimi, Duke Math. J. 30, 81-94 (1963; Zbl 0125.03201)] \(V_\sigma= \{x\in m\mid\lim_q t_{qn}(x)=\sigma\)-\(\lim x\) uniformly in \(n\}\), where \(t_{qn}(x):= (x_n+ (Tx)_n+\cdots+ (T^qx)_n)/(q+ 1)\) for \(q= 0,1,\dots, n= 1,2,\dots\), and \(Tx:= (x_{\sigma(n)})\). A real infinite matrix \(A\) is said to be (1) \(\sigma\)-regular if \(A: c\to V_\sigma\) and \(\sigma\)-\(\lim Ax= \lim x\) for all \(x\in c\), (2) \(V_\sigma\)-regular if \(A: V_\sigma\to V_\sigma\) and \(\sigma\)-\(\lim Ax= \sigma\)-\(\lim x\) for all \(x\in V_\sigma\), and (3) \(\sigma\)-uniformly positive if \(\lim_q \sum_k{1\over q+1} \sum^q_{j= 0} a^-_{\sigma^j(n)k}= 0\) uniformly in \(n\) (here \(a^-_{\sigma^j(n)k}:= \max\{- a_{\sigma^j(n)k}, 0\}\)). Let \(L(x):= \limsup_n x_n\), \(V(x):= \sup_n \limsup_q t_{qn}(x)\), and \(v(x):= \inf_n \liminf_q t_{qn}(x)\) for \(x\in m\). For matrix transformations \(A\) and \(B\), the authors discuss the inequalities \(V(Ax)\leq L(Bx)\), \(V(Ax)\leq V(Bx)\) and \(V(Ax)\leq v(Bx)\) and generalize some results of S. Mishra, B. Satapathy and N. Rath [J. Indian Math. Soc., New Ser. 60, No. 1-4, 151-158 (1994; Zbl 0882.40004)]. A main result of the paper is the following Theorem 2: Let \(B\) be a normal matrix and \(A\) any matrix. In order that, whenever \(Bx\) is bounded, \(Ax\) exists, is bounded and satisfies \(V(Ax)\leq V(Bx)\) for all \(x\in m\), it is necessary and sufficient that the matrix \(C:= AB^{-1}\) exists, is \(\sigma\)-regular, \(V_\sigma\)-regular and \(\sigma\)-uniformly positive.

MSC:

40C05 Matrix methods for summability
46A45 Sequence spaces (including Köthe sequence spaces)
40J05 Summability in abstract structures
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