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Open Access Published by De Gruyter Open Access July 29, 2013

On extended eigenvalues and extended eigenvectors of truncated shift

  • Hasan Alkanjo EMAIL logo
From the journal Concrete Operators

Abstract

In this paper we consider the truncated shift operator Su on the model space K2u := H2 θ uH2. We say that a complex number λ is an extended eigenvalue of Su if there exists a nonzero operator X, called extended eigenvector associated to λ, and satisfying the equation SuX = λXSu. We give a complete description of the set of extended eigenvectors of Su, in the case of u is a Blaschke product..

References

[1] H. Bercovici. Operator theory and arithmetic in H1, volume 26 of Mathematical Surveys and Monographs. American Mathematical Society, Providence, RI, 1988. 10.1090/surv/026/01Search in Google Scholar

[2] A. Biswas and S. Petrovic. On extended eigenvalues of operators. Integral Equations Operator Theory, 55(2):233–248, 2006. 10.1007/s00020-005-1381-5Search in Google Scholar

[3] N. K. Nikol0ski˘ı. Treatise on the shift operator, volume 273 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer-Verlag, Berlin, 1986. Spectral function theory, With an appendix by S. V. Hrušcev [S. V. Khrushchëv] and V. V. Peller, Translated from the Russian by Jaak Peetre. 10.1007/978-3-642-70151-1Search in Google Scholar

[4] M. Rosenblum. On the operator equation BX − XA = Q. Duke Math. J., 23:263–269, 1956. 10.1215/S0012-7094-56-02324-9Search in Google Scholar

[5] D. Sarason. Free interpolation in the Nevanlinna class. In Linear and complex analysis, volume 226 of Amer. Math. Soc. Transl. Ser. 2, pages 145–152. Amer. Math. Soc., Providence, RI, 2009. 10.1090/trans2/226/12Search in Google Scholar

[6] B. Sz.-Nagy and C. Foias. Harmonic analysis of operators on Hilbert space. Translated from the French and revised. North-Holland Publishing Co., Amsterdam, 1970.Search in Google Scholar

Received: 2013-02-20
Accepted: 2013-07-18
Published Online: 2013-07-29

©2013 Versita Sp. z o.o.

This content is open access.

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