Abstract. We consider operator-valued Herglotz functions and their applications to self-adjoint perturbations of self-adjoint operators and self-adjoint extensions of densely defined closed symmetric operators. Our applications include model operators for both situations, linear fractional transformations for Herglotz operators, results on Friedrichs and Krein extensions, and realization theorems for classes of Herglotz operators. Moreover, we study the concrete case of Schrödinger operators on a half-line and provide two illustrations of Livsic's result [44] on quasi-hermitian extensions in the special case of densely defined symmetric operators with deficiency indices (1, 1).

We compute the Fredholm index, index(D

For a wide class of two-body energy operators h(k) on the three-dimensional lattice Z 3 , k being the two-particle quasi-momentum, we prove that if the following two assumptions (i) and (ii) are satisfied, then for all nontrivial values k, k = 0, the discrete spectrum of h(k) below its threshold is non-empty. The assumptions are: (i) the twoparticle Hamiltonian h(0) corresponding to the zero value of the quasi-momentum has either an eigenvalue or a virtual level at the bottom of its essential spectrum and (ii) the oneparticle free Hamiltonians in the coordinate representation generate positivity preserving semi-groups.

We provide additional results in connection with Krein's formula, which describes the resolvent difference of two self-adjoint extensions A 1 and A 2 of a densely defined closed symmetric linear operatorȦ with deficiency indices n n n ∈ ∪ ∞ . In particular, we explicitly derive the linear fractional transformation relating the operator-valued Weyl-Titchmarsh M-functions M 1 z and M 2 z corresponding to A 1 and A 2 .

Abstract. We obtain a new representation for the solution to the operator Sylvester equation in the form of a Stieltjes operator integral. We also formulate new sufficient conditions for the strong solvability of the operator Riccati equation that ensures the existence of reducing graph subspaces for block operator matrices. Next, we extend the concept of the Lifshits-Krein spectral shift function associated with a pair of self-adjoint operators to the case of pairs of admissible operators that are similar to self-adjoint operators. Based on this new concept we express the spectral shift function arising in a perturbation problem for block operator matrices in terms of the angular operators associated with the corresponding perturbed and unperturbed eigenspaces.