Estimate of Kato -Temple type for a spectral measure on R^{n} S. T. Kuroda

Let A be a selfadjoint operator with the spectrum σ(A). Suppose one knows that σ(A) ∩ (a, b) consitsts of at most one point. Given the Rayleigh quotient η = (Au, u) of a trial vector u with ∥u∥ = 1, the Kato -Temple theorem ([1], [2]) gives a sharper estimate a < λ_{−} < λ_{0} < λ_{+} < b for the sole eigenvalue λ_{0} ∈ (a, b), where λ_{±} are explicitly given in terms of a, b, η, and the error ε = ∥Au − ηu∥. A natural generalization is to prove a similar estimate for a normal operator A. (For a partial result see [3].) More generally, we will prove in this talk that an estimate of Kato -Temple type holds for a general spectral measure defined on R^{n}. Once the result is formulated, the proof is a simple generalization of the original one and is elementary.

When n = 2, the case that A is a normal operator, a part of the proof can be related to the Rayleigh principle applied to the resolvent (A − z)^{−1}.

References

[1] G. Temple, The theory of Rayleigh’s principle as applied to continuous systems, Proc. Royal Soc. London Ser. A 119(1928), 276–293.

[2] T. Kato, On the upper and lower bounds of eigenvalues, J. Phys. Soc. Japan 4(1949), 334–339.

[3] E. M. Harrell, Generalizations of Temple’s inequality, Proc. Amer. Math. Soc. 69(1978), 271–276.