Showing papers in "Mathematische Zeitschrift in 1962"

On the zeros of functions with finite Dirichlet integral and some related function spaces

Abstract: (A set {z~} is called a set o/ uniqueness for the class D if there is no /~ D ( / ~ 0 ) vanishing at all these points.) He noted that a n earlier result of LOKm [9] was incorrect. (The 1955 Ergebnisse tract of W:TX:CH [13] quotes only the LOKK: result.) CARLESON also pointed out that if the {z,} all lie on one radius, then the necessary and sufficient condition for the existence of an lED with these zeros is: Y0-1z l)< oo. Indeed, taking the radius to be the unit interval and letting B(z) be the Blaschke product formed from these points, then [(z) = ( t z)~B (z) has a bounded derivative and so is in D. Our contribution is to show that in (A) one may take e = 0 and that in (B) the expression ( log ( t [z~l)) -* may be replaced by any function tending to zero. Our methods are different from those of CARLESON and make greater use of the fact that D is a Hilbert space. We prove similar theorems

Criteria for the reality of matrix eigenvalues

Abstract: In this note we are, essentially, concerned with generalizations of the (known) fact tha t an n • matr ix with n linearly independent eigenvectors all corresponding to real eigenvalues is similar to a hermitian matrix, and can consequently be transformed into its conjugate transpose by a positive definite hermitian similarity. We first establish, fo r ' any positive integer m, an analogous necessary and sufficient condition tha t a given square complex matr ix A should have a set of real eigenvalues, not necessarily all distinct, to which therecorrespond at least m linearly,independent eigenvectors; this of course implies a corresponding result about pure imaginary eigenvalues. We also obtain an analogous result concerning eigenvalues of modulus unity: As a simple application of our more general results, we establish, in Theorem 4, the reality of the eigenvalues of a certain rather special type of matrix. Throughout, we shall use A = (aij) to denote an arbi t rary n • n complex matrix, and A* will denote the transposed conjugate matr ix; we denote the rank of A by r(A).