Symmetric norm ideals and relative conjugate ideals

Salinas, Norberto

doi:10.1090/S0002-9947-1974-0336371-3

Symmetric norm ideals and relative conjugate ideals
HTML articles powered by AMS MathViewer

by Norberto Salinas

Trans. Amer. Math. Soc. 188 (1974), 213-240

DOI: https://doi.org/10.1090/S0002-9947-1974-0336371-3

PDF | Request permission

Abstract:

In this paper some aspects of the algebraic structure of the ring of all bounded linear operators on an infinite dimensional separable complex Hilbert space are discussed. In particular, a comparison criterion for maximal and minimal norm ideals is established. Also, a general notion of the conjugate of an ideal relative to another ideal is studied and some questions concerning joins and intersections of ideals are solved.

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
Ky Fan, Maximum properties and inequalities for the eigenvalues of completely continuous operators, Proc. Nat. Acad. Sci. U.S.A. 37 (1951), 760–766. MR 45952, DOI 10.1073/pnas.37.11.760
D. J. H. Garling, On ideals of operators in Hilbert space, Proc. London Math. Soc. (3) 17 (1967), 115–138. MR 208398, DOI 10.1112/plms/s3-17.1.115
I. C. Gohberg and M. G. Kreĭn, Introduction to the theory of linear nonselfadjoint operators, Translations of Mathematical Monographs, Vol. 18, American Mathematical Society, Providence, R.I., 1969. Translated from the Russian by A. Feinstein. MR 0246142, DOI 10.1090/mmono/018
I. C. Gohberg and M. G. Kreĭn, On the theory of triangular representations of nonselfadjoint operators, Dokl. Akad. Nauk SSSR 137 (1961), 1034–1037 (Russian). MR 0139946
Alfred Horn, On the singular values of a product of completely continuous operators, Proc. Nat. Acad. Sci. U.S.A. 36 (1950), 374–375. MR 45316, DOI 10.1073/pnas.36.7.374
B. S. Mitjagin, Normed ideals of intermediate type, Izv. Akad. Nauk SSSR Ser. Mat. 28 (1964), 819–832 (Russian). MR 0173935
B. S. Mitjagin and A. S. Švarc, Functors in categories of Banach spaces, Uspehi Mat. Nauk 19 (1964), no. 2 (116), 65–130 (Russian). MR 0166593
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, Ideals of commutators of compact operators, Acta Sci. Math. (Szeged) 36 (1974), 131–144. MR 358425
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, DOI 10.1007/978-3-642-87652-3
Hermann Weyl, Inequalities between the two kinds of eigenvalues of a linear transformation, Proc. Nat. Acad. Sci. U.S.A. 35 (1949), 408–411. MR 30693, DOI 10.1073/pnas.35.7.408
Adriaan Cornelis Zaanen, Linear analysis. Measure and integral, Banach and Hilbert space, linear integral equations, Interscience Publishers Inc., New York; North-Holland Publishing Co., Amsterdam; P. Noordhoff N. V., Groningen, 1953. MR 0061752