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Isometries and Hermitian Operators on Zygmund Spaces

Published online by Cambridge University Press:  20 November 2018

Fernanda Botelho
Fernanda Botelho*
Affiliation:
Department of Mathematical Sciences, The University of Memphis, Memphis, TN 38152, USA. e-mail: mbotelho@memphis.edu
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Abstract

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In this paper we characterize the isometries of subspaces of the little Zygmund space. We show that the isometries of these spaces are surjective and represented as integral operators. We also show that all hermitian operators on these settings are bounded

Keywords

46E1547B1547B38Zygmund spacesthe little Zygmund spaceHermitian operatorssurjective linear isometriesgenerators of one-parameter groups of surjective isometries
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 58 , Issue 2 , 01 June 2015 , pp. 241 - 249
Copyright
Copyright © Canadian Mathematical Society 2015

References

[1] Berkson, E. and Porta, H., Hermitian operators and one-parameter groups in Hardy spaces. Trans. Amer. Math. Soc. 185 (1973), 331–344 . http://dx.doi.Org/10.1090/S0002-9947-1973-0338833-0 Google Scholar
[2] Blasco, O., Contreras, M., Diaz-Madrigal, S., Martinez, J., and Siskakis, A., Semigroups of composition operators in BMOA and the extension of a theorem ofSarason. Michigan Math. J. 25 (1978), 101–115.Google Scholar
[3] Blasco, O., Contreras, M., Diaz-Madrigal, S., Martinez, J., Papadimitrakis, M., and Siskakis, A., Semigroups of composition operators and integral operators in spaces of analytic functions. Ann. Acad. Sci. Fenn. Math. 38 (2013), no. 1, 67–89. http://dx.doi.Org/10.5186/aasfm.2O13.3806 Google Scholar
[4] Bonsall, F. F. and Duncan, J., Complete normed algebras. Ergebnisse der Mathematik und ihrer Grenzgebiete, 80, Springer-Verlag, New York-Heidelberg, 1973.Google Scholar
[5] Botelho, F., Fleming, R., and Jamison, J., Extreme points and isometries on vector-valued Lipschitz spaces. J. Math. Anal. Appl. 381 (2011), no. 2, 821–832. http://dx.doi.Org/10.1016/j.jmaa.2O11.03.062 Google Scholar
[6] Botelho, F. and Jamison, J., Generalized bi-circular projections on spaces of analytic functions. Acta Sci. Math. (Szeged) 75 (2009), no. 3–4, 527–546.Google Scholar
[7] Cima, J. and Wogen, W., On isometries of the Bloch space. Illinois J. Math. 24 (1980), no. 2, 313–316.Google Scholar
[8] Colonna, F. and Li, S., Weighted composition operators from H ∞ into the Zygmund spaces. Complex Anal. Oper. Theory 7 (2013), no. 5,1495–1512. http://dx.doi.Org/10.1007/s11785-012-0260-8 Google Scholar
[9] de Leeuw, K., Rudin, W., and Wermer, J., The isometries of some functions spaces. Proc. Amer. Math. Soc. 11 (1960), 694–698. http://dx.doi.Org/10.1090/S0002-9939-1960-0121646-9 Google Scholar
[10] Engel, K-J. and Nagel, R., A short course on operator semigroups. Universitext, Springer, New York, 2006.Google Scholar
[11] Fleming, R. J. and Jamison, J. E., Isometries on Banach spaces: function spaces. Chapman & Hall/CRC Monographs and Surveys in Pure and Applied Mathematics, 129 Chapman & Hall/CRC, Boca Raton, FL, 2003.Google Scholar
[12] Fleming, R. J. and Jamison, J. E., Hermitian operators and isometries on sums of Banach spaces. Proc. Edinburgh Math. Soc. 32 (1989), no. 2,169–191. http://dx.doi.Org/10.1017/S0013091500028583 Google Scholar
[13] Hoffman, K., Banach spaces of analytic functions. Prentice-Hall Series in Modern Analysis, Prentice-Hall, Inc., Englewood Cliffs, NJ, 1962.Google Scholar
[14] Hornor, W. and Jamison, J. E., Isometries of some Banach spaces of analytic functions. Integral Equations Operator Theory 41 (2001), no. 4, 410–425. http://dx.doi.Org/10.1007/BF01202102 Google Scholar
[15] Lotz, H., Uniform convergence of operators on L°° and similar spaces. Math. Z. 190 (1985), no. 2, 207–220. http://dx.doi.Org/10.1007/BF01160459 Google Scholar
[16] Ye, S. and Hu, Q., Weighted composition operators on the Zygmund space. Abstr. Appl. Anal. 2012, Art. ID 462482. http://dx.doi.Org/10.1155/2012/462482 Google Scholar
[17] Zygmund, A., Trigonometric series. Vol. I, II. Third éd., Cambridge Mathematical Library, Cambridge University Press, 2002.Google Scholar