On perturbations of the isometric semigroup of shifts on the semiaxis
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G. G. Amosov, A. D. Baranov and V. V. Kapustin
Translated by: the authors - St. Petersburg Math. J. 22 (2011), 515-528
- DOI: https://doi.org/10.1090/S1061-0022-2011-01156-2
- Published electronically: May 2, 2011
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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 $p>1$.References
- G. G. Amosov and A. D. Baranov, On the dilation of contractive cocycles and cocycle perturbations of the translation group on the line, Mat. Zametki 79 (2006), no. 1, 3–18 (Russian, with Russian summary); English transl., Math. Notes 79 (2006), no. 1-2, 3–17. MR 2249142, DOI 10.1007/s11006-006-0001-2
- G. G. Amosov and A. D. Baranov, On the dilation of contractive cocycles and cocycle perturbations of the translation group on the line. II, Mat. Zametki 79 (2006), no. 5, 779–780 (Russian); English transl., Math. Notes 79 (2006), no. 5-6, 719–720. MR 2249135, DOI 10.1007/s11006-006-0081-z
- A. D. Baranov, Embeddings of model subspaces of the Hardy class: compactness and Schatten–von Neumann ideals, Izv. Ross. Akad. Nauk Ser. Mat. 73 (2009), no. 6, 3–28 (Russian, with Russian summary); English transl., Izv. Math. 73 (2009), no. 6, 1077–1100. MR 2640976, DOI 10.1070/IM2009v073n06ABEH002473
- K\B{o}saku Yosida, Functional analysis, Classics in Mathematics, Springer-Verlag, Berlin, 1995. Reprint of the sixth (1980) edition. MR 1336382, DOI 10.1007/978-3-642-61859-8
- Tosio Kato, Perturbation theory for linear operators, Classics in Mathematics, Springer-Verlag, Berlin, 1995. Reprint of the 1980 edition. MR 1335452
- N. K. Nikol′skiĭ, Lektsii ob operatore sdviga, “Nauka”, Moscow, 1980 (Russian). MR 575166
- O. G. Parfënov, On the properties of embedding operators of some classes of analytic functions, Algebra i Analiz 3 (1991), no. 2, 199–222 (Russian); English transl., St. Petersburg Math. J. 3 (1992), no. 2, 425–446. MR 1137528
- O. G. Parfënov, Weighted estimates for the Fourier transform, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 222 (1995), no. Issled. po Lineĭn. Oper. i Teor. Funktsiĭ. 23, 151–162, 309 (Russian, with English and Russian summaries); English transl., J. Math. Sci. (New York) 87 (1997), no. 5, 3878–3885. MR 1359997, DOI 10.1007/BF02355829
- A. G. Poltoratskiĭ, Boundary behavior of pseudocontinuable functions, Algebra i Analiz 5 (1993), no. 2, 189–210 (Russian, with Russian summary); English transl., St. Petersburg Math. J. 5 (1994), no. 2, 389–406. MR 1223178
- Béla Sz.-Nagy and Ciprian Foiaş, Harmonic analysis of operators on Hilbert space, North-Holland Publishing Co., Amsterdam-London; American Elsevier Publishing Co., Inc., New York; Akadémiai Kiadó, Budapest, 1970. Translated from the French and revised. MR 0275190
- Einar Hille and Ralph S. Phillips, Functional analysis and semi-groups, American Mathematical Society Colloquium Publications, Vol. 31, American Mathematical Society, Providence, R.I., 1957. rev. ed. MR 0089373
- D. R. Yafaev, Mathematical scattering theory, Translations of Mathematical Monographs, vol. 105, American Mathematical Society, Providence, RI, 1992. General theory; Translated from the Russian by J. R. Schulenberger. MR 1180965, DOI 10.1090/mmono/105
- P. R. Ahern and D. N. Clark, Radial limits and invariant subspaces, Amer. J. Math. 92 (1970), 332–342. MR 262511, DOI 10.2307/2373326
- P. R. Ahern and D. N. Clark, On functions orthogonal to invariant subspaces, Acta Math. 124 (1970), 191–204. MR 264385, DOI 10.1007/BF02394571
- Grigori G. Amosov, Cocycle perturbation of quasifree algebraic $K$-flow leads to required asymptotic dynamics of associated completely positive semigroup, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 3 (2000), no. 2, 237–246. MR 1812699, DOI 10.1142/S0219025700000170
- G. G. Amosov and A. D. Baranov, On perturbations of the group of shifts on the line by unitary cocycles, Proc. Amer. Math. Soc. 132 (2004), no. 11, 3269–3273. MR 2073301, DOI 10.1090/S0002-9939-04-07423-4
- William Arveson, Continuous analogues of Fock space, Mem. Amer. Math. Soc. 80 (1989), no. 409, iv+66. MR 987590, DOI 10.1090/memo/0409
- Douglas N. Clark, One dimensional perturbations of restricted shifts, J. Analyse Math. 25 (1972), 169–191. MR 301534, DOI 10.1007/BF02790036
- E. B. Davies, Lipschitz continuity of functions of operators in the Schatten classes, J. London Math. Soc. (2) 37 (1988), no. 1, 148–157. MR 921753, DOI 10.1112/jlms/s2-37.121.148
- Vladimir Kapustin and Alexei Poltoratski, Boundary convergence of vector-valued pseudocontinuable functions, J. Funct. Anal. 238 (2006), no. 1, 313–326. MR 2253018, DOI 10.1016/j.jfa.2006.04.006
- Nikolai K. Nikolski, Operators, functions, and systems: an easy reading. Vol. 1, Mathematical Surveys and Monographs, vol. 92, American Mathematical Society, Providence, RI, 2002. Hardy, Hankel, and Toeplitz; Translated from the French by Andreas Hartmann. MR 1864396
- Donald Sarason, Sub-Hardy Hilbert spaces in the unit disk, University of Arkansas Lecture Notes in the Mathematical Sciences, vol. 10, John Wiley & Sons, Inc., New York, 1994. A Wiley-Interscience Publication. MR 1289670
Bibliographic 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
- 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
- © Copyright 2011 American Mathematical Society
- 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
- MathSciNet review: 2768959