On conservative and coercive SM-methods (Q2744620)

Let \({\mathcal V}\) be a space of double sequences \(y= (y_{mn})\) converging with respect to a linear notion of convergence \({\mathcal V}\)-\(\lim: {\mathcal V}\to\mathbb{K}\) and let \(B= (b_{mnk})\) be a 3-dimensional matrix. The summability method \(B\) induced by the summability domain NEWLINE\[NEWLINE{\mathcal V}_B:= \Biggl\{z= (z_k)\;\Biggl|\;Bz:= \Biggl(\sum_k b_{mnk} z_k\Biggr)_{m,n}\text{ exists and belongs to }{\mathcal V}\Biggr\}NEWLINE\]NEWLINE and the limit functional \({\mathcal V}\)-\(\lim_B:{\mathcal V}_B\to \mathbb{K}\), \(z\mapsto{\mathcal V}\)-\(\lim Bz\) is called a \({\mathcal V}\)-SM-method. A \({\mathcal V}\)-SM-method \(B\) is said to be conservative and coercive if \({\mathcal V}_B\) includes \(c\), the space of all convergent sequences, and \(\ell^\infty\), the space of all bounded sequences, respectively. Let \({\mathcal C}_e\) denote the space of all double sequences \(y\) satisfying the statement \(\lim_n\varlimsup_m|y_{mn}- a|= 0\) for some number \(a:={\mathcal C}_e\)-\(\lim y\). The subspace of the double sequences \(y\in{\mathcal C}_e\) with \((y_{mn})_m\in \ell^\infty\) for all \(n\in\mathbb{N}\) is denoted by \({\mathcal C}_{be}\). Using gliding hump arguments, the author gives necessary and sufficient conditions for \({\mathcal C}_e\)-SM-methods and \({\mathcal C}_{be}\)-SM-methods to be conservative or coercive. The main result is Theorem 3.1: A \({\mathcal C}_e\)-SM-method \(B\) is coercive if and only if each of the following conditions holds: (i) the limit \(b_k:={\mathcal C}_e\)-\(\lim_{m,n} b_{mnk}\) exists for all \(k\in\mathbb{N}\), (ii) \(\sum_k|b_{mnk}|< \infty\) for all \(m,n\in\mathbb{N}\), (iii) there exists \(N\in\mathbb{N}\) such that \(\sup_{n\geq N}\varlimsup_m \sum_k|b_{mnk}|< \infty\), and (iv) \(\lim_n\varlimsup_m \sum_k|b_{mnk}- b_k|= 0\). Under these circumstances, \(\sum_k|b_k|< \infty\) and \({\mathcal C}_e\)-\(\lim B x= \sum_k b_k x_k\) for all \(x\in\ell^\infty\).