DILATIONS FOR POLYNOMIALLY BOUNDED OPERATORS

Title & Authors
DILATIONS FOR POLYNOMIALLY BOUNDED OPERATORS
EXNER, GEORGE R.; JO, YOUNG SOO; JUNG, IL BONG;

Abstract
We discuss a certain geometric property X$\small{\_{}$ of dual algebras generated by a polynomially bounded operator and property ($\small{\mathbb{A}}$, $\small{\aleph}$, =$\small{\aleph}$) these are central to the study of $\small{\aleph}$$\small{\_{0}}$$\small{\times}$$\small{\aleph}$$\small{\_{0}}$-systems of simultaneous equations of weak$\small{^{*}}$-continuous linear functionals on a dual algebra. In particular, we prove that if T $\small{\in}$ $\small{\mathbb{A}}$$\small{^{M}}$ satisfies a certain sequential property, then T $\small{\in}$ $\small{\mathbb{A}}$$\small{^{M}}$$\small{\_{}$(H) if and only if the algebra A$\small{\_{T}}$ has property X$\small{\_{0, 1/M}}$, which is an improvement of Li-Pearcy theorem in [8].
Keywords
polynomially bounded operator;dual operator algebra.;
Language
English
Cited by
References
1.
H. Bercovici, C. Foias, and C. Pearcy, Dual algebras with applications to invariant subspaces and dilation theory, CBMS regional Conference Series, no. 56, Amer. Math. Soc., Providence, R. I. 1985

2.
L. Carleson, An interpolating problem for bounded analytic functions, Amer. J. Math. 80 (1958), 921-930

3.
K. Davidson and V. Paulsen, Polynomially bounded operators, J. Reine Angew.Math. 487 (1997), 153-170

4.
G. Exner and I. Jung, Compressions of absolutely continuous contractions, Acta Sci. Math. (Szeged) 59 (1994), 545-553

5.
J. Garnett, Bounded analytic functions, Academic Press, 1981

6.
P. Halmos, Ten problems in Hilbert space, Bull. Amer. Math. Soc. 76 (1970), 887-933

7.
W. Li, On polynomially bounded operators, I, Houston J. Math. 18 (1992), 73-96

8.
W. Li and C. Pearcy, On polynomially bounded operators, II, Houston J. Math. 21 (1995), 719-733

9.
J. Mujica, Linearization of holomorphic mappings on Banach spaces, Trans. Amer. Math. Soc. 324 (1991), 867-887

10.
J. Mujica, Linearization of holomorphic mappings on infinite dimensional spaces, Rev. Un. Mat. Argentina 37 (1991), 127-134

11.
G. Pisier, A polynomially bounded operator on Hilbert space which is not similar to a contraction, J. Amer. Math. Soc. 10 (1997), 351-369

12.
B. Sz.-Nagy and C. Foias, Harmonic analysis of operators on Hilbert space, North Holland Akademiai Kiado, Amsterdam/Budapest, 1970