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Generalizations of Temple's Inequality

Evans M. Harrell

1978
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Proceedings of the American Mathematical Society
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T. Kato's little-known generalization of a classic variational inequality for eigenvalues is extended to the case of normal operators and briefly discussed. It is usually not possible to evaluate precisely the eigenvalues of the linear operators which occur in realistic models in the physical sciences. It is thus a problem of great practical importance to have formulae for approximate evaluation of eigenvalues and for the errors of those approximations. The most important approximate formula
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... an eigenvalue is the Rayleigh-Ritz inequality, which gives an upper bound for the lowest eigenvalue of a selfadjoint operator. This is the prototype of a variational estimate, whereby a set of approximate eigenfunctions is guessed at and used to estimate the eigenvalues. The problem of obtaining lower bounds for the lowest eigenvalue of a selfadjoint operator is notoriously more difficult than the discovery of upper bounds, but some methods are widely known, though not so widely as the Rayleigh-Ritz inequality. The best such bound which relies only on the selfadjointness of the operator and the isolation of the lowest eigenvalue from the rest of the spectrum is due to G. Temple [7 Theorem 1]. The proofs of the Rayleigh-Ritz inequality and Temple's inequality show them to be straightforward applications of the spectral theorem [5], [6], and similar arguments can extend these inequalities to give useful estimates for any isolated eigenvalue of a selfadjoint operator-without the necessity of first estimating all the lower eigenvalues, as with the min-max principle. It is peculiar and unfortunate that more than two decades elapsed between Temple's original paper and the discovery of the generalization of Temple's inequality to arbitrary eigenvalues, and that this generalization has remained but little known for almost three more decades. In this paper Temple's inequality is generalized still further to the case of normal operators. Its use is not only for numerical computation, but also for the proofs of many abstract theorems about perturbation expansions and convergence of operator-valued functions [1], [2], [4], [5]. The classical result is

doi:10.2307/2042610
fatcat:ij3t7zolqrgopbamd3can2r4qy