Core equality theorems for real bounded sequences (Q1574268)

Let \(m,c,c_0\) be the linear spaces of real bounded, convergent and null sequences \(x=\{x_n\}\) respectively. Let \(l(x)\) and \(L(x)\) be defined on \(m\) by \(l(x)=\liminf x_n\) and \(L(x)=\limsup x_n\). The famous Knopp theorem determines a class of regular matrices \(A\) for which \(L(Ax)\leq L(x)\) for all \(x \in m\), i.e. \(\text{K-core}\{A(x)\} \subseteq \text{K-core} \{x\}\) where \(\text{K-core}\{x\}\) denotes Knopp's core of \(x\in m\) defined by the closed interval \([l(x),L(x)]\). \(\beta\text{-core}\{x\}\) denote the Banach core of \(x\in m\). The author has proved necessary and sufficient conditions for \(\text{K-core} \{A(x)\}= \beta\text{-core}\{x\}\). Further he has proved two other such theorems establishing the core equality theorems for real bounded sequences.