TO THE BIRMAN–KREIN–VISHIK THEORY

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 ∈ G_∞ ⇒ (\({{\hat {A}}_{F}}\) )^–1 ∈ G_∞(ℌ) holds true or not? It turns out that under condition A^–1 ∈ G_∞ 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 G, 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.