Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Multipliers and essential norm on the Drury-Arveson space


Authors: Quanlei Fang and Jingbo Xia
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
Published electronically: December 16, 2010
Corrigendum: Proc. Amer. Math. Soc. 141 (2013), 363-368
MathSciNet review: 2784815
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: It is well known that for multipliers $ f$ of the Drury-Arveson space $ H_{n}^{2}$, $ \Vert f\Vert _{\infty}$ does not dominate the operator norm of $ M_{f}$. We show that in general $ \Vert f\Vert _{\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 [Enhancements On Off] (What's this?)

  • 1. N. Arcozzi, R. Rochberg and E. Sawyer, Carleson measures for the Drury-Arveson Hardy space and other Besov-Sobolev spaces on complex balls, Advances in Math, 218(4) (2008), 1107-1180. MR 2419381 (2009j:46062)
  • 2. W. Arveson, Subalgebras of $ C^*$-algebras. III. Multivariable operator theory, Acta Math. 181 (1998), 159-228. MR 1668582 (2000e:47013)
  • 3. W. Arveson, The curvature invariant of a Hilbert module over $ {\mathbf C}[z_{1}, \dots, z_{d}]$, J. Reine Angew. Math. 522 (2000), 173-236. MR 1758582 (2003a:47013)
  • 4. J. A. Ball and V. Bolotnikov, Interpolation problems for multipliers on the Drury-Arveson space: From Nevanlinna-Pick to abstract interpolation problem, Integr. Equ. Oper. Theory 62 (2008), 301-349. MR 2461123 (2010a:47034)
  • 5. J. A. Ball, T. T. Trent and V. Vinnikov, Interpolation and commutant lifting for multipliers on reproducing kernel Hilbert spaces, Op. Th. Adv. and App. 122 (2001), 89-138. MR 1846055 (2002f:47028)
  • 6. Z. Chen, Characterizations of Arveson's Hardy space, Complex Var. Theory Appl. 48 (2003), 453-465. MR 1974382 (2004c:32010)
  • 7. 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.
  • 8. S. W. Drury, A generalization of von Neumann's inequality to the complex ball, Proc. Amer. Math. Soc. 68 (1978), 300-304. MR 480362 (80c:47010)
  • 9. Q. Fang and J. Xia, Commutators and localization on the Drury-Arveson space, preprint, 2009.
  • 10. D. Greene, S. Richter and C. Sundberg, The structure of inner multipliers on spaces with complete Nevanlinna-Pick kernels, J. Funct. Anal. 194 (2002), 311-331. MR 1934606 (2003h:46038)
  • 11. P. Halmos, A Hilbert space problem book, Second edition. Graduate Texts in Mathematics, 19, Springer-Verlag, New York-Berlin, 1982. MR 675952 (84e:47001)
  • 12. S. McCullough and T. T. Trent, Invariant subspaces and Nevanlinna-Pick kernels, J. Funct. Anal. 178 (2000), 226-249. MR 1800795 (2002b:47006)
  • 13. W. Rudin, Function theory in the unit ball of $ {\text{\bf C}}^n$, Springer-Verlag, New York-Berlin, 1980. MR 601594 (82i:32002)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 47B10, 47B32, 47B38

Retrieve articles in all journals with MSC (2010): 47B10, 47B32, 47B38


Additional Information

Quanlei Fang
Affiliation: Department of Mathematics, State University of New York at Buffalo, Buffalo, New York 14260
Email: fangquanlei@gmail.com

Jingbo Xia
Affiliation: Department of Mathematics, State University of New York at Buffalo, Buffalo, New York 14260
Email: jxia@acsu.buffalo.edu

DOI: https://doi.org/10.1090/S0002-9939-2010-10680-9
Keywords: Multiplier, Drury-Arveson space
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
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society