Multipliers and essential norm on the Drury-Arveson space
HTML articles powered by AMS MathViewer
- by Quanlei Fang and Jingbo Xia PDF
- Proc. Amer. Math. Soc. 139 (2011), 2497-2504 Request permission
Corrigendum: Proc. Amer. Math. Soc. 141 (2013), 363-368.
Abstract:
It is well known that for multipliers $f$ of the Drury-Arveson space $H_{n}^{2}$, $\|f\|_{\infty }$ does not dominate the operator norm of $M_{f}$. We show that in general $\|f\|_{\infty }$ does not even dominate the essential norm of $M_{f}$. A consequence of this is that there exist multipliers $f$ of $H_{n}^{2}$ for which $M_f$ fails to be essentially hyponormal; i.e., if $K$ is any compact, self-adjoint operator, then the inequality $M_f^\ast M_f - M_fM_f^\ast + K \geq 0$ does not hold.References
- N. Arcozzi, R. Rochberg, and E. Sawyer, Carleson measures for the Drury-Arveson Hardy space and other Besov-Sobolev spaces on complex balls, Adv. Math. 218 (2008), no. 4, 1107–1180. MR 2419381, DOI 10.1016/j.aim.2008.03.001
- William Arveson, Subalgebras of $C^*$-algebras. III. Multivariable operator theory, Acta Math. 181 (1998), no. 2, 159–228. MR 1668582, DOI 10.1007/BF02392585
- William Arveson, The curvature invariant of a Hilbert module over $\textbf {C}[z_1,\cdots ,z_d]$, J. Reine Angew. Math. 522 (2000), 173–236. MR 1758582, DOI 10.1515/crll.2000.037
- Joseph A. Ball and Vladimir Bolotnikov, Interpolation problems for Schur multipliers on the Drury-Arveson space: from Nevanlinna-Pick to abstract interpolation problem, Integral Equations Operator Theory 62 (2008), no. 3, 301–349. MR 2461123, DOI 10.1007/s00020-008-1626-1
- Joseph A. Ball, Tavan T. Trent, and Victor Vinnikov, Interpolation and commutant lifting for multipliers on reproducing kernel Hilbert spaces, Operator theory and analysis (Amsterdam, 1997) Oper. Theory Adv. Appl., vol. 122, Birkhäuser, Basel, 2001, pp. 89–138. MR 1846055
- Zeqian Chen, Characterizations of Arveson’s Hardy space, Complex Var. Theory Appl. 48 (2003), no. 5, 453–465. MR 1974382, DOI 10.1080/0278107031000103421
- S. Costea, E. Sawyer and B. Wick, The Corona Theorem for the Drury-Arveson Hardy space and other holomorphic Besov-Sobolev spaces on the unit ball in $\mathbf {C}^{n}$, preprint.
- S. W. Drury, A generalization of von Neumann’s inequality to the complex ball, Proc. Amer. Math. Soc. 68 (1978), no. 3, 300–304. MR 480362, DOI 10.1090/S0002-9939-1978-0480362-8
- Q. Fang and J. Xia, Commutators and localization on the Drury-Arveson space, preprint, 2009.
- Devin C. V. Greene, Stefan Richter, and Carl Sundberg, The structure of inner multipliers on spaces with complete Nevanlinna-Pick kernels, J. Funct. Anal. 194 (2002), no. 2, 311–331. MR 1934606, DOI 10.1006/jfan.2002.3928
- Paul Richard Halmos, A Hilbert space problem book, 2nd ed., Encyclopedia of Mathematics and its Applications, vol. 17, Springer-Verlag, New York-Berlin, 1982. MR 675952
- Scott McCullough and Tavan T. Trent, Invariant subspaces and Nevanlinna-Pick kernels, J. Funct. Anal. 178 (2000), no. 1, 226–249. MR 1800795, DOI 10.1006/jfan.2000.3664
- Walter Rudin, Function theory in the unit ball of $\textbf {C}^{n}$, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 241, Springer-Verlag, New York-Berlin, 1980. MR 601594
Additional Information
- Quanlei Fang
- Affiliation: Department of Mathematics, State University of New York at Buffalo, Buffalo, New York 14260
- MR Author ID: 698351
- Email: fangquanlei@gmail.com
- Jingbo Xia
- Affiliation: Department of Mathematics, State University of New York at Buffalo, Buffalo, New York 14260
- MR Author ID: 215486
- Email: jxia@acsu.buffalo.edu
- Received by editor(s): April 11, 2010
- Received by editor(s) in revised form: July 1, 2010
- Published electronically: December 16, 2010
- Communicated by: Richard Rochberg
- © Copyright 2010
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 139 (2011), 2497-2504
- MSC (2010): Primary 47B10, 47B32, 47B38
- DOI: https://doi.org/10.1090/S0002-9939-2010-10680-9
- MathSciNet review: 2784815