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)

 
 

 

An extension of the Fuglede-Putnam theorem to subnormal operators using a Hilbert-Schmidt norm inequality


Author: Takayuki Furuta
Journal: Proc. Amer. Math. Soc. 81 (1981), 240-242
MSC: Primary 47B20; Secondary 47A05, 47B10
DOI: https://doi.org/10.1090/S0002-9939-1981-0593465-4
MathSciNet review: 593465
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We prove that if $ A$ and $ {B^ * }$ are subnormal operators acting on a Hubert space, then for every bounded linear operator $ X$, the Hilbert-Schmidt norm of $ AX - XB$ is greater than or equal to the Hilbert-Schmidt norm of $ {A^ * }X - X{B^ * }$. In particular, $ AX = XB$ implies $ {A^ * }X = X{B^ * }$. In addition, if we assume $ X$ is a Hilbert-Schmidt operator, we can relax the subnormality conditions to hyponormality and still retain the inequality.


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

  • [1] S. K. Berberian, Note on a theorem of Fuglede and Putnam, Proc. Amer. Math. Soc. 10 (1959), 175-182. MR 0107826 (21:6548)
  • [2] -, Extensions of a theorem of Fuglede and Putnam, Proc. Amer. Math. Soc. 71 (1978), 113-114. MR 0487554 (58:7176)
  • [3] T. Furuta, Relaxation of normality in the Fuglede-Putnam theorem, Proc. Amer. Math. Soc. 77 (1979), 324-328. MR 545590 (80i:47037)
  • [4] P. R. Halmos, Shifts on Hilbert spaces, J. Reine Angew. Math. 208 (1961), 102-112. MR 0152896 (27:2868)
  • [5] -, A Hilbert space problem book, Van Nostrand, Princeton, NJ., 1967. MR 0208368 (34:8178)
  • [6] C. R. Putnam, On normal operators in Hilbert space, Amer. J. Math. 73 (1951), 357-362. MR 0040585 (12:717f)
  • [7] M. Rosenblum, On a theorem of Fuglede and Putnam, J. London Math. Soc. 33 (1958), 376-377. MR 0099598 (20:6037)
  • [8] G. Weiss, The Fuglede commutativity theorem modulo operator ideals (to appear). MR 619994 (82k:47037)
  • [9] -, Fuglede's commutativity theorem modulo the Hilbert-Schmidt class and generating functions for matrix operators. II (to appear).

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 47B20, 47A05, 47B10

Retrieve articles in all journals with MSC: 47B20, 47A05, 47B10


Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1981-0593465-4
Keywords: Subnormal operator, hyponormal operator, Hilbert-Schmidt class
Article copyright: © Copyright 1981 American Mathematical Society

American Mathematical Society