Bounds on characteristic values
Edward W. Barankin · Bulletin of the American Mathematical Society · 1948
In a previous paper [l] 1 the author obtained several upper bounds for the characteristic roots of a finite matrix.The purpose of this note is to establish the corresponding results for characteristic values of certain linear transformations in certain infinite spaces.1. Infinite matrices.We consider a non-null infinite matrix, A = (a s t), s, 2 = 1, 2, • • -, of complex numbers.(The present considerations pertain also to the finite case when a 5 * = 0 for s>n and t>n.)We shall say that A is absolutely summable (abbreviated a.s.) when the double series ^8,t \ a 9t \ converges.In general, if ^s,t | a 8t \ p converges for a positive number p, A will be said to be absolutely summable (p).If A is a.s.(£), then X^l a *«| p converges for all s, X^«| a st\ v converges for all t, and si«i.i'-zz:i«..i'-E5:i«..i'-