Abstract
In this paper we shall establish necessary and sufficient conditions for a system of characteristic vectors of a nonunitary operator in Hilbert space to form an unconditional basis (i.e., a basis similar to an orthonormal basis). These conditions will be formulated in terms of the characteristic function of the operator and its functional model, as developed by B. S. Nagy and C. Foias [1]. Therefore, we shall consider only operators of contraction, which are in some (sufficiently weak) sense close to unitary, i.e., operators which are well-placed in the scheme of Nagy and Foias.
Keywords
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Literature Cited
B. S. Nagy and C. Foias, Analysé harmonique des operateurs de l’éspace de Hilbert, Budapest (1967).
N. K. Nikol’skii and B. S. Pavlov, “Bases of characteristic vectors, the characteristic function, and problems of interpolation in the Hardy space H2,” Dokl. Akad. Nauk SSSR (in press).
H. Helson, Lectures on Invariant Subspaces, Academic Press, New York (1964).
K. Hoffman, Banach Spaces of Analytic Functions, Prentice-Hall, Englewood Cliffs, N. J. (1962).
I. Ts. Gokhberg and M. G. Krein, Introduction to the Theory of Linear Non-self-adjoint Operators in Hilbert Space [in Russian], “Nauka,” Moscow (1965).
B. R. Mukminov, “On the expansion in terms of characteristic functions of dissipative kernels,” Dokl. Akad. Nauk SSSR, Vol. 99, No. 4, pp. 499–502 (1954).
I. M. Glazman, “On the expandibility with respect to a system of characteristic elements of dissipative operators,” Uspekh. Matem. Nauk, Vol. 13, No. 3 (81), pp. 179–181 (1958).
A. S. Markus, “On the basis of the root vectors of a dissipative operator,” Dokl. Akad. Nauk SSSR, Vol. 132, No. 3, (1960).
V. E. Katsnel’son, “On the conditions for the basis property of the system of root vectors of some classes of operators,” Funktsional. Analiz i Ego Prilozhen., Vol. 1, No. 2 (1967).
D. Sarason, “Generalized interpolation in H°°,” Trans. Am. Math. Soc., Vol. 127, No. 2, pp. 179–203 (1967).
H. S. Shapiro and A. L. Shields, “On some interpolation problems for analytic functions,” Am. J. Math., Vol. 83, No. 3, pp. 513–532 (1961).
L. Carleson, “An interpolation problem for bounded analytic functions,” Am. J. Math.,Vol. 80, No. 4, pp. 921–930 (1958).
D. Newman, “Interpolation in H°°,” Trans. Am. Math. Soc., Vol. 92, No. 3, pp. 501–507 (1959).
J. T. Schwartz and N. Dunford, Linear Operators, Vol. 1, Interscience, New York (1958).
L. Carleson, “Interpolations by bounded analytic functions and the corona problem,” Annals of Math., Vol. 76, No. 3, pp. 547–559 (1962).
V. P. Potapov, “Multiplicative structure of J-Nonexpanding matrix-functions,” Trudy Mosk. Matem. Obshch., Vol. 4, pp. 125–236 (1955).
Yu. P. Ginzburg, “A maximum principle for J-nonexpanding operator-functions and some of its consequences,” Izv. VUZov, Matem., No. 1, pp. 42–53 (1963).
Editor information
Rights and permissions
Copyright information
© 1970 Springer Science+Business Media New York
About this chapter
Cite this chapter
Nikol’skii, N.K., Pavlov, B.S. (1970). Expansions in Characteristic Vectors of Nonunitary Operators and the Characteristic Function. In: Ladyzhenskaya, O.A. (eds) Boundary Value Problems of Mathematical Physics and Related Aspects of Function Theory. Seminars in Mathematics, vol 11. Springer, Boston, MA. https://doi.org/10.1007/978-1-4757-4666-2_5
Download citation
DOI: https://doi.org/10.1007/978-1-4757-4666-2_5
Publisher Name: Springer, Boston, MA
Print ISBN: 978-1-4757-4668-6
Online ISBN: 978-1-4757-4666-2
eBook Packages: Springer Book Archive