To Birman–Krein–Vishik theory

Let $A\ge m_A>0$ be a closed positive definite symmetric operator in a Hilbert space $\mathcal H$, let $\hat{A}_F$ and $\hat{A}_K$ be its Friedrichs and Krein extensions, and let $\mathfrak S_\infty$ be the ideal of compact operators in $\mathcal H$. The following problem has been posed by M.S. Birman: Is the implication $A^{-1}\in\mathfrak S_\infty\Rightarrow (\hat{A}_F)^{-1}\in\mathfrak S_\infty(\mathcal H)$ holds true or not? It turns out that under condition $A^{-1}\in\mathfrak{S}_\infty$ the spectrum of Friedrichs extension $\hat{A}_F$ might be of arbitrary nature. This gives a complete negative solution to the Birman problem. Let $\hat{A}'_K$ be the reduced Krein extension. It is shown that certain spectral properties of the operators $(I_{\mathfrak{M}_0}+\hat{A}'_K)^{-1}$ and $P_1(I+A)^{-1}$ are close. For instance, these operators belong to a symmetrically normed ideal $\mathfrak S$, say are compact, only simultaneously. Moreover, it turns out that under a certain additional condition the eigenvalues of these operators have the same asymptotic. Besides we complete certain investigations by Birman and Grubb regarding the equivalence of semiboubdedness property of selfadjoint extensions of $A$ and the corresponding boundary operators.