Absolutely continuous and singular subspaces of a non-self-adjoint operator

Abstract

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 ...

Equipped absolutely continuous subspaces and stationary construction of the wave operators in the non-self-adjoint scattering theory

“…It is easy to see that for X E Bor the space A/'+(X) is positive with respect to Na(X) and A/'-(X) := Z-(X)N'-is the corresponding negative space (cf. [22]). Now let us recall the definition of the characteristic function of an operator L proposed in [28].…”

“…In presenting the material in Sec. 1, we take as known the main propositions of the theory of equipped spaces and freely use the terminology of [1] (see also [2,18] The following theorem holds (see, for example, [11,14,22]): Theorem 1.1. /n the notation introduced, …”

“…In Sec. 1 we briefly present some information from [13,14,11,22] that we need-in the sequel. Moreover, the same section contains an interpretation of the absolutely continuous subspace of a non-self-adjoint operator (probably going back to the papers [12,10,19,20]), as an equipped (Hilbert) space.…”

“…Definitions of absolutely continuous and singular subspaces of such operators are introduced according to the results of [13,14,11]. Some additional properties of these subspaces were investigated in [22]. In Sec.…”

Functional Model of a Closed Non-Selfadjoint Operator

“…The paper offers such a model, free of limitations described above. It was already successfully applied (without a proof) to the study of nondissipative operators from a fairly wide class in [42,43], where the notion of local absolutely continuous and singular subspaces was examined and utilized for the subsequent study of scattering theory for a pair of nonselfadjoint operators. Owing to the generic form of operator under consideration, results obtained in the current paper cover both cases of the model for nonselfadjoint additive perturbations [28] and for extensions of symmetric operators [44].…”