Joins and intersections of ideals of compact operators

Schweinsberg, Allen

doi:10.1090/S0002-9939-1976-0454670-9

Joins and intersections of ideals of compact operators
HTML articles powered by AMS MathViewer

by Allen Schweinsberg PDF

Proc. Amer. Math. Soc. 59 (1976), 268-272 Request permission

Abstract:

It is shown that certain well-known ideals of compact operators are the intersection of a decreasing, countable family of strictly larger ideals. Also, it is shown that if ${T_1}$ and ${T_2}$ are compact operators, neither of which lies in the principal ideal generated by the other, and if $\mathcal {I}$ is an arbitrary countably generated ideal, then there exist ideals ${\mathcal {J}_1}$ and ${\mathcal {J}_2}$ such that $\mathcal {I} \subseteq {\mathcal {J}_1} \vee {\mathcal {J}_2}$ and ${T_i} \notin {\mathcal {J}_i},\;i = 1, 2$.

References

Arlen Brown, Carl Pearcy, and Norberto Salinas, Ideals of compact operators on Hilbert space, Michigan Math. J. 18 (1971), 373–384. MR 291819
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
Norberto Salinas, Ideal sets and ideals of compact operators on Hilbert space, Indiana Univ. Math. J. 22 (1972/73), 505–521. MR 0317094, DOI 10.1512/iumj.1972.22.22043
Norberto Salinas, Symmetric norm ideals and relative conjugate ideals, Trans. Amer. Math. Soc. 188 (1974), 213–240. MR 336371, DOI 10.1090/S0002-9947-1974-0336371-3
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
Allen Schweinsberg, Principal ideals of compact operators, Indiana Univ. Math. J. 25 (1976), no. 3, 229–233. MR 399882, DOI 10.1512/iumj.1976.25.25019