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)

 
 

 

On normal derivations


Author: Joel Anderson
Journal: Proc. Amer. Math. Soc. 38 (1973), 135-140
MSC: Primary 47B47; Secondary 47D99
DOI: https://doi.org/10.1090/S0002-9939-1973-0312313-6
MathSciNet review: 0312313
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Let $ {\Delta _T}$ be the derivation on $ \mathfrak{B}(\mathcal{H})$ defined by $ {\Delta _T}(X) = TX - XT(T,X \in \mathfrak{B}(\mathcal{H}))$. We prove that if $ T$ is an isometry or a normal operator, then the range of $ {\Delta _T}$ is orthogonal to the null space of $ {\Delta _T}$. Also, we prove that if $ T$ is normal with an infinite number of points in its spectrum then the closed linear span of the range and the null space of $ {\Delta _T}$ is not all of $ \mathfrak{B}(\mathcal{H})$.


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

  • [1] I. Colojoara and C. Foiaş, Theory of generalized spectral operators, Gordon and Breach, New York, 1968. MR 0394282 (52:15085)
  • [2] J. Hakeda and J. Tomiyama, On some extension properties of von Neumann algebras, Tôhoku Math. J. (2) 19 (1967), 315-323. MR 36 #5706. MR 0222656 (36:5706)
  • [3] P. R. Halmos, A Hilbert space problem book, Van Nostrand, Princeton, N.J., 1967. MR 34 #8178. MR 0208368 (34:8178)
  • [4] G. Lumer, Semi-inner-product spaces, Trans. Amer. Math. Soc. 100 (1961), 29-43. MR 24 #A2860. MR 0133024 (24:A2860)
  • [5] J. P. Williams, Finite operators, Proc. Amer. Math. Soc. 26 (1970), 129-136. MR 41 #9039. MR 0264445 (41:9039)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 47B47, 47D99

Retrieve articles in all journals with MSC: 47B47, 47D99


Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1973-0312313-6
Keywords: Derivation, commutant of a normal operator, commutant of an isometry, orthogonality, complemented subspace
Article copyright: © Copyright 1973 American Mathematical Society

American Mathematical Society