Estimates of the Spectrum in the Interval

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 $${\mathsf {N}}(A;\tau _1,\tau _2)=\infty $$ .