On some inequalities for multiplicative matrices (Q875651)

This paper is devoted to some inequalities for multiplicative matrices, and their characterizations. Theorem 1: Let \(L(x)= \limsup x\) for \(x\in m=\) \{all bounded sequences\}. Then \(LA(x)\leq rL(x)\) \(\forall x\in m\Leftrightarrow A\in (c,c)\) and \(\lim_n\,\sum_k|a_{nk}|= r\) where \(r\geq 0\). Theorem 2: Let \(t_{pk}(x)= {x_k+ Tx_k+\dots+ T^p x_k\over p+1}\), \(t_{-1,n}(x)= 0\) where \((Tx_k)= (x_{\sigma(k)})\). Let \(q_\sigma(x)= \limsup_p\,\sup_n t_{pn}(x)\). Then \[ q_\sigma(Ax)\leq r\,L(x)\;\forall x\in m\Leftrightarrow\lim_p\,\sup_n \sum_k{1\over p+1} \Biggl|\sum^p_{i=0} a_{\sigma^i(n),k}\Biggr|= r. \] Theorem 3: \(q_\sigma(Ax)\leq rq_\mu(x)\) \(\forall x\in m\Leftrightarrow A\in (V_\mu, V_\sigma)\) and the condition of Theorem 2.