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

Gilles Pisier

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

Functional Analysis 2016-09-06 v1

Authors: Gilles Pisier

Abstract

Let $\eps >0$ . We prove that there exists an operator $T_\eps:\ell_2\to\ell_2$ , such that for any polynomial $P$ we have $\|{P(T)}\| \leq(1+\eps)\|{P}\|_\infty$ , but which is not similar to a contraction, {\it i.e.} there does not exist an invertible operator $S:\ \ell_2\to\ell_2$ such that $\|{S^{-1}T_\eps S}\|\leq 1$ . This answers negatively a question attributed to Halmos after his well known 1970 paper (``Ten problems in Hilbert space").

Keywords

operator theory on hilbert spaces operator theory numerical range

Cite

@article{arxiv.math/9602207,
  title  = {A polynomially bounded operator on Hilbert space which is not similar to a contraction},
  author = {Gilles Pisier},
  journal= {arXiv preprint arXiv:math/9602207},
  year   = {2016}
}

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

Abstract

Keywords

Cite

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