Convergence Fields of Row-finite and Row-infinite Toeplitz Transformations

During a recent conversation, R . P. Agnew suggested a determination of the validity of the proposition that row-infinite Toeplitz transformations are more powerful than row-finite transformations . Before this proposition is examined, it is necessary to assign a precise meaning to it. Corresponding to every sequence s a regular rowfinite Toeplitz transformation A can be constructed such that the transform A s converges to a finite limit . In order to be of interest, the proposition must therefore be interpreted in terms of convergence fields of individual transformations . Also, because every row-infinite regular Toeplitz matrix is the sum of two Toeplitz matrices A and B, where A is regular and row-finite and B has the norm zero, the convergence field in the space of bounded sequences of every regular Toeplitz transformation coincides with that of a row-finite regular transformation ; in other words, the problem is trivial except in its reference to unbounded sequences .