Uniform Toeplitz matrices (Q808345)

For functions \(f_ k: T\to X\), where T is an infinite set and X is a Banach space, one defines a sequence of functions \(f_ n\Rightarrow f\) uniformly on T. Let \(A=(A_{nk})\) be a matrix of operators where each \(A_{nk}\in B(X)\). A is a uniform Toeplitz matrix if A takes uniformly convergent sequences to uniformly convergent sequences which converge to the same limit. A Silverman-Toeplitz type theorem is proved which characterizes the uniform Toeplitz matrices. Let C be the Cesàro matrix and \(D=(d_{nk})\) be the Kuttner-Maddox matrix where \(D_{11}=1\) and for \(n>1\), \(d_{nk}=(1/2^{n-1})\) for \(2^{n-1}\leq k<2^ n\) and 0 otherwise. A sequence of functions \((s_ k)\) has uniform strong slow oscillation iff \(s_ n-s_ k\Rightarrow 0\) whenever \(n>k\) and \(n/k=O(1)\). The relationship \(f_ n\Rightarrow f(C)\) implies \(f_ n\Rightarrow f(D)\) is demonstrated along with some Tauberian conditions relating C and D. It is also shown that uniformly strong slow oscillation along with \(s_ k\Rightarrow f(D)\) imply \(s_ k\Rightarrow f\).