К теории Бирмана–Крейна–Вишика

Let A ≥ m_A > 0 be a closed positive definite symmetric operator in a Hilbert space ℌ, let ${{\hat {A}}_{F}}$ and ${{\hat {A}}_{K}}$ be its Friedrichs and Krein extensions, and let ?_∞ be the ideal of compact operators in ℌ. The following problem has been posed by M.S. Birman: Is the implication A^–1 ∈ ?_∞ ⇒ (${{\hat {A}}_{F}}$ )^–1 ∈ ?_∞(ℌ) holds true or not? It turns out that under condition A^–1 ∈ ?_∞ 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 P₁(I + A)^–1 are close. For instance, these operators belong to a symmetrically normed ideal ?, 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.