A subideal is an ideal of an ideal of B(H) and a principal subideal is a principal ideal of an ideal of B(H). We determine necessary and sufficient conditions for a principal subideal to be an ideal of B(H). This generalizes to arbitrary ideals the 1983 work of Fong and Radjavi characterizing principal subideals of the ideal of compact operators that are also ideals of B(H). We then characterize all principal subideals. We also investigate the lattice structure of subideals as part of the general study of ideal lattices such as the often studied lattice structure of ideals of B(H). This study of subideals and the study of elementary operators with coefficient constraints are closely related.There are three natural kinds of principal J-ideals, namely, the classical principal J-ideals (S) J which we call principal linear J-ideals; principal J-ideals S J and principal real linear J-ideals (S) R J (Definition 2.1). Standard notation then dictates that we denote (S) = (S) B(H) . It is immediate that B(H)-ideals are always J-ideals but, as we shall see later, often not conversely ([4], see also Example 1.3 below).The main results of this paper, generalizing the 1983 work of Fong and Radjavi [4] and characterizing the principal subideals of B(H), are summarized in the following two theorems.For compact operators S, T, s(S) denotes the sequence of singular numbers (s-numbers) for S, and the product s(S)s(T) denote their pointwise product. THEOREM 1.1. For S ∈ J, the following are equivalent.(i) Any of the three types of principal J-ideals generated by S, (STHEOREM 1.2. The principal J-ideal, the principal linear J-ideal and the principal real linear J-ideal generated by S ∈ J are respectively given by S J = ZS + JS + SJ + J(S)J (S) J = CS + JS + SJ + J(S)J (S) R J = RS + JS + SJ + J(S)J. So J(S)J ⊆ JS + SJ + J(S)J ⊆ S J ⊆ (S) R J ⊆ (S) J ⊆ (S) which first two, J(S)J and JS + SJ + J(S)J respectively, are a B(H)-ideal and a J-ideal. Consequently, each of these three kinds of principal J-ideals have the common B(H)-ideal "nucleus" J(S)J, with the common J-ideal JS + SJ + J(S)J containing it. All these principal J-ideals, S J ⊆ (S) R J ⊆ (S) J , are distinct except under the following equivalent conditions. They all collapse to merely J(S)J = (S) = S J = (S) J = (S) R J if and only if the principal B(H)-ideal (S) is J-soft (that is, (S) = J(S) (Definition 2.5) in which case (S) = J(S) = J(S)J) if and only if any, and hence all of them, is a B(H)-ideal.Remark on Theorem 1.1-Proof (i)⇒(ii): Proof 2 may seem simpler or shorter but Proof 1 keeps the analysis in the same Hilbert space, it is more constructive, and it appears to us more useful. Proof of Theorem 1.2 Corollary 2.4 gives directly the explicit descriptions of (S) J , S J and (S) R J : S J = ZS + JS + SJ + J(S)J (S) J = CS + JS + SJ + J(S)J (S) R J = RS + JS + SJ + J(S)J from which it follows that J(S)J ⊆ JS + SJ + J(S)J ⊆ S J ⊆ (S) R J ⊆ (S) J ⊆ (S) SUBIDEALS OF OPERATORS All these finitely generated J-ideals, S J ⊆ (S) R J ⊆ (S) J , collapse to merely J(S)J = (S) = ...

Abstract. Denote by U I (H) the group of all unitary operators in 1+I where H is a separable infinite-dimensional complex Hilbert space and I is any twosided ideal of B(H). A Cartan subalgebra C of I is defined in this paper as a maximal abelian self-adjoint subalgebra of I and its conjugacy class is defined herein as the set of Cartan subalgebras {V CV * | V ∈ U I (H)}. For nonzero proper ideals I we construct an uncountable family of Cartan subalgebras of I with distinct conjugacy classes. This is in contrast to the by now classical observation of P. de La Harpe who noted that when I is any of the Schatten ideals, there is precisely one conjugacy class under the action of the full group of unitary operators on H. Our perspective is that the action of the full unitary group on Cartan subalgebras of I is transitive, while by shrinking to U I (H) we obtain an action with uncountably many orbits if {0} = I = B(H).In the case when I is a symmetrically normed ideal and is the dual of some Banach space, we show how the conjugacy classes of the Cartan subalgebras of I become smooth manifolds modeled on suitable Banach spaces. These manifolds are endowed with groups of smooth transformations given by the action of the group U I (H) on the orbits, and are equivariantly diffeomorphic to each other. We then find that there exists a unique diffeomorphism class of full flag manifolds of U I (H) and we give its construction. This resembles the case of compact Lie groups when one has a unique full flag manifold, since all the Cartan subalgebras are conjugated to each other.

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Operator ideals in B(H) are well understood and exploited but ideals inside them have only recently been studied starting with the 1983 seminal work of Fong and Radjavi and continuing with two recent articles by the authors of this survey. This article surveys this study embodied in these three articles. A subideal is a two-sided ideal of J (for specificity also called a J-ideal) for J an arbitrary ideal of B(H). In this terminology we alternatively call J a B(H)-ideal.This surveys [5], [13] and [14] in which we developed a complete characterization of all J-ideals generated by sets of cardinality strictly less than the cardinality of the continuum. So a central theme is the impact of generating sets for subideals on their algebraic structure. This characterization includes in particular finitely and countably generated J-ideals. It was obtained by first generalizing to arbitrary principal Jideals the 1983 work of Fong-Radjavi who determined which principal K(H)-ideals are also B(H)-ideals. A key property in our investigation turned out to be J-softness of a B(H)-ideal I inside J, that is, IJ = I, a generalization of a recent notion of K(H)-softness of B(H)-ideals introduced by Kaftal-Weiss and earlier exploited for Banach spaces by Mityagin and Pietsch. This study of subideals and the study of elementary operators with coefficient constraints are closely related. Here we also introduce and study a notion of subideal-traces where classical traces (unitarily invariant linear functionals) need not make sense for subideals that are not B(H)-ideals.