proceedings of the american mathematical society Volume 121, Number 4, August 1994 A CLASS OF MAPS IN AN ALGEBRA WITH INDEFINITE METRIC ANGELO B. MINGARELLI (Communicated by Palle E. T. Jorgensen) Dedicated to the memory of David Conibear Abstract. We study a class of hermitian maps on an algebra endowed with an indefinite inner product. We show that, in particular, the existence of a non- real eigenvalue is incompatible with the existence of a real eigenvalue having a right-invertible eigenvector. It also follows that for this class of maps the exis- tence of an appropriate extremal for an indefinite Rayleigh quotient implies the nonexistence of nonreal eigenvalues. These results are intended to complement the Perron-Fröbenius and Kreln-Rutman theorems, and we conclude the paper by describing applications to ordinary and partial differential equations and to tridiagonal matrices. 1. Introduction Let (K, [ , ]) be an indefinite inner product algebra over C with identity, e ; that is, K is an algebra and [ , ] : K x K -> C is a sesquilinear hermitian (symmetric) form which is indefinite on K (i.e., there exists x, y e K with [x, x] > 0 and [y, y] < 0). Thus K is, in particular, an indefinite inner product space [4, p. 3]. In the sequel A: 2(A) ç K -> K, 21(A) ^ 0, is a nontrivial hermitian map, i.e., [Af,g] = [f,Ag], f,g£2(A), and y/ ^ 0 denotes a right-invertible element in K. Note that A is not nec- essarily linear (since 2(A) may be a discrete set in an infinite-dimensional space). Furthermore, / is the identity map and "multiplication by tp" is de- fined by (4>I)(u) = tpu, u £ K. We always assume that the inner product [ , ] is indefinite on 2(A). The map A is homogeneous if, for every m £ C, m/0, A(mu) = mAu, u e 2(A). We study maps A on K with the property that, whenever the display is real, (1.1) [(A-(Ay/)y/~xI)u,u]>0 Received by the editors November 9, 1992; presented during a special session talk at the confer- ence Operator Theory and Boundary Eigenvalue Problems, Technical University of Vienna, Vienna, Austria, August 1993. 1991 Mathematics Subject Classification. Primary 46C20, 47B50; Secondary 34C10, 34B24, 35P05, 39A70, 15A18. Key words and phrases. Krein space, hermitian maps, Perron-Fröbenius. Partially funded by a grant from N.S.E.R.C. Canada. ©1994 American Mathematical Society 0002-9939/94 $1.00+ $.25 per page 1177 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use