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St. Petersburg Mathematical Journal
St. Petersburg Mathematical Journal
ISSN 1547-7371(online) ISSN 1061-0022(print)

 

Cauchy-type integrals and singular measures


Author: V. V. Kapustin
Translated by: the author
Original publication: Algebra i Analiz, tom 24 (2012), nomer 5.
Journal: St. Petersburg Math. J. 24 (2013), 743-757
MSC (2010): Primary 47B35; Secondary 30H10
Published electronically: July 24, 2013
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Abstract: In an earlier paper by the author it was shown that, in the case of rank-two commutators the problem of existence of an averaged wave operator for a pair of unitary operators whose spectral measures are singular with respect to the Lebesgue measure can be rewritten in terms of Cauchy-type integrals. The approach to the problem presented in the paper is based upon truncated Toeplitz operators, convergence is analyzed in terms of their symbols, and the results obtained are applied to the boundary behavior of functions belonging to $ \ast $-invariant subspaces of the Hardy class $ H^2$.


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Additional Information

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

DOI: http://dx.doi.org/10.1090/S1061-0022-2013-01263-5
PII: S 1061-0022(2013)01263-5
Keywords: Cauchy-type integrals, Clark measures, truncated Toeplitz operators, averaged wave operators
Received by editor(s): March 1, 2012
Published electronically: July 24, 2013
Additional Notes: The author was partially supported by RFBR (grant no. 11-01-00584)
Article copyright: © Copyright 2013 American Mathematical Society