Estimates of Kato-Temple type for \(n\)-dimensional spectral measures (Q2459939)

Kato--Temple estimates [see, e.g., \textit{F.\,Chatelin}, ``Spectral approximation of linear operators'' (Computer Science and Applied Mathematics; New York etc.:\ Academic Press) (1983; Zbl 0517.65036)]) give an interval that contains only one point from the spectrum of a selfadjoint operator on a Hilbert space. The boundary points of that interval and the center and the length of an enclosing interval can be estimated in terms of Rayleigh quotients. In this paper, these results are generalized to \(n\)-dimensional spectral measures. That means that a spectral measure is defined on \({\mathbb R}^n\) and also associated spectral measures along the different coordinate axes. The latter define a set of commuting selfadjoint operators. Given a ball in \({\mathbb R}^n\) that intersects the spectrum, the results give another ball whose center and radius depend on \(n\) Rayleigh quotients defined along the coordinate axes and that will contain the subset isolated by the first ball. As a special case, it might isolate a single point from the spectrum. It is also shown that the latter ball is invariant under a reflection in the unit sphere of \({\mathbb R}^n\).