Book contents
- Frontmatter
- Contents
- Preface
- 1 Introduction
- 2 Positive Maps
- 3 Completely Positive Maps
- 4 Dilation Theorems
- 5 Commuting Contractions on Hilbert Space
- 6 Completely Positive Maps into Mn
- 7 Arveson's Extension Theorems
- 8 Completely Bounded Maps
- 9 Completely Bounded Homomorphisms
- 10 Polynomially Bounded and Power-Bounded Operators
- 11 Applications to K-Spectral Sets
- 12 Tensor Products and Joint Spectral Sets
- 13 Abstract Characterizations of Operator Systems and Operator Spaces
- 14 An Operator Space Bestiary
- 15 Injective Envelopes
- 16 Abstract Operator Algebras
- 17 Completely Bounded Multilinear Maps and the Haagerup Tensor Norm
- 18 Universal Operator Algebras and Factorization
- 19 Similarity and Factorization
- Bibliography
- Index
10 - Polynomially Bounded and Power-Bounded Operators
Published online by Cambridge University Press: 24 November 2009
- Frontmatter
- Contents
- Preface
- 1 Introduction
- 2 Positive Maps
- 3 Completely Positive Maps
- 4 Dilation Theorems
- 5 Commuting Contractions on Hilbert Space
- 6 Completely Positive Maps into Mn
- 7 Arveson's Extension Theorems
- 8 Completely Bounded Maps
- 9 Completely Bounded Homomorphisms
- 10 Polynomially Bounded and Power-Bounded Operators
- 11 Applications to K-Spectral Sets
- 12 Tensor Products and Joint Spectral Sets
- 13 Abstract Characterizations of Operator Systems and Operator Spaces
- 14 An Operator Space Bestiary
- 15 Injective Envelopes
- 16 Abstract Operator Algebras
- 17 Completely Bounded Multilinear Maps and the Haagerup Tensor Norm
- 18 Universal Operator Algebras and Factorization
- 19 Similarity and Factorization
- Bibliography
- Index
Summary
Polynomially bounded and power-bounded operators have played an important role in the development of this area, and there are a number of interesting results, counterexamples, and open questions about these operators. In particular, we will present Foguel's example [98] of a power-bounded operator and Pisier's example [191] of a polynomially bounded operator that are not similar to contractions.
Recall that an operator T is power-bounded provided that there is a constant M such that ||T n|| ≤ M for all n ≥ 0. Clearly, if T = S-1CS with C a contraction, then T is power-bounded with ||T n|| ≤ ||S-1|| ||S||.
It is fairly easy to see (Exercise 10.1), by using the Jordan form, that a matrix T ∈ Mn is power-bounded if and only if it is similar to a contraction. Sz.-Nagy [229] proved that the same characterization holds when T is a compact operator. This led naturally to the conjecture that an arbitrary operator is similar to a contraction if and only if it is power-bounded. Foguel provided the first example of a power-bounded operator that is not similar to a contraction.
Recall that an operator is polynomially bounded provided there is a constant K such that ||p(T)|| ≤ K||p||∞ for every polynomial p, where the ∞-norm is the supremum norm over the unit disk.
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- Completely Bounded Maps and Operator Algebras , pp. 135 - 149Publisher: Cambridge University PressPrint publication year: 2003