A Quadratic Eigenvalue Problem

Let P, Q be compact selfadjoint operators in a Hilbert space. It is proven that the characteristic and associated vectors of the quadratic eigenvalue problem, x=\Px + (\¡X)Qx, form a Riesz basis for the cartesian product of the closure of the range of P and the closure of the range of Q. 1. Investigations in the theory of hydrodynamic stability (cf. [3, Chapter X] ; [8]) lead to the search for expansion theorems associated with the following characteristic value problem. Let £, Q be compact selfadjoint operators in a Hilbert space H and for x £ H let A(X)x = XPx + (l¡X)Qx. a, the spectrum of A(X), is defined as the set of all complex numbers X for which I—A(X) does not have a bounded inverse defined on all of H. p, the resolvent of A(X), is the complement of a in the complex plane. A number X^O is a characteristic value of A(X) if the equation (1) x = A(X0)x = X0Px + (l/X0)Qx has nonzero solutions. Any such x is called a characteristic vector of A(X) corresponding to X=X0. Since any nonzero number on the imaginary axis is in p, it follows that a is at most countable (cf. [5, p. 21]). All nonzero points in o are isolated characteristic values whose only possible limit points are 0 and oo. Thus, via the transformation X~*aX, a>0, it may be assumed without loss of generality that ±1 e />. Following [4] the characteristic values are poles of the function (I-A(X))"1, which is analytic on p. Presented to the Society, January 17, 1972 under the title Bases associated with a quadratic eigenvalue problem; received by the editors September 25, 1972. AMS (MOS) subject classifications (1970). Primary 47B50, 35P10, 76E99; Secondary 46D05.