Functional Model of a Closed Non-Selfadjoint Operator

Abstract

Abstract. We construct the symmetric functional model for an arbitrary closed operator with a non-empty resolvent set acting on a separable Hilbert space. The main techniques of the study are based on the explicit form of the Sz.-Nagy-Foiaş model for a closed dissipative operator, the Potapov-Ginzburg transform of characteristic functions, and certain resolvent identities. All considerations are carried out under minimal assumptions, and obtained results are directly applicable to problems typically arising in mathematical physics. Explicit formulae for all objects relevant to the model construction are provided.

The functional model for maximal dissipative operators (translation form): An approach in the spirit of operator knots

“…These are constructed using the Sz.-Nagy-Foias functional model involving square roots, as discussed at the end of Section 4. We show that for the special choice of λ = i, our model can be recovered from the results in [35]. However, the method will not reproduce our explicit formulae, as transformations that use square roots of operators are involved.…”

“…Comparison to the Kudryashov/Ryzhov model. Based on the work of Kudryashov, in [35], Ryzhov discusses two selfadjoint dilations (which are then shown to coincide) of a dissipative operator A. These are constructed using the Sz.-Nagy-Foias functional model involving square roots, as discussed at the end of Section 4.…”

“…Kudryashov [20] presented an explicit form of the selfadjoint dilation of a general maximal dissipative operator, announcing only the result without any details of the proof. Later, Ryzhov [35] reconstructed this form of the selfadjoint dilation and gave a complete proof. In both cases the explicit construction of the dilation is purely abstract and contains the square roots of some complicated operators which can only be explicitly calculated in a useful form in the case of a rank one perturbation of a selfadjoint operator.…”

Functional Model for Boundary‐value Problems

“…Any completely non‐selfadjoint dissipative operator L admits a self‐adjoint dilation [55], which is unique up to a unitary transformation, under an assumption of minimality, see (3.2) below. There are numerous approaches to an explicit construction of the named dilation [13, 40–42, 46, 47, 51, 52, 54]. In applications, one is compelled to seek a realisation corresponding to a particular setup.…”

“…We refer the reader to the paper[52], where a three-component model is constructed for a dissipative operator with at least one regular point in the upper half-plane.…”