Remote Access Journal of the American Mathematical Society
Green Open Access

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?)

  • [AP] A. B. Aleksandrov and V. V. Peller, Hankel operators and similarity to a contraction, Internat. Math. Res. Notices 1996, no. 6, 263-275. CMP 96:11
  • [BR] O. Bratteli and D. Robinson, Operator algebras and quantum statistical mechanics II, Springer Verlag, New York, 1981. MR 82k:82013
  • [Bo1] J. Bourgain, New Banach space properties of the disc algebra and $H^{\infty }$, Acta Math. 152 (1984), 1-48. MR 85j:46091
  • [Bo2] J. Bourgain, On the similarity problem for polynomially bounded operators on Hilbert space, Israel J. Math. 54 (1986), 227-241. MR 88h:47024
  • [BP] D. Blecher and V. Paulsen, Tensor products of operator spaces, J. Funct. Anal. 99 (1991), 262-292. MR 93d:46095
  • [Bu] D. Burkholder, Distribution function inequalities for martingales, Ann. Probab. 1 (1973), 19-42. MR 51:1944
  • [D] P. Duren, Theory of $H^{p}$ spaces, Academic Press, New York, 1970. MR 42:3552
  • [Du] R. Durrett, Brownian motion and martingales in analysis, Wadsworth Math. Series, Belmont (California) (1984). MR 87a:60054
  • [ER] E. Effros and Z. J. Ruan, A new approach to operator spaces, Canadian Math. Bull. 34 (1991), 329-337. MR 93a:47045
  • [Fo] S. Foguel, A counterexample to a problem of Sz. Nagy, Proc. Amer. Math. Soc. 15 (1964), 788-790. MR 29:2646
  • [FS] C. Fefferman and E. Stein, $H^{p}$-spaces of several variables, Acta Math. 129 (1972), 137-193. MR 56:6263
  • [FW] C. Foias and J. P. Williams, On a class of polynomially bounded operators, Preprint (unpublished, 1979 or 1980?).
  • [Ga] A. Garsia, Martingale inequalities: seminar notes on recent progress, Benjamin, Reading, MA, 1973. MR 56:6844
  • [H] U. Haagerup, Injectivity and decomposition of completely bounded maps, in Operator Algebras and their Connection with Topology and Ergodic Theory, Springer Lecture Notes in Math. 1132 (1985), 170-222. MR 87i:46133
  • [Ha1] P. Halmos, Ten problems in Hilbert space, Bull. Amer. Math. Soc. 76 (1970), 887-933. MR 42:5066
  • [Ha2] P. Halmos, On Foguel's answer to Nagy's question, Proc. Amer. Math. Soc. 15 (1964), 791-793. MR 29:2647
  • [JP] M. Junge and G. Pisier, Bilinear forms on exact operator spaces and $B(H)\otimes B(H)$, Geometric and Functional Analysis (GAFA Journal) 5 (1995), 329-363. MR 96i:46071
  • [Le] A. Lebow, A power bounded operator which is not polynomially bounded, Mich. Math. J. 15 (1968), 397-399. MR 38:5047
  • [LPP] F. Lust-Piquard and G. Pisier, Noncommutative Khintchine and Paley inequalities, Ark. Mat. 29 (1991), 241-260. MR 94b:46011
  • [vN] J. von Neumann, Eine Spektraltheorie für allgemeine Operatoren eines unitären Raumes, Math. Nachr. 4 (1951), 49-131. MR 13:254a
  • [Ni] N. Nikolskii, Treatise on the shift operator, Springer Verlag, Berlin, 1986. MR 87i:47042
  • [Pa1] V. Paulsen, Completely bounded maps and dilations, Pitman Research Notes in Math., vol. 146, Longman, Wiley, New York, 1986. MR 88h:46111
  • [Pa2] V. Paulsen, Every completely polynomially bounded operator is similar to a contraction, J. Funct. Anal. 55 (1984), 1-17. MR 86c:47021
  • [Pa3] V. Paulsen, The maximal operator space of a normed space, Proc. Edinburgh Math. Soc. (2) 39 (1996), no. 2, 309-323. CMP 96:14
  • [Pag] L. Page, Bounded and compact vectorial Hankel operators, Trans. Amer. Math. Soc. 150 (1970), 529-540. MR 42:8327
  • [Pe1] V. Peller, Estimates of functions of power bounded operators on Hilbert space, J. Oper. Theory 7 (1982), 341-372. MR 83i:47019
  • [Pe2] V. Peller, Estimates of functions of Hilbert space operators, similarity to a contraction and related function algebras, Research Problems, Springer Lecture Notes 1043 (199) (1984), 199-204. MR 85k:46001
  • [Pe3] V. Peller, Vectorial Hankel operators, commutators and related operators of the Schatten von Neumann classes, Integral Equations and Operator Theory 5 (1982), 244-272. MR 83f:47024
  • [Pet] K. Petersen, Brownian motion, Hardy spaces and bounded mean oscillation, LMS Lecture Notes Series 28 (1979). MR 58:31383
  • [Pi1] G. Pisier, Factorization of linear operators and the Geometry of Banach spaces, CBMS (Regional Conferences of the A.M.S.) 60 (1986), Reprinted with corrections 1987. MR 88a:47020
  • [Pi2] G. Pisier, Factorization of operator valued analytic functions, Advances in Math. 93 (1992), 61-125. MR 93g:46075
  • [Pi3] G. Pisier, Multipliers and lacunary sets in non-amenable groups, Amer. J. Math. 117 (1995), 337-376. MR 96e:46078
  • [Pi4] G. Pisier, Similarity problems and completely bounded maps, Springer Lecture Notes 1618 (1995).
  • [Ro] R. Rochberg, A Hankel type operator arising in deformation theory, Proc. Symp. Pure Math. 35 (1979), 457-458. MR 80f:42001a
  • [Sa] D. Sarason, Generalized interpolation in $H^{\infty }$, Trans. Amer. Math. Soc. 127 (1967), 179-203. MR 34:8193
  • [SN] B. Sz.-Nagy, Completely continuous operators with uniformly bounded iterates, Publ. Math. Inst. Hungarian Acad. Sci. 4 (1959), 89-92. MR 21:7436
  • [SNF] B. Sz.-Nagy and C. Foias, Harmonic analysis of operators on Hilbert space, Akademiai Kiadó, Budapest, 1970. MR 43:947
  • [St] J. Stafney, A class of operators and similarity to contractions, Michigan Math. J. 41 (1994), 509-521. MR 95m:47061
  • [TJ] N. Tomczak-Jaegermann, Banach-Mazur distances and finite dimensional operator ideals, Longman, Pitman Monographs and Surveys in Pure and Applied Math. 38 (1989). MR 90k:46039
  • [Tr] S. Treil, Geometric methods in spectral theory of vector valued functions: some recent results, Operator Theory: Adv. Appl., vol. 42, Birkhauser, 1989, pp. 209-280. MR 91j:47036

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

Received by editor(s): March 11, 1996
Received by editor(s) in revised form: October 11, 1996
Article copyright: © Copyright 1997 American Mathematical Society

American Mathematical Society