Perturbations which are infinitely small in the strong operator topology (Q909211)

The author uses the nonstandard analysis in the \textit{E. Nelson}'s form [Bull. Am. Math. Soc. 83, No.6, 1165-1198 (1977; Zbl 0373.02040)]. In a standard Hilbert space the oprator \(T\in {\mathcal B}(H)\) is the perturbation in the strong operator topology of the standard compact operator \(S\in {\mathcal B}(H)\), when \(\| T-{\mathcal P}S{\mathcal P}\| \approx 0\). Here \(\approx\) is the relation of infinite proximity and \({\mathcal P}\) is the nonstandard projector such that \(\| {\mathcal P}x-x\| \approx 0\) for \(\forall^{st}x\in H\). The author investigates the variation of the standard spectrum and the corresponding spectral projectors for the perturbation T.