Speaker
Description
Let $A\ge 0$ be a closed densely defined non-negative symmetric operator in a Hilbert space ${\frak H}$, let ${\frak H}_1 := \text{ran}(A+I)$, and let $P_1$ be the orthoprojection in $\frak H$ onto $\frak H_1$.
Let also $A_{F}$ and $A_{K}$ be, respectively, the maximal (Friedrichs') and minimal (Krein's) non-negative selfadjoint extensions of $A$.
Next, assuming $A$ to be positive definite, $A \ge m_A > 0$, Krein characterized the extension $A_K$ as follows:
$$\text{dom} A_K = \text{dom} A + {\frak N}_0,\,\text{where}\, {\frak N}_0 = \text{ker}A*.$$ Therefore Krein's extension $A_{K}$ admits the following representation: $$A_K = A'_K \oplus (O | {\frak N}_0),$$ where $$A'_{K} := A_K | {\frak M}_0, \text{ and } {\frak M}_0 := {\frak N}_0^{\perp} = \text{ran} A$$. The operator $ A_{K}'$ is called the reduced Krein extension. We will discuss relations between certain spectral properties of the operators $A_{F}, A_{K}'$ and $A$ assuming the operator $(A +I)^{-1}$ to be compact. First we discuss the validity of the following equivalence: $$ P_1 (I_{\frak H} + A)^{-1} \in \frak S(\frak H_1) \quad \Longleftrightarrow \quad (I_{{\frak M}_0} + A'_K)^{-1} \in \frak S(\frak M_0 ), $$ which improves and complements the known Krein's result. Here $\frak S$ is arbitrary symmetrically normed ideal $\frak S$ including Neumann-Schatten ideals $\frak S = \frak S_{p}$, $p\in (0,\infty]$, as well as ideals $\Sigma_p$ (a compact operator $T$ is put in the class $\Sigma_p(\frak H),$ if $s_n(T))= O(n^{-1/p}), p\in(0,\infty)$). Secondary we will discuss the improvement of above equivalence for $\frak S = \Sigma_p$. It happens that the inclusion $P_1 (I_{\frak H} + A)^{-1} \in \Sigma_p(\frak H_1)$ for some $p\in(0,\infty)$ does not ensure coincidence of the eigenvalues asymptotics of operators, i.e. the following equivalence with some $a\ge 0:$ $$ \lambda_n\bigl(P_1(I_{\frak H} + A)^{-1}\bigr) = an^{-1/p}\left(1 + o(1)\right)\quad \Longleftrightarrow \quad \lambda_n \bigl((I_{{\frak M}_0} + \widehat A'_K)^{-1}\bigr) = a n^{-1/p}\left(1 + o(1)\right). $$ In fact, it will be explained that the validity of this equivalence as $n\to \infty$ depends on $A_F$. We will also discuss the abstract Alonso-Simon problem [1] on the eigenvalues asymptotics of $A_{F}$ and $A_{K}'$, and the explicit solution to the Birman problem. Besides, we discuss improvement of Birman's and Grubb's results (see [2], [3]) regarding equivalence of semiboundedness properties of an extension $\widetilde A = \widetilde A^*$ of $A$ and the corresponding boundary operator. A part of results of the talk are announced in [4] and published in [5]. $$\,$$ References: $$\,$$ 1. $A. Alonso, B. Simon$, The Birman-Krein-Vishik theory of self-adjoint extensions of semibounded operators, J. Operator Theory, 4 (1980), 251--270. $$\,$$ 2. $M.S. Birman$, On the self-adjoint extensions of positive definite operators, Mat. Sb., 38, 431--450 (1956). $$\,$$ 3. $G. Grubb$, A characterization of the non-local boundary value problems associated with an elliptic operator, Ann. Scuola Norm. Sup. Pisa (3), 22 (1968), 425--513. $$\,$$ 4. $M.M. Malamud$, To Birman--Krein--Vishik theory, Doklady Mathematics, 10, No.1 (2023), 44--48. $$\,$$ 5. $M.M. Malamud$, Explicit solution to the Birman problem for the $2D$-Laplace operator, Russian Journal of Mathematical Physics, 31, No. 3 (2024), 495--503.
