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)

 
 

 

The Fuglede-Putnam theorem
and a generalization of Barría's lemma


Authors: Toshihiro Okuyama and Keiichi Watanabe
Journal: Proc. Amer. Math. Soc. 126 (1998), 2631-2634
MSC (1991): Primary 47A62, 47A99; Secondary 47B20
DOI: https://doi.org/10.1090/S0002-9939-98-04355-X
MathSciNet review: 1451824
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Let $A$ and $B$ be bounded linear operators, and let $C$ be a partial isometry on a Hilbert space. Suppose that (1) $CA=BC$, (2) $\|A\|\ge \|B\|$, (3) $(C^*C)A=A(C^*C)$ and (4) $C(\|A\|^2-AA^*)^{1/2}=0$. Then we have $CA^*=B^*C$.


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

  • 1. J. Barría, The commutative product $V^*_1V_2=V_2V^*_1$ for isometries $V_1$ and $V_2$, Indiana Univ. Math. J. 28 (1979), 581-585. MR 80h:47021
  • 2. S. K. Berberian, Note on a theorem of Fuglede and Putnam, Proc. Amer. Math. Soc. 10 (1959), 175-182. MR 21:6548
  • 3. J. B. Conway, A Course in Functional Analysis (2nd ed.), Springer-Verlag, New York, 1990. MR 91e:46001
  • 4. B. Fuglede, A commutativity theorem for normal operators, Proc. Nat. Acad. Sci. 36 (1950), 35-40. MR 11:371c
  • 5. T. Furuta, On relaxation of normality in the Fuglede-Putnam theorem, Proc. Amer. Math. Soc. 77 (1979), 324-328. MR 80i:47037
  • 6. P. R. Halmos and L. J. Wallen, Powers of partial isometries, J. Math. Mech. 19 (1970), 657-663. MR 40:4801
  • 7. R. L. Moore, D. D. Rogers and T. T. Trent, A note on intertwining $M$-hyponormal operators, Proc. Amer. Math. Soc. 83 (1981), 514-516. MR 82j:47033
  • 8. C. R. Putnam, On normal operators in Hilbert space, Amer. J. Math. 73 (1951), 357-362. MR 12:717f
  • 9. J. G. Stampfli and B. L. Wadhwa, On dominant operators, Monatsh. Math. 84 (1977), 143-153. MR 56:16428
  • 10. K. Takahashi, On the converse of the Fuglede-Putnam theorem, Acta. Sci. Math. 43 (1981), 123-125. MR 82g:47018
  • 11. T. Yoshino, Remark on the generalized Fuglede-Putnam theorem, Proc. Amer. Math. Soc. 95 (1985), 571-572. MR 87i:47034

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (1991): 47A62, 47A99, 47B20

Retrieve articles in all journals with MSC (1991): 47A62, 47A99, 47B20


Additional Information

Toshihiro Okuyama
Affiliation: Department of Mathematics, Faculty of Science, Niigata University, Niigata 950-21, Japan
Address at time of publication: Tsuruoka Minami Highschool, 26-31 Wakaba-cho, Tsuruoka Yamagata-ken 997-0037, Japan
Email: wtnbk@scux.sc.niigata-u.ac.jp

Keiichi Watanabe
Affiliation: Department of Mathematics, Faculty of Science, Niigata University, Niigata 950-21, Japan
Address at time of publication: Department of Mathematics, Indiana University, Bloomington, Indiana 47405

DOI: https://doi.org/10.1090/S0002-9939-98-04355-X
Received by editor(s): October 19, 1995
Received by editor(s) in revised form: January 27, 1997
Communicated by: Palle E. T. Jorgensen
Article copyright: © Copyright 1998 American Mathematical Society

American Mathematical Society