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Direct and Inverse Spectral Theorems for a Class of Canonical Systems with Two Singular Endpoints

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Notes

  1. 1.

    A proof is given in [51] (in Russian).

  2. 2.

    A detailed account on the operator representation of scalar-valued generalized Nevanlinna functions can be found in [72, §1]. For a slightly different viewpoint and results for operator- (or matrix-) valued functions, see [18, 71, §3], or some of the vast more recent literature on operator models.

  3. 3.

    If the integral in (4.4) is 0, then \(\uptheta (x;\cdot )\) and \(\upvarphi (x;\cdot )\) are either of minimal exponential type or of order less than 1.

  4. 4.

    Here ‘\(l,1\)’ stands for ‘left vector, first entry’. This is a generic notation: for example, \(\pi _l\) is the projection from \(\mathbb C^2\times \mathbb C^2\) onto the first vector component, \(\pi _{r,2}\) onto the lower entry of the second vector component, etc. The use of ‘left’ and ‘right’ is motivated by the fact that the vectors correspond to boundary values at the left and right endpoint, respectively.

  5. 5.

    Remember in the following that \(T(H)\) is self-adjoint, and hence \(T(H)=T_{\max }(H)=S(H)\) in the notation of several previous papers like [59].

  6. 6.

    In [76] bounded operators are treated. The extension to the case of relations is provided in [21]. In [76] results are formulated for \(\Delta \) being an interval whose endpoints are not critical points. In our case, the only critical point is \(\infty \). Moreover, with the usual measure-theoretic extension process one can define \(E(\Delta )\) for each bounded Borel set; cf. the first paragraph in the proof of Lemma 5.4 below. We tacitly use this fact and often formulate results for bounded Borel sets although in the original references only intervals were considered.

  7. 7.

    The set on the right-hand side of [84, Theorem 3.9 (v)] is certainly non-empty.

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Acknowledgements

The first author gratefully acknowledges the support of the Nuffield Foundation, grant no. NAL/01159/G, and the Engineering and Physical Sciences Research Council (EPSRC), grant no. EP/E037844/1. The second author was supported by the joint project I 4600 of the Austrian Science Fund (FWF) and the Russian Foundation for Basic Research (RFBR).

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  1. Department of Mathematics and Statistics, University of Strathclyde, Glasgow, UK

    Matthias Langer

  2. Institute for Analysis and Scientific Computing, Vienna University of Technology, Wien, Austria

    Harald Woracek

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    Damir Kinzebulatov

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Langer, M., Woracek, H. (2023). Direct and Inverse Spectral Theorems for a Class of Canonical Systems with Two Singular Endpoints. In: Binder, I., Kinzebulatov, D., Mashreghi, J. (eds) Function Spaces, Theory and Applications. Fields Institute Communications, vol 87. Springer, Cham. https://doi.org/10.1007/978-3-031-39270-2_5

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