Some structural properties of Hausdorff matrices

B. E. Rhoades
1959 Bulletin of the American Mathematical Society  
Definitions. Let A ~{a n k) denote an infinite matrix. A is called conservative if A has finite norm, ak = limn-«> cbnk exists for each fe, and lim«^oo ^kdnk exists. A is called multiplicative if A is conservative and ajfc = 0 for each k. s denotes the space of sequences, m the subspace of bounded sequences, and c the subspace of convergent sequences. E\ is the field of complex numbers and £«> the set of sequences, each of which possesses only a finite number of nonzero terms. Let x be a fixed
more » ... equence. Then c®x = {y+x\yÇzc}. Let H=(hnk) denote a Hausdorff matrix generated by a sequence ju. I shall use (H, /x) to denote the convergence domain of H, iJ M to denote the matrix, and H^p to denote the method. A matrix A = (a n k) is said to be of property P, displacement m (written c m A is of property P) if, for all k^m y a n k possesses property P. A corridor matrix is a matrix with the property that there exists a positive integer r such that a W fc = 0 for all n and k with \n -k\ >r. The smallest such r denotes the width of A.
doi:10.1090/s0002-9904-1959-10256-1 fatcat:ptpzt7clcjdfnlgjh4pnjzmnke