Functional Models of Operators and Their Multivalued Extensions in Hilbert Space

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Integral Equations and Operator Theory

Functional Models of Operators and Their Multivalued Extensions in Hilbert Space Damir Z. Arov and Harry Dym Abstract. This paper presents the first part of a study of functional  of symmetric and models of selfadjoint and nonselfadjoint extensions A nonsymmetric operators A in a Hilbert space H. The extensions will be considered in the framework of linear relations (which may also be interpreted as the graphs of multivalued operators) that are required  In these models H is to have a nonempty set of regular points ρ(A). modelled by a reproducing kernel Hilbert space H of vector valued holomorphic functions that are defined on some nonempty open set Ω ⊆  and H is invariant under the action of the (generalized) backward ρ(A) shift operator Rα for every α ∈ Ω; A is modelled by the operator A of multiplication by the independent variable [i.e., (Af )(λ) = λf (λ) for   is modelled by a linear relation A f ∈ H for which Af ∈ H]; and A −1  with the property that (A − αI) = Rα for all points α ∈ Ω. Mathematics Subject Classification. 46E22, 47A20, 47A06, 47B25, 47B32. Keywords. Linear relations, Extensions of operators in Hilbert space, Functional models, Reproducing kernel Hilbert spaces, Vector valued de Branges spaces, Multiplication operators.

1. Introduction This paper is the first part of a study of selfadjoint and nonselfadjoint extensions of symmetric and nonsymmetric operators in Hilbert spaces and their functional models. The development was motivated by the earlier investigations in [30] and [15], both of which contain a number of relevant references to the literature. The operators A under consideration in this paper will belong to the linear space L(H) of linear operators acting in a complex separable Hilbert  of A in the class LR(H) space H. However, we shall consider extensions A of linear relations. The main properties of this class will be discussed briefly D. Arov acknowledges with thanks the support of a Belkin visiting Professorship from the Weizmann Institute of Science. 0123456789().: V,-vol

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in Sect. 3. For the moment, it suffices to note that every linear relation  ∈ LR(H) can be identified with a subspace of H2 = H × H and that LR(H) A is closed under addition, scalar multiplication and inversion. Moreover, the  = (A  − λI)−1 is well defined in the class LR(H) for every resolvent R(λ; A)  ∈ LR(H) and λ ∈ C; and adjoints A∗ of A ∈ L(H) exist within choice of A the class LR(H), even if the domain of A is not dense. The notation dom A, ker A, rng A and crng A denote the domain of A, the kernel of A, the range of A and the closure of the range of A, respectively; clspan stands for closed linear span. The acronyms mvf and vvf stand for matrix valued function and vector valued function, respectively. Let B(H) = B(H, H), where B(H1 , H2 ) denotes the space of bounded linear operators acting from the Hilbert space H1 into the Hilbert space H2 ; all Hilbert spaces considered in this paper will be complex and separab