## A characterization and sum decomposition for operator ideals

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- by Andreas Blass and Gary Weiss PDF
- Trans. Amer. Math. Soc.
**246**(1978), 407-417 Request permission

## Abstract:

Let $L(H)$ be the ring of bounded operators on a separable Hubert space. Assuming the continuum hypothesis, we prove that in $L(H)$ every two-sided ideal that contains an operator of infinite rank is the sum of two smaller two-sided ideals. The proof involves a new combinatorial description of ideals of $L(H)$. This description is also used to deduce some related results about decompositions of ideals. Finally, we discuss the possibility of proving our main theorem under weaker assumptions than the continuum hypothesis and the impossibility of proving it without the axiom of choice.## References

- Arlen Brown, Carl Pearcy, and Norberto Salinas,
*Ideals of compact operators on Hilbert space*, Michigan Math. J.**18**(1971), 373β384. 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), 839β873. MR**5790**, DOI 10.2307/1968771
F. Hausdorff, - Kenneth Kunen,
*Ultrafilters and independent sets*, Trans. Amer. Math. Soc.**172**(1972), 299β306. MR**314619**, DOI 10.1090/S0002-9947-1972-0314619-7 - Kenneth Kunen,
*Some points in $\beta N$*, Math. Proc. Cambridge Philos. Soc.**80**(1976), no.Β 3, 385β398. MR**427070**, DOI 10.1017/S0305004100053032 - D. A. Martin and R. M. Solovay,
*Internal Cohen extensions*, Ann. Math. Logic**2**(1970), no.Β 2, 143β178. MR**270904**, DOI 10.1016/0003-4843(70)90009-4 - David Morris and Norberto Salinas,
*Semiprime ideals and irreducible ideals of the ring of bounded operators on Hilbert space*, Indiana Univ. Math. J.**23**(1973/74), 575β589. MR**326414**, DOI 10.1512/iumj.1974.23.23048 - Norberto Salinas,
*Symmetric norm ideals and relative conjugate ideals*, Trans. Amer. Math. Soc.**188**(1974), 213β240. MR**336371**, DOI 10.1090/S0002-9947-1974-0336371-3 - Robert M. Solovay,
*A model of set-theory in which every set of reals is Lebesgue measurable*, Ann. of Math. (2)**92**(1970), 1β56. MR**265151**, DOI 10.2307/1970696
G. Weiss,

*Die Graduierung nach dem Endverlauf*, Abh. SΓ€chs. Ges. Wiss.

**31**(1909), 295-334.

*Commutators and operator ideals*, Thesis, Univ. of Michigan, Ann Arbor, Mich., 1975.

## Additional Information

- © Copyright 1978 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**246**(1978), 407-417 - MSC: Primary 47D25; Secondary 03E50
- DOI: https://doi.org/10.1090/S0002-9947-1978-0515547-X
- MathSciNet review: 515547