Abstract
In the present paper a subclass of scalar Nevanlinna functions is studied, which coincides with the class of Weyl functions associated to a nonnegative symmetric operator of defect one in a Hilbert space. This class consists of all Nevanlinna functions that are holomorphic on (−∞, 0) and all those Nevanlinna functions that have one negative pole a and are injective on \({(-\infty, a)\,\cup\, (a, 0)}\) . These functions are characterized via integral representations and special attention is paid to linear fractional transformations which arise in extension and spectral problems of symmetric and selfadjoint operators.
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Communicated by Guest Editors L. Littlejohn and J. Stochel.
Dedicated to our friend Franek Szafraniec on the occasion of his seventieth birthday.
This research was supported by the grants from the Academy of Finland (project 139102) and the German Academic Exchange Service (DAAD project D/08/08852). The third author thanks the Deutsche Forschungsgemeinschaft (DFG) for the Mercator visiting professorship at the Technische Universität Berlin. The authors would like to thank also an anonymous referee on some constructive comments, especially for paying their attention to group actions.
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Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
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Behrndt, J., Hassi, S., de Snoo, H. et al. Linear Fractional Transformations of Nevanlinna Functions Associated with a Nonnegative Operator. Complex Anal. Oper. Theory 7, 331–362 (2013). https://doi.org/10.1007/s11785-011-0197-3
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DOI: https://doi.org/10.1007/s11785-011-0197-3