The Twisted Hilbert Space Ideals

We study those operators on a Hilbert space that can be lifted or extended to any twisted Hilbert space. We prove that these form an ideal of operators which contains all the Schatten classes. We characterize those multiplication operators on ℓp\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ell _p$$\end{document} that are liftable/extensible through centralizers.


The Object of Study
A twisted Hilbert space is a short exact sequence of (quasi) Banach spaces and operators in which H and K are Hilbert spaces. Since "exact" means that the kernel of each arrow agrees with the image on the preceding, we see that ı is an isomorphic embedding and that π induces an isomorphism between H and T /ı [K]. Thus, less technically, we can regard T as a (quasi) Banach space containing K as a closed subspace in such a way that the corresponding quotient is H. This already implies that T is (isomorphic to) a Banach space, i.e., its quasinorm is equivalent to a convex norm; see [11,Theorem 4 Being clear that K is complemented in T if and only if T is itself isomorphic to a Hilbert space (namely, to K ⊕ 2 H), the first thing one must know on the subject is that nontrivial twisted Hilbert spaces do exist. We refer the reader to [10,15] for the early constructions and to [8], [1,Chapter 16] and the references therein for further developments. Readers who are not familiar with any of those references can stop here. From now on all (quasi) Banach spaces are assumed to be complex and all Hilbert spaces separable.

The Ideal of Liftable Operators
Let us begin with the elementary observation that operators in the Hilbert-Schmidt class S 2 (H) factorize through 1 and, by the lifting property of 1 and the fact that twisted Hilbert spaces are (isomorphic to) Banach spaces, are liftable:  is an orthonormal basis of H to define a surjective isometry I : H −→ H sending x n to −|x n , that is, I(ξ), η = n ξ n η n , where ξ = n ξ n x n and η = n η n x n . The basis (y n ) gives another isometry J : H −→ H sending y n to −|y n . Since f (y n ) = s n x n one has f ( −|x n ) = s n −|y n , that is, the following square is commutative: In order to gain a deeper understanding of L(H) we need the notion of a quasilinear map. A homogeneous mapping φ : X −→ Y acting between quasinormed spaces is said to be quasilinear if it obeys an estimate for some constant Q and all x, y ∈ X. The least constant for which the preceding inequiality holds is called the quasilinearity constant of φ and is denoted by Given a quasilinear map φ : called, with good reason, the sequence generated by φ. The preceding con- in which u is an isomorphism. (There is an equivalent description of extensions by means "factor systems" [16]; the connection between factor systems and quasilinear maps is explained in [6, Note 3.13.2].) Now suppose we intend to lift a given operator f : From now on we shall use the following notation: given a homogeneous map h : X −→ Y , acting between quasinormed spaces, we put and we say that h is bounded if h < ∞. In this way the inequality (2.2) Since all separable Hilbert spaces are isometrically isomorphic and all twisted Hilbert spaces arise from quasilinear maps we have proved: For an operator f ∈ B(H) the following conditions are equivalent: Let p ∈ (0, ∞). The Schatten class S p consists of those operators on H whose sequence of singular numbers (s n (f )) belongs to p . It is a quasi Banach space under the quasinorm f p = |(s n (f ))| p = n |(s n (f ))| p 1/p , and a Banach space if 1 ≤ p < ∞.
We now prove that L(H) is a quite large ideal: where C depends only on p and the quasilinearity constant of φ.
The key point is that if φ and Φ are as before, then for every finite-rank f the operator Φ(f ) is a good approximation of the quasilinear map φ • f : Pick x, y ∈ H and consider the rank-one operator y ⊗ x. Note that f (y ⊗ x) = y ⊗ f (x) for any f ∈ B(H) and that the operator norm of y ⊗ x is y x . We have by (a). Combining, Finally, for general f ∈ S p we can write f = n≥0 f n with f n ∈ F(H) and f n p ≤ 2 −n f p . If F n is a lifting of f n with F n ≤ M f n p then F = n≥0 F n is a lifting of f , which completes the proof.
Note that F may depend not only on the norms of the summands, but also on the Banach-Mazur distance between H ⊕ φ H and its Banach envelope which, in turn, depends only on Q(φ).
It is clear that an operator belongs to some S p if and only if its singular numbers are O(n −α ) for some α > 0, which does not necessarily agree with 1/p. An ideal that contains all the Schatten classes and is commonly regarded as reasonably small is the Macaev ideal S ω of those operators whose singular values satisfy n s n /n < ∞, with the obvious norm, see [9,17]. This ideal appears naturally in a surprisingly wide variety of situations, problems, results and applications, as a quick internet search reveals.
Let us denote by L unif and E unif the spaces of uniformly liftable and uniformly extensible operators on the ground Hilbert space H, respectively. These become normed ideals when equipped with The proof of Proposition 2.4 shows that L unif contains every S p and the inclusion is continuous. One has (we omit the proofs): • L unif = E unif , with equivalent norms.
• L unif is complete.
• The norms defined above do not vary if one considers bounded quasilinear maps and bounded linear maps only.
Actually one can "eliminate" the linear map from the definitions the existence of a universal constant C such that for every Φ ∈ Q(H) the infimum inf L Φ− L .
An immediate consequence of the completeness of L unif and Proposition 2.4 is that L unif is strictly larger than p<∞ S p .

Diagonal Operators and Centralizers
An idea that could work to show that liftable implies uniformly liftable is to find a single quasilinear map φ able to test liftability in the sense that f is liftable if (and only if) φ • f − L < ∞ for some linear map L. While we do not know if such a test exists, it turns out that this path is practicable when working with centralizers and multiplication operators, that is, moving from the linear category to the category of ∞ -modules. To avoid unnecessary complications we will not consider ∞ -modules explicitly and we give a simplified definition of the notion of centralizer that suffices to work with sequence spaces. From now on, if X is a (quasi) Banach sequence space we denote by X 0 the (usually dense) subspace of finitely supported elements of X.

Definition 3.1.
A centralizer on X is a homogenenous mapping Φ : X 0 −→ X satisfying the estimate Φ(ax) − aΦ(x) ≤ C a ∞ x for some C and all a ∈ ∞ , x ∈ X 0 .
Centralizers are automatically quasilinear maps and actually one has Q(Φ) ≤ MC(Φ), where M is a constant depending only (on the modulus of concavity of) X; see [13,Lemma 4.2], [14,Proposition 3.1] or [6,Lemma 3.12.2]. The notion of a centralizer was invented by Kalton isolating a crucial property that most derivations appearing in interpolation theory share. If you feel curious about why centralizers are not defined on the whole space X, take a look at [2,Corollary 2].
Let us present some important examples of centralizers. Let x : N −→ C be a sequence converging to zero. The rank-sequence of x is defined as that is, r x (n) is the place that |x(n)| occupies in the decreasing rearrangement of |x|. If ϕ : R 2 + −→ C is a Lipschitz function vanishing at the origin, then the map Φ : 0 is a centralizer; this is a specialization of [13,Theorem 3.1]. Actually these centralizers are symmetric in the sense that Φ(x • σ) = Φ(x) • σ when σ is a permutation of the integers. Taking ϕ(s, t) = s and ϕ(s, t) = t one obtains the Kalton-Peck and Kalton-alone maps given by respectively. A multiplication operator on a sequence space X is one of the form x −→ ax for some (necessarily bounded) sequence a. We denote by a • the multiplication operator induced by a. Note that a • : X −→ X = a ∞ and that multiplication operators leave X 0 invariant.
where dv denotes the Haar measure on the "Cantor group" Δ = {±1} N and the integral is taken in the Bochner sense, satisfies the same estimate as L and commutes with every v ∈ Δ in the sense thatL(vx) = vL(x). It quickly follows thatL is implemented by some (perhaps unbounded) sequence λ : N −→ C in the sense that L(x) = λx for every x ∈ 0 p . When 0 < p < 1 one can simply take λ(n) = (Le n )(n).
We emphasize that we are really interested on this notion only for p = 2. The reason for this sudden urge for generalization is: Let a be a bounded sequence. If the operator a • : p −→ p is liftable through centralizers for some 0 < p < ∞, then so does for all 0 < p < ∞.
Proof. This is a consequence of the fact that centralizers can be "shifted" from one p to any other. Precisely, if 0 < q < p < ∞ and Φ : 0 p −→ p is a centralizer, then the map Φ q : 0 q −→ q defined by Φ q (x) = u|x| q/s Φ(|x| q/p ), where q −1 = p −1 + s −1 and x = u|x| is the polar decomposition, is a centralizer on q ; cf. [3,Lemma 5]. Moreover, every centralizer on q arises in this way, up to strong equivalence [3,Corollary 3]. Here, two homogeneous maps are said to be strongly equivalent if their difference is bounded.
It therefore suffices to see that, given a ∈ ∞ , 0 < q < p < ∞ and a centralizer Φ : 0 p −→ p , the multiplication operator induced by a on 0 p lifts through Φ if and only if the corresponding operator on q lifts through Φ q , with the same witness sequence.
(=⇒) Let λ be a sequence witnessing that a • lifts through Φ, so that λx − aΦ(x) p ≤ K x p for all x ∈ 0 p . Given a finitely supported x ≥ 0 we can write x = x q/s x q/p , with x q = x q/s s x q/p p and, by Hölder inequality, λx − aΦ q (x) q = λx q/s x q/p − ax q/s Φ(x q/p ) q ≤ x q/s s λx q/p − aΦ(x q/p ) p ≤ K x q .
(⇐=) We need the following additional property of Φ q : there is a constant M so that for finitely supported x, y one has Φ q (xy) − xΦ(y) q ≤ M x s y p ; see [3, Proof of Lemma 5(b)]. Now, if we assume λz−aΦ(z) q ≤ M z q then for any finitely supported x, y one has λxy − axΦ(y) q ≤ M x s y p and since g p = sup f s≤1 fg q we are done.
Experience dictates that, more often than not, the most efficient way to prove things is to connect them with some result of Nigel Kalton. A minimal extension of 1 is an exact sequence of quasi Banach spaces