## $$\sigma$$-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

### Citations:

Zbl 0125.03201; Zbl 0882.40004