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The paper defines the Weyl-Titchmarsh function for an abstract boundary value problem and shows that it coincides with the transfer function of some explicitly described linear boundary control system. On the ground of obtained results we explore interplay among boundary value problems, operator colligations, and the linear systems theory that suggests an approach to the study of boundary value problems based on the open systems theory founded in works of M. S. Livšic. Examples of boundary value problems for partial differential equations and calculations of their Weyl-Titchmarsh functions are offered as illustration. In particular, we give an independent derivation of the Weyl-Titchmarsh function for the three dimensional Schrödinger operator introduced by W. O. Amrein and D. B. Pearson. Relationships to the Schrödinger operator with singular potential supported by the unit sphere are clarified and other possible applications of the developed approach in mathematical physics are noted. Mathematics Subject Classification (2000). Primary 47A48, 47B25, 47F05; Secondary 35J10, 47N70.Keywords. Abstract boundary value problem, Green formula, Weyl-Titchmarsh function, operator colligations, linear systems with boundary control, Schrödinger operator. V. RyzhovComp.an.op.th.

For a non-self-adjoint operator with a characteristic function that has boundary values almostThis paper is devoted to the investigation of (local) absolutely continuous (a.c) and singular (s.) subspaces (that correspond to measurable subsets of the real axis) for a non-self-adjoint operator. This paper extends the investigations of [4, 9,10,11,16,28,36,37]. We continue to investigate the spectral structure of operators that are "similar" to some extent to self-adjoint ones. The principal results obtained here are direct generalizations of the corresponding propositions of the "self-adjoint theory." Some of them were published in [20].Let L be a closed completely non-self-adjoint operator in a separable Hilbert space with a characteristic operator-function having measurable boundary values in the weak operator topology almost everywhere on the real axis. In this work, we discuss the following problems:1. Equivalent descriptions of local a.c. and s. subspaces of the operator L. 2. Formulation of conditions for a subset of the real axis that are sufficient for the separability and completeness of the corresponding a.c. and s. subspaces.3. Criteria for similarity of (parts) of the operator L to a self-adjoint one. 4. Construction of a functional calculus consistent with the available one, for the absolutely continuous part of the operator L.5. Localization of the absolutely continuous spectrum of the operator L. As in the above-mentioned works, our principal tool is a functional model that is a generalization of the known Sz.-Nagy-Foia, 2 model for conractions [13] in the case of closed non-dissipative operators. Construction of such a model 1 by the scheme of [9, 10] is given in detail in [19].In Sec. 1, according to [10,36,19], we briefly present some information that we need below. In the same section, we give definitions of the investigated objects that are based on the model representation. In Sec. 2, we formulate the principal restriction on the operator L. We assume that this condition holds in the sequel of this work. Section 3 contains results of the investigation of problems of similarity and also a discussion of criteria for separability and completeness of a.c. and s. subspaces. Section 4 is devoted to "non-model" descriptions of a.c. and s. subspaces. Here we formulate the principal result on functional calculus (Theorem 4.1). Finally, in Sec. 5 we present information on the absolutely continuous spectrum of the operator L.In general, the notation applied in this paper is standard. Note that C+ := {z E C [ =t= Im z > 0} are the open upper and lower half-planes of the complex plane C. If H1, /-/2 are Hilbert spaces and A is a linear operator from H1 in/-/2, :D(A) and T~(A) are the domain of its definition and the set of values, respectively, and p(A) is the resolvent set of A. The notation A: H1 ~ H2 means that A belongs to the Banach space B(H1, H2) of bounded operators from H1 to/-/2 defined everywhere on H1. We denote by 11 9 1[ H~--~H2 a norm on B(H1,//2). Concepts of measure, measurability, "almost ...

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