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Functional models for commutative systems of linear operators and de Branges spaces on a Riemann surface. (English. Russian original) Zbl 1188.47009

Sb. Math. 200, No. 3, 339-356 (2009); translation from Mat. Sb. 200, No. 3, 31-48 (2009).
To any given commuting pair of bounded linear nonselfadjoint operators \(\{A_1,A_2\}\) in a Hilbert space \(H\), one can associate the commutative (regular) colligation
\[ \Delta=(\{A_1,A_2\};H;\varphi;E;\{\sigma_1,\sigma_2\};\gamma^-;\gamma^+) \]
with the following properties: \(2\operatorname{Im} A_k=\varphi^*\sigma_k\varphi\;(k=1,2)\), \(\gamma^-\varphi=\sigma_1\varphi A_2^*-\sigma_2\varphi A_1^*\) and \(\gamma^+-\gamma^-=-2\operatorname{Im}(\sigma_1\varphi\varphi^*\sigma_2)\), where \(E\) is a Hilbert space, \(\varphi\) is a bounded linear operator on \(H\) into \(E\), and \(\sigma_1,\sigma_2,\gamma^-,\gamma^+\) are bounded selfadjoint operators on \(E\).
Functional models for pairs \(\{A_1,A_2\}\) such that \(A_t=t_1A_1+t_2A_2\) is dissipative for some \(t=(t_1,t_2)\in{\mathbb R}^2\) were constructed in [M.S.Livšic, N.Kravitsky, A.S.Markus and V.Vinnikov, “Theory of commuting nonselfadjoint operators” (Math.Appl.(Dordrecht) 332; Dordrecht:Kluwer) (1995; Zbl 0834.47004)], V.A.Zolotarev [Math.USSR, Sb.70, No.2, 399–429 (1991); translation from Mat.Sb.181, No.7, 965–994 (1990; Zbl 0738.47009)] using certain techniques of Fourier transforms.
In the paper under review, the author presents several functional models for a commuting pair \(\{A_1,A_2\}\) (without the explicit assumption that it contains a dissipative operator) such that the corresponding commutative colligation \(\Delta\) is simple (i.e., \(H=\text{span}\,\{A_1^{n_1}A_2^{n_2}\varphi^*E:n_k\in{\mathbb Z}_+\), \(k=1,2\}\)), the spectrum of \(A_1\) is concentrated at the origin, the Livšic characteristic function of \(A_1\) has certain properties related to its multiplicative decomposition [M.Livshits and A.A.Yantsevich, “Operator colligations in Hilbert spaces” (Washington/DC:V.H.Winston & Sons; New York etc.:John Wiley & Sons) (1979; Zbl 0435.47002), \(\dim E=2n < \infty\) and \(\sigma_1\) (an involution), \(\sigma_2\) and \(\gamma^+\) have a particular form.
The given constructions have realizations in special spaces of meromorphic functions on Riemann surfaces, leading to analogues of de Branges spaces on these Riemann surfaces. An interesting observation is that the functions determining the order of growth in de Branges spaces on Riemann surfaces are exactly the Baker-Akhiezer functions of B.A.Dubrovin [Russ.Math.Surv.36, No.2, 11–92 (1981); translation from Usp.Mat.Nauk 36, No.2(218), 11–80 (Russian) (1981; Zbl 0478.58038)].

MSC:

47A45 Canonical models for contractions and nonselfadjoint linear operators
46E20 Hilbert spaces of continuous, differentiable or analytic functions
47A48 Operator colligations (= nodes), vessels, linear systems, characteristic functions, realizations, etc.
30F99 Riemann surfaces
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