Estimates of the Spectrum in the Interval

Abstract

Here $E(A;\tau _1,\tau _2)$ is the spectral projector corresponding to the open interval $(\tau _1,\tau _2)$ rather than to more standard semi-open interval $[\tau _1,\tau _2)$ as before and ${\mathsf {N}}(A;\tau _1,\tau _2)$ is number of eigenvalues in $(\tau _1,\tau _2)$ counting their multiplicities, unless $(\tau _1,\tau _2)\cap {\text {Spec}}_{{\mathsf {ess}}}(A)\ne \emptyset$ in which case ${\mathsf {N}}(A;\tau _1,\tau _2)=\infty$.