A general scheme based on a commutation relation is proposed to give rise to a definition of generalized Toeplitz operators on a Banach space. Under suitable conditions the existence of a symbol is proved and its continuation to algebras generated by generalized Toeplitz operators is constructed. A stability theorem for the point spectrum of an operator from generalized Toeplitz algebras is established; as examples one considers the standard and operator valued Toeplitz operators on weighted Hardy spaces and on spaces of functions (distributions) with weighted Fourier transforms.
@article{bwmeta1.element.bwnjournal-article-smv116i1p1bwm,
author = {Lioudmila Nikolskaia},
title = {Stabilit\'e du spectre ponctuel d'op\'erateurs de Toeplitz g\'en\'eralis\'es},
journal = {Studia Mathematica},
volume = {113},
year = {1995},
pages = {1-22},
zbl = {0847.47022},
language = {fra},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-smv116i1p1bwm}
}
Nikolskaia, Lioudmila. Stabilité du spectre ponctuel d'opérateurs de Toeplitz généralisés. Studia Mathematica, Tome 113 (1995) pp. 1-22. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-smv116i1p1bwm/
[00000] [1] J. Bergh and J. Löfström, Interpolation Spaces. An Introduction, Springer, 1976. | Zbl 0344.46071
[00001] [2] A. Bottcher and B. Silbermann, Analysis of Toeplitz Operators, Akademie-Verlag, Berlin, 1989. | Zbl 0689.45009
[00002] [3] A. Devinatz and M. Shinbrot, General Wiener-Hopf operators, Trans. Amer. Math. Soc. 145 (1969), 467-494. | Zbl 0193.10901
[00003] [4] R. Douglas, Banach Algebra Techniques in the Theory of Toeplitz Operators, CBMS Regional Conf. Ser. in Math. 15, Amer. Math. Soc., 1973. | Zbl 0252.47025
[00004] [5] R. Douglas, Banach Algebra Techniques in Operator Theory, Academic Press, New York, 1972. | Zbl 0247.47001
[00005] [6] D. A. Herrero, T. J. Taylor and L. V. Wang, Variations of the point spectrum under compact perturbations, dans : Oper. Theory Adv. Appl. 32, Birkhäuser, 1988, 113-157.
[00006] [7] T. Kato, Perturbation Theory for Linear Operators, Springer, 1976. | Zbl 0342.47009
[00007] [8] L. N. Nikol'skaya, Criteria for stability of the point spectrum under completely continuous perturbations, Math. Notes 19 (1975), 946-949.
[00008] [9] L. N. Nikol'skaya, Stability of the eigenvalues of certain types of singular integral equations, J. Soviet Math. 32 (1986), 513-519.
[00009] [10] L. N. Nikol'skaya, Stability of characteristic values of singular integral equations, ibid. 50 (1990), 2027-2031.
[00010] [11] N. Nikolski, Treatise on the Shift Operator, Springer, 1986.
[00011] [12] L. Page, Bounded and compact vectorial Hankel operators, Trans. Amer. Math. Soc. 150 (1970), 529-539. | Zbl 0203.45701
[00012] [13] R. Rochberg, Toeplitz operators on weighted spaces, Indiana Univ. Math. J. 26 (1977), 291-298. | Zbl 0373.47018
[00013] [14] M. Rosenblum, Vectorial Toeplitz operators and the Fejér-Riesz Theorem, J. Math. Anal. Appl. 23 (1968), 139-147. | Zbl 0159.43102
[00014] [15] I. Singer, Theory of Bases, Vol. 1, Springer, 1975.
[00015] [16] S. Treil, Geometric aspects of the spectral function theory, dans : Oper. Theory Adv. Appl. 42, Birkhäuser, 1989, 209-280.
[00016] [17] I. E. Verbitski, Sur les multiplicateurs dans les espaces pondérés, dans : Propriétés spectrales des opérateurs, Mat. Issled. 45, Shtiintsa, Kishinev, 1977 (en russe).