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)

 
 

 

Norms of elementary operators


Authors: Hong-Ke Du, Yue-Qing Wang and Gui-Bao Gao
Journal: Proc. Amer. Math. Soc. 136 (2008), 1337-1348
MSC (2000): Primary 47B47, 47A30
DOI: https://doi.org/10.1090/S0002-9939-07-09112-5
Published electronically: December 24, 2007
MathSciNet review: 2367107
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Let $ A_i$ and $ B_i$, $ 1\leq i\leq n$, be bounded linear operators acting on a separable Hilbert space $ \mathcal H$. In this note, we prove that $ \sup\{\parallel\sum_{i=1}^n A_iXB_i\parallel~: X\in \mathcal{B(H)}, \parallelX... ...{\parallel\sum_{i=1}^n A_iUB_i\parallel : UU^*=U^*U=I, U\in {\mathcal{B(H)}}\}.$ Moreover, we prove that there exists an operator $ X_0$ with $ \parallel X_0\parallel =1$ such that $ \parallel\sum_{i=1}^n A_iX_0B_i\parallel\ =\sup\{\parallel\sum_{i=1}^n A_iXB_i\parallel : X\in {\mathcal{B(H)}}, \parallelX\parallel \leq 1\}$ if and only if there exists a unitary $ U_0\in \mathcal{B(H)}$ such that $ \parallel\sum_{i=1}^n A_iU_0B_i\parallel\ =$ $ \sup\{\parallel\sum_{i=1}^n A_iXB_i\parallel : X\in {\mathcal{B(H)}}, \parallelX\parallel \leq 1\}.$


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

  • 1. M.D. Choi and C.K. Li, The ultimate estimate of the upper norm bound for the summation of operators, J. Funct. Anal. 232 (2006), 455-476. MR 2200742 (2006j:47010)
  • 2. H.K. Du and G.X. Ji, Norm attainability of elementary operators and derivations, Northeast. Math. J. 10(1994), No. 3, 396-400. MR 1319103 (96a:47060)
  • 3. B.P. Duggal, On the range closure of an elementary operator, Linear Algebra and Appl. 402(2005), 199-206. MR 2141084 (2005k:47073)
  • 4. B.P. Duggal, Range-kernel orthogonality of the elementary operator $ X\rightarrow\sum_{i=1}^nA_iXB_i-X,$ Linear Algebra and Appl. 337(2001), 79-86. MR 1856852 (2002i:47044)
  • 5. D.A. Herrero, Approximation of Hilbert space operators, Vol. I, Research Notes in Mathematics No. 72, Pitman Advanced Publishing Program, 1982. MR 676127 (85m:47001)
  • 6. J.G. Stampfli, The norm of a derivation, Pacific J. Math. 33(1970), 737-747. MR 0265952 (42:861)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 47B47, 47A30

Retrieve articles in all journals with MSC (2000): 47B47, 47A30


Additional Information

Hong-Ke Du
Affiliation: College of Mathematics and Information Science, Shaanxi Normal University, Xi’an 710062, People’s Republic of China
Email: hkdu@snnu.edu.cn

Yue-Qing Wang
Affiliation: College of Mathematics and Information Science, Shaanxi Normal University, Xi’an 710062, People’s Republic of China
Email: wangyq@163.com

Gui-Bao Gao
Affiliation: College of Mathematics and Information Science, Shaanxi Normal University, Xi’an 710062, People’s Republic of China
Email: gaoguibao@stu.snnu.edu.cn

DOI: https://doi.org/10.1090/S0002-9939-07-09112-5
Keywords: Elementary operator, norm-attainability, unitary
Received by editor(s): May 19, 2006
Received by editor(s) in revised form: February 12, 2007
Published electronically: December 24, 2007
Additional Notes: This research was partially supported by the National Natural Science Foundation of China (10571113).
Communicated by: Joseph A. Ball
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society