Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS

Remote Access
Green Open Access
Journal of the American Mathematical Society
Journal of the American Mathematical Society
ISSN 1088-6834(online) ISSN 0894-0347(print)


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

Author: Gilles Pisier
Journal: J. Amer. Math. Soc. 10 (1997), 351-369
MSC (1991): Primary 47A20, 47B35, 47D25, 47B47; Secondary 47A56, 42B30
MathSciNet review: 1415321
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Let $\varepsilon >0$. We prove that there exists an operator $T_{\varepsilon }:\ell _{2}\to \ell _{2}$ such that for any polynomial $P$ we have $\|{P(T_{\varepsilon })}\| \leq (1+\varepsilon ) \|{P}\|_{\infty }$, but $T_{\varepsilon }$ is not similar to a contraction, i.e. there does not exist an invertible operator $S:\ \ell _{2}\to \ell _{2}$ such that $\|{S^{-1}T_{\varepsilon }S}\|\leq 1$. This answers negatively a question attributed to Halmos after his well-known 1970 paper (``Ten problems in Hilbert space"). We also give some related finite-dimensional estimates.

References [Enhancements On Off] (What's this?)

Similar Articles

Retrieve articles in Journal of the American Mathematical Society with MSC (1991): 47A20, 47B35, 47D25, 47B47, 47A56, 42B30

Retrieve articles in all journals with MSC (1991): 47A20, 47B35, 47D25, 47B47, 47A56, 42B30

Additional Information

Gilles Pisier
Affiliation: Department of Mathematics, Texas A&M University, College Station, Texas 77843; Université Paris VI, Equipe d’Analyse, Case 186, 75252 Paris Cedex 05, France

PII: S 0894-0347(97)00227-0
Received by editor(s): March 11, 1996
Received by editor(s) in revised form: October 11, 1996
Article copyright: © Copyright 1997 American Mathematical Society

Comments: Email Webmaster

© Copyright , American Mathematical Society
Contact Us · Sitemap · Privacy Statement

Connect with us Facebook Twitter Google+ LinkedIn Instagram RSS feeds Blogs YouTube Podcasts Wikipedia