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Let $\mathcal{B}(H)$ denote the C*-algebra of all bounded operators on a separable infinite dimensional complex Hilbert space $H$. It is a well-known fact that $\mathcal{B}_0(H)$, the ideal of compact operators, is the unique, nontrivial, proper closed ideal in $\mathcal{B}(H)$. Furthermore, any proper ideal in $\mathcal{B}(H)$ must be contained in $\mathcal{B}_0(H)$.

Is there any reasonable characterization of all (not necessarily closed) ideals in $\mathcal{B}(H)$ which contain the ideal of finite rank operators (possibly under some definability constraints, like being Borel in the strong operator topology, etc)?

Also, not being an operator theorist, are there important ideals of this type other than $\mathcal{B}_0(H)$ and the Schatten $p$-ideals (including the trace-class and Hilbert-Schmidt operators)?

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  • $\begingroup$ Have you checked the book by A. Pietsch, "Operator Ideals" (North-Holland, 1980)? $\endgroup$ – Pedro Lauridsen Ribeiro Jun 20 '14 at 4:23
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    $\begingroup$ Consult R. Schatten [Norm ideals of completely continuous operators, Springer, Berlin, 1960; MR0119112 (22 #9878)] for various characterizations. $\endgroup$ – Bill Johnson Jun 20 '14 at 5:34
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    $\begingroup$ In fact, every non-zero two sided ideal of $B(H)$ contains the ideal of finite rank operators. $\endgroup$ – user23860 Jun 20 '14 at 11:31
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One of my ancient papers, joint work with Gary Weiss, concerns the question whether the ideal of compact operators is the sum of two properly smaller (and therefore not closed) ideals. We proved that the answer is affirmative if one assumes the continuum hypothesis (or various weaker assumptions). The paper contains a description of ideals that "reduces" this sort of question to set theory. Later work showed that the use of assumptions (like the continuum hypothesis) beyond the usual axioms of set theory is needed; it's consistent with ZFC that $K(H)$ is not the sum of two properly smaller ideals.

The joint paper with Gary Weiss is "A characterization and sum decomposition of operator ideals" [Trans. A. M. S. 246 (1978) 407-417]. For the later work, see my paper "Near Coherence of Filters II: Applications to operator ideals, the Stone-Cech remainder of a half-line, order-ideals of sequences, and slenderness of groups" [Trans. A. M. S. 300 (1987) 557-581] and the references there.

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If you are interested in closed ideals, then

  • for separable $H$ the only ideal is $K(H)$

  • for non-separable $H$ complete characterization may be read here

If you are inrested in all ideals, then they are between $F(H)$ and $K(H)$ and consisist of operators whose singular values belong to some order ideal in $c_0^+$.

For details see section I.8.7 here.

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    $\begingroup$ There are lots of (non-closed) ideals between the ideal of finite-rank operators and the ideal of compact operators. Some examples were mentioned already in the question. $\endgroup$ – Andreas Blass Jun 20 '14 at 14:15
  • $\begingroup$ @AndreasBlass, ok $\endgroup$ – Norbert Jun 20 '14 at 14:29

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