Invariant linear functionals and infinite matrices (Q1097420)

Let A be a sequence of matrices A \(i=(a_{nk}(i))\) of real numbers. A real sequence \(x=(x_ k)\) is said to be summable by the method (A) to the number s (denoted by \(x\to s(A))\) if \(\lim_{n}\sum_{k}a_{nk}(i)x_ k=s\) uniformly in i. The method (A) is called regular if \(X_ k\to s\Rightarrow x\to s(A).\) A regular method (A) is called almost positive if \(\lim_{n}\sum_{k}a_{nk}(i)\) \(-=0\) uniformly in i, where \(a_{nk}(i)\) \(-=\max (-a_{nk}(i),0)\). The methods (A) and (B) are said to be absolutely equivalent on a set u, if for all \(x\in\) u \(\lim_{n}\sum_{k}(a_{nk}(i)-b_{nk}(i))x_ k=0\) uniformly in i. A method (A) is called translative on u if for all \(x\in u\lim_{n}\sum_{k}(a_{nk}(i)-a_{n,k+1}(i))x_ k=0\) uniformly in i. The methods (A) and (B) are said to be consistent for x if \(x\to s(A)\) and \(x\to s'(B)\) implies \(s=s!\) Let m be the linear space of real bounded sequences and let w be the maximal set of almost positive, regular methods such that for any pair of methods (A) and (B) the methods (AB) and (BA) are absolutely equivalent. For \(x\in m\) let \(V(x)=\inf_{A\in W}\nu_ A(x)\), where \(\nu_ A:m\to {\mathbb{R}}\) is defined by \(\nu_ A(x)=\overline{\lim}_{n}\sup_{i}\sum_{k}a_{nk}(i)x_ k.\) The authors prove the following: (1) \(V(Ax)=V(x)\) for all \(x\in m\) and \(A\in W\). (2) In order that \(\phi\) in the algebraic dual of m satisfies \(\phi (x)\leq \overline{\lim}_{p}\sup_{n}(p+1)^{- 1}\sum^{p}_{i=0}x_{n+i}\) and \(\phi (Ax)=\phi (x)\), \(A\in W\) it is necessary and sufficient that \(\phi\) (x)\(\leq V(x)\), \(x\in m\). (3) If W contains a translative method on m, then V generates Banach limits which are invariant under W. (4) W consists of consistent methods for bounded sequences.