PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 100. Number 2. June 1987 AN INEQUALITY FOR SELFADJOINT OPERATORS ON A HILBERT SPACE HERBERT J. BERNSTEIN ABSTRACT. An elementary inequality of use in testing convergence of eigen- vector calculations is proven. If e\ is a unit eigenvector corresponding to an eigenvalue A of a selfadjoint operator A on a Hilbert space H, then '^ * \\(A-Xl)gV for all g in H for which Ag ^ Xg. Equality holds only when the component of g orthogonal to e\ is also an eigenvector of A. When computing eigenvectors of real symmetric matrices by iterative techniques, convergence is usually assumed on the basis of stagnation of Rayleigh quotients, (e.g., see [2]). In most cases this is satisfactory. However, when dealing with pathologically close eigenvalues, significant components orthogonal to the desired vector may remain [3, p. 174]. In such cases one may try to estimate the size of those orthogonal components to decide if, say, Richardson's purificatin is needed. In this paper we present an estimator which has been of value in such calculations. The proof is elementary and the author suspects that it is not new, but is able to find no prior publication nor use within standard eigenvector programs. Before proving the inequality, we state a lemma which can be derived directly from the equations used in proving Cauchy's or Schwartz' inequality [1, p. 16]. We include an abstract proof here for completeness. LEMMA. Let H be a Hilbert space, A a selfadjoint operator on H, x a vector in H, and let r be real. Then ||x||2||Ax||2 - (x, Ax)2 = \\x\\2\\(A - tI)x\\2 - (x, (A - rl)x)2. Proof. ||x||2||(A-rI)x||2-(x,(A-r/)x)2 = ||x||2(||Ax||2-2r(x,Ax)+r2||x||2) -((x,Ax)2-2t(x,Ax)||x||2 + t2||x||4) = ||x||2||Ax||2-2r(x,Ax)||x||2+r2||x||4 - (x, Ax)2 + 2r(x, Ax)||x||2 - r2||x||4 = ||x||2||Ax||2-(x,Ax)2. Q.E.D. Now we can prove the desired inequality. Received by the editors January 27, 1986. 1980 Mathematics Subject Classification (1985 Revision). Primary 65F15; Secondary 65J10. This research was partially supported by the Department of Energy under contract DEACO276ERO3077-V, the National Science Foundation, under contract DCR-8511302, and the Office of Naval Research, under contract N00014-82-K-0381. ©1987 American Mathematical Society 0002-9939/87 $1.00 + $.25 per page 319 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use