Abstract
In this chapter we shall consider the inverse problem for commutative two-operator vessels, as formulated in Section 10.1, in the case when the discriminant curve is a real smooth cubic. We shall use all the notation of Chapter 11 regarding smooth cubics, their determinantal representations and semiexpansive functions on them. We start the solution of the inverse problem by investigating the simplest case when the inner space H is one-dimensional and the joint spectrum consists of a single (non-real) point; we have already taken a first look at these one-dimensional vessels in Section 4.4.
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Notes
Chapters 10–12 were written by V. Vinnikov.
Chapter 10. The inverse problem for commutative two-operator vessels has been first posed and solved, for the finite-dimensional dissipative case, by Livšic [37]; the general formulation of the problem in Section 10.1 is from Vinnikov [58], The joint characteristic function was introduced by Livšic [39]; the restoration formula (10.52), and Theorem 10.3.1, Theorem 10.3.6 and Theorem 10.4.3 were established in [39] for the dissipative case. The results of Section 10.5 were surveyed in Vinnikov [58].
Chapter 11. The description of self-adjoint determinantal representations of real smooth cubics was obtained by Vinnikov [56]; see Vinnikov [57] for the generalization to the case of arbitrary real smooth plane curves. The results of Sections 11.2–11.3 are the specification to the case of genus 1 of the general theory of normalized joint characteristic functions on a smooth discriminant curve, see Vinnikov [58,60].
Chapter 12. Most results of Sections 12.1–12.2 were obtained in the dissipative case, for an arbitrary discriminant curve but without direct formulas such as (12.43), (12.45) for the solution of the system of recursive equations (12.22), by Livšic [37,38]. Commuting triangular integral operators (12.168) and the corresponding system of differential equations (12.113) were first considered by Waksman [61], and then by Kravitsky [29] and Livšic [39]. Proposition 12.4.1 is from [29]; solvability of the system of differential equations (12.113) in the dissipative case and when the image of the function c(t) consists of a single point, for an arbitrary smooth discriminant curve but without explicit formulas for the solution, was obtained in [39]. The calculation of the resolvents and the complete characteristic functions in Sections 12.3–12.4 follows the single-operator case in Brodskii, Livšic [7]. The results of this Chapter are the specification to the case of genus 1 of the general theory of triangular models for commutative two-operator vessels on real smooth plane curves, see Vinnikov [58,59].
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© 1995 Springer Science+Business Media Dordrecht
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Livšic, M.S., Kravitsky, N., Markus, A.S., Vinnikov, V. (1995). Triangular Models for Commutative Two-Operator Vessels on Real Smooth Cubics. In: Theory of Commuting Nonselfadjoint Operators. Mathematics and Its Applications, vol 332. Springer, Dordrecht. https://doi.org/10.1007/978-94-015-8561-3_12
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DOI: https://doi.org/10.1007/978-94-015-8561-3_12
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