- PII
- 10.31857/S2686954322600574-1
- DOI
- 10.31857/S2686954322600574
- Publication type
- Status
- Published
- Authors
- Volume/ Edition
- Volume 509 / Issue number 1
- Pages
- 54-59
- Abstract
- Let A ≥ mA > 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.
- Keywords
- positive definite symmetric operator Friedrichs and Krein extensions compactness of resolvent asymptotic of spectrum
- Date of publication
- 17.09.2025
- Year of publication
- 2025
- Number of purchasers
- 0
- Views
- 13
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