Abstract
AbstractGiven a symmetric linear transformation on a Hilbert space, a natural problem to consider is the characterization of its set of symmetric extensions. This problem is equivalent to the study of the partial isometric extensions of a fixed partial isometry. We provide a new function theoretic characterization of the set of all self-adjoint extensions of any symmetric linear transformation B with finite equal indices and inner Livšic characteristic function θ
Highlights
The purpose of this paper is to study of the family of all closed symmetric extensions of a given closed simple symmetric linear transformation B with equal deficiency indices (n, n), 1 ≤ n < ∞ defined on a domain in a separable Hilbert space in the case where the Livsic characteristic function of B is an inner function
More generally Sn will denote the family of all closed simple symmetric linear transformations with indices (n, n) defined in some separable Hilbert space, and S the set of all closed simple symmetric linear transformations with equal indices defined in some separable Hilbert space
Our goal is to provide a new characterization the set of extensions, canonical and non-canonical in the special case where the characteristic function ΘB is inner. (Recall that in this case B is unitarily equivalent to multiplication by z in a model subspace KΘ2 B = H2(C+) ⊖ ΘBH2(C+) of Hardy space [2, 3, 4].)
Summary
The purpose of this paper is to study of the family of all closed symmetric extensions of a given closed simple symmetric linear transformation B with equal deficiency indices (n, n), 1 ≤ n < ∞ defined on a domain in a separable Hilbert space in the case where the Livsic characteristic function of B is an inner function. Application of the Cayley transform, which is a bijection from S onto V , the set of all partial isometries with equal indices, converts this into a partial order on V Modulo unitary equivalence, this is the same as the partial order previously defined by Halmos and McLaughlin on partial isometries in [6]. In the case where ΘB1 is an inner function, we provide necessary and sufficient conditions on ΘB2 so that B1 B2 in Theorem 9.5 Many of these results will be achieved using the concept of a generalized model. This is a reproducing kernel Hilbert space theory approach which generalizes the concept of a model for a symmetric operator as defined in [4]
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