Almost isometric operators: functional model, invariant subspaces, commutant. (English. Russian original) Zbl 0846.47007

Kapustin, V. V.

J. Math. Sci., New York 78, No. 2, 181-194 (1996); translation from Zap. Nauchn. Semin. POMI 201, 95-116 (1992).

Summary: A new function model for an arbitrary bounded operator on a Hilbert space is constructed. This model generalizes the model of Sz.-Nagy and Foiaş for contractions and seems to be useful for operators close to an isometry (in a sense). All the model spaces are Hilbert spaces, but instead of dilation a generalization of it is used. The model admits a symmetry with respect to the map \(z\mapsto 1/z\) of the complex plane. In terms of the model the question of lifting the commutant is investigated, a relationship between invariant subspaces of a unitary operator is established, and the characteristic function of the model operator is calculated. Some other problems are solved as well.

MSC:

47A45	Canonical models for contractions and nonselfadjoint linear operators
47A15	Invariant subspaces of linear operators

Keywords:

function model for an arbitrary bounded operator on a Hilbert space; operators close to an isometry; lifting the commutant

References:

[1]	A. A. Borovkov,Mathematical Statistics [in Russian], Nauka, Moscow (1984). · Zbl 0575.62002
[2]	H. Cramér, ”On asymptotic expansions for sums of independent random variables with a limiting stable distribution,”Sankhya Indian J. Stat. Ser. A 25, 13–24 (1963). · Zbl 0129.31105
[3]	G. Cristoph and W. Wolf,Convergence Theorems with a Stable Limit Law, Akad. Verl., Berlin (1993).
[4]	I. A. Ibragimov and Yu. V. Linnik,Independent and Stationary Sequences of Random Variables, Wolters-Noordhoff Publishing Gröningen (1971). · Zbl 0219.60027
[5]	M. Lipschutz, ”On the magnitude of the error in the approach to stable distributions,”Proc. Konikl. Nederl. Akad. Wetensch. A,59, No. 3, 281–294 (1956). · Zbl 0072.34501
[6]	A. V. Nagaev, ”Remarks on multidimensional local limit theorems,”Math. Notes,14, No. 4, 559–563 (1973).
[7]	A. V. Nagaev, ”New theorems on large deviations under the Cramér condition,” in:4th Vilnius Conference on Probab. Theory and Math. Statist. Abstracts,1, [in Russian], Inst. Math. and Cybernetics, Vilnius (1985), pp. 32–33.
[8]	H. I. Pereira, ”Tests for the characteristic exponent and the scale parameter of symmetric stable distributions,”Commun. Statist.-Simula.,19(4), 1465–1475 (1990). · Zbl 0850.62218 · doi:10.1080/03610919008812929
[9]	V. M. Zolotarev, ”Analog of the asymptotic distribution of Edgewort-Cramér for the case of approach to stable distributions laws,”Selected Transl. Math. Statist. and Probab.,9, American Mathematical Society, Providence (1971).
[10]	V. M. Zolotarev,One-Dimensional Stable Distributions. American Mathematical Society, Providence, RI (1986). · Zbl 0589.60015