Estimates of Kato–Temple Type for $n$-dimensional Spectral Measures

Abstract

The Kato–Temple estimate for an isolated eigenvalue of a selfadjoint operator is extended to spectral measures on ${\mathbf{R}}^n$, or equivalently, a commuting set of $n$ selfadjoint operators. The proof depends on a general variational characterization of the spectrum. In the case of normal operators this corresponds to Rayleigh bounds applied to resolvents. It is also shown that the obtained estimate has a certain invariance property under an inversion of the space.