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On perturbations of the isometric semigroup of shifts on the semiaxis


Authors: G. G. Amosov, A. D. Baranov and V. V. Kapustin
Translated by: the authors
Original publication: Algebra i Analiz, tom 22 (2010), nomer 4.
Journal: St. Petersburg Math. J. 22 (2011), 515-528
MSC (2010): Primary 47D03, 47B37, 47B10
DOI: https://doi.org/10.1090/S1061-0022-2011-01156-2
Published electronically: May 2, 2011
MathSciNet review: 2768959
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Abstract: Perturbations $ (\widetilde\tau_t)_{t\ge 0}$ of the semigroup of shifts $ (\tau_t)_{t\ge 0}$ on $ L^2(\mathbb{R}_+)$ are studied under the assumption that $ \widetilde\tau_t - \tau_t$ belongs to a certain Schatten-von Neumann class $ \mathfrak{S}_p$ with $ p\ge 1$. It is shown that, for the unitary component in the Wold-Kolmogorov decomposition of the cogenerator of the semigroup $ (\widetilde\tau_t)_{t\ge 0}$, any singular spectral type may be achieved by $ \mathfrak{S}_1$-perturbations. An explicit construction is provided for a perturbation with a given spectral type, based on the theory of model spaces of the Hardy space $ H^2$. Also, it is shown that an arbitrary prescribed spectral type may be obtained for the unitary component of the perturbed semigroup by a perturbation of class $ \mathfrak{S}_p$ with $ p>1$.


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Additional Information

G. G. Amosov
Affiliation: Moscow Institute of Physics and Technology, Moscow, Russia
Email: gramos@mail.ru

A. D. Baranov
Affiliation: Department of Mathematics and Mechanics, St. Petersburg State University, Stary Petergof, Bibliotechnaya Pl. 2, St. Petersburg 198504, Russia
Email: anton.d.baranov@gmail.com

V. V. Kapustin
Affiliation: St. Petersburg Branch, Steklov Mathematical Institute, Russian Academy of Sciences, Fontanka 27, St. Petersburg 191023, Russia
Email: kapustin@pdmi.ras.ru

DOI: https://doi.org/10.1090/S1061-0022-2011-01156-2
Keywords: Semigroup of shifts, trace-class perturbation, Schatten–von Neumann ideals, Hardy space, inner function.
Received by editor(s): January 20, 2010
Published electronically: May 2, 2011
Additional Notes: Partially supported by the Federal Program 2.1.1/1662, by RFBR (grant no. 08-01-00723), and by the President of Russian Federation grant no. NSH 2409.2008.1
Article copyright: © Copyright 2011 American Mathematical Society

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