A polynomially bounded operator on Hilbert space which is not similar to a contraction

A Polynomially Bounded Operator on Hilbert Space Which Is Not Similar to a Contraction

Let ε > 0. We prove that there exists an operator Tε : `2 → `2such that for any polynomial P we have ‖P (Tε)‖ ≤ (1 +ε)‖P‖∞, but Tε isnot similar to a contraction, i.e. there does not exist an invertible operatorS : `2 → `2 such that‖S−1TεS‖ ≤ 1. This answers negatively a question at-tributed to Halmos after his well-known 1970 paper (“Ten problems in Hilbertspace”). ...

The Curvature of a Single Contraction Operator on a Hilbert Space ∗

This note studies Arveson’s curvature invariant for d-contractions T = (T1, T2, . . . , Td) for the special case d = 1, referring to a single contraction operator T on a Hilbert space. It establishes a formula which gives an easy-to-understand meaning for the curvature of a single contraction. The formula is applied to give an example of an operator with nonintegral curvature. Under the additio...

Operator Extensions on Hilbert Space

Polynomially Bounded Minimization Problems which are Hard to Approximate

Min PB is the class of minimization problems whose objective functions are bounded by a polynomial in the size of the input. We show that there exist several problems that are Min PB-complete with respect to an approximation preserving reduction. These problems are very hard to approximate; in polynomial time they cannot be approximated within nε for some ε > 0, where n is the size of the input...

A Hilbert space approach to self-similar systems

This paper investigates the structural properties of linear self-similar systems, using an invariant subspace approach. The self-similar property is interpreted in terms of invariance of the corresponding transfer function space to a given transformation in a Hilbert space, in a same way as the time invariance property for linear systems is related to the shift-invariance of the Hardy spaces. T...