Search

Let A be a densely defined simple symmetric operator in $${\mathfrak{H}}$$ , let $${\Pi=\{\mathcal{H},\Gamma_0, \Gamma_1}\}$$ be a boundary triplet for A * and let M(·) be the corresponding Weyl function. It is known that the Weyl function M(·) determines the boundary triplet Π, in particular, the pair {A, A 0}, uniquely up to the unitary similarity. Here $${A_0 := A^* \upharpoonright \text{ker}\, \Gamma_0 ( = A^*_0)}$$ . At the same time the Weyl function corresponding to a boundary triplet for a dual pair of operators defines it uniquely only up to the weak similarity. We consider a symmetric dual pair {A, A} with symmetric $${A \subset A^*}$$ and a special boundary triplet $${\widetilde{\Pi}}$$ for{A, A} such that the corresponding Weyl function is $${\widetilde{M}(z) = K^*(B-M(z))^{-1} K}$$ , where B is a non-self-adjoint bounded operator in $${\mathcal{H}}$$ . We are interested in the problem whether the result on the unitary similarity remains valid for $${\widetilde{M}(\cdot)}$$ in place of M(·). We indicate some sufficient conditions in terms of the operators A 0 and $${A_B= A^* \upharpoonright \text{ker}\, (\Gamma_1-B \Gamma_0)}$$ , which guaranty an affirmative answer to this problem. Applying the abstract results to the minimal symmetric 2nth order ordinary differential operator A in $${L^2(\mathbb{R}_+)}$$ , we show that $${\widetilde{M}(\cdot)}$$ defined in $${\Omega_+ \subset \mathbb{C}_+}$$ determines the Dirichlet and Neumann realizations uniquely up to the unitary equivalence. At the same time similar result for realizations of Dirac operator fails. We obtain also some negative abstract results demonstrating that in general the Weyl function $${\widetilde{M}(\cdot)}$$ does not determine A B even up to the similarity.

A completeness theorem is proved involving a system of integro-differential equations with some λ-depending boundary conditions. Also some sufficient conditions for the root functions to form a Riesz basis are established.

In this paper operator-valued Q-functions of Krein-Ovcharenko type are introduced. Such functions arise from the extension theory of Hermitian contractive operators A in a Hilbert space h. The definition is related to the investigations of M.G. Krein and I.E. Ovcharenko of the so-called Q(mu) and Q(M)-functions. It turns out that their characterizations of such functions hold true only in the matrix valued case. The present paper extends the corresponding properties for wider classes of selfadjoint contractive extensions of A. For this purpose some peculiar but fundamental properties on the behaviour of operator ranges of positive operators will be used. Also proper characterizations for Q(mu)- and Q(M)-functions in the general operator-valued case are given. Shorted operators and parallel sums of positive operators will be needed to give a geometric understanding of the function-theoretic properties of the corresponding Q-functions.