Lie and Jordan ideals of operators on Hilbert space

Fong, C. K.; Miers, C. R.; Sourour, A. R.

doi:10.1090/S0002-9939-1982-0643740-0

Lie and Jordan ideals of operators on Hilbert space
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by C. K. Fong, C. R. Miers and A. R. Sourour PDF

Proc. Amer. Math. Soc. 84 (1982), 516-520 Request permission

Abstract:

A linear manifold $\mathfrak {L}$ in $\mathfrak {B}(\mathfrak {H})$ is a Lie ideal in $\mathfrak {B}(\mathfrak {H})$ if and only if there is an associative ideal $\mathfrak {J}$ such that $[\mathfrak {J},\mathfrak {B}(\mathfrak {H})] \subseteq \mathfrak {L} \subseteq \mathfrak {J} + {\mathbf {C}}I$. Also $\mathfrak {L}$ is a Lie ideal if and only if it contains the unitary orbit of every operator in it. On the other hand, a subset of $\mathfrak {B}(\mathfrak {H})$ is a Jordan ideal if and only if it is an associative ideal.

References

Joel Anderson, Commutators of compact operators, J. Reine Angew. Math. 291 (1977), 128–132. MR 442742, DOI 10.1515/crll.1977.291.128
Arlen Brown and Carl Pearcy, Compact restrictions of operators, Acta Sci. Math. (Szeged) 32 (1971), 271–282. MR 305119
J. W. Calkin, Two-sided ideals and congruences in the ring of bounded operators in Hilbert space, Ann. of Math. (2) 42 (1941), 839–873. MR 5790, DOI 10.2307/1968771
P. A. Fillmore, Sums of operators with square zero, Acta Sci. Math. (Szeged) 28 (1967), 285–288. MR 221301
Paul R. Halmos, A Hilbert space problem book, D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto, Ont.-London, 1967. MR 0208368
Paul R. Halmos and Shizuo Kakutani, Products of symmetries, Bull. Amer. Math. Soc. 64 (1958), 77–78. MR 100225, DOI 10.1090/S0002-9904-1958-10156-1
I. N. Herstein, Topics in ring theory, University of Chicago Press, Chicago, Ill.-London, 1969. MR 0271135
Carl Pearcy and David Topping, On commutators in ideals of compact operators, Michigan Math. J. 18 (1971), 247–252. MR 284853
H. Radjavi, The group generated by involutions, Proc. Roy. Irish Acad. Sect. A 81 (1981), no. 1, 9–12. MR 635572
Robert Schatten, Norm ideals of completely continuous operators, Ergebnisse der Mathematik und ihrer Grenzgebiete, (N.F.), Heft 27, Springer-Verlag, Berlin-Göttingen-Heidelberg, 1960. MR 0119112

Einige Sätze über Matrizen

13

A. R. Sourour, A note on integral operators, Acta Sci. Math. (Szeged) 41 (1979), no. 3-4, 375–379. MR 555431
Ian Stewart, The Lie algebra of endomorphisms of an infinite-dimensional vector space, Compositio Math. 25 (1972), 79–86. MR 318253
David M. Topping, On linear combinations of special operators, J. Algebra 10 (1968), 516–521. MR 231215, DOI 10.1016/0021-8693(68)90077-X

The unitarily invariant subspaces of

Gary Weiss, The Fuglede commutativity theorem modulo the Hilbert-Schmidt class and generating functions for matrix operators. II, J. Operator Theory 5 (1981), no. 1, 3–16. MR 613042