Calkin correspondence

In mathematics, the Calkin correspondence, named after mathematician John Williams Calkin, is a bijective correspondence between two-sided ideals of bounded linear operators of a separable infinite-dimensional Hilbert space and Calkin sequence spaces (also called rearrangement invariant sequence spaces). The correspondence is implemented by mapping an operator to its singular value sequence.

It originated from John von Neumann's study of symmetric norms on matrix algebras.^[1] It provides a fundamental classification and tool for the study of two-sided ideals of compact operators and their traces, by reducing problems about operator spaces to (more resolvable) problems on sequence spaces.

Definitions

A two-sided ideal J of the bounded linear operators B(H) on a separable Hilbert space H is a linear subspace such that AB and BA belong to J for all operators A from J and B from B(H).

A sequence space j within l_∞ can be embedded in B(H) using an arbitrary orthonormal basis {e_n }_n=0^∞. Associate to a sequence a from j the bounded operator

${\rm diag}(a) = \sum_{n=0}^\infty a_n | e_n \rangle \langle e_n |,$

where bra–ket notation has been used for the one-dimensional projections onto the subspaces spanned by individual basis vectors. The sequence of absolute values of the elements of a in decreasing order is called the decreasing rearrangement of a. The decreasing rearrangement can be denoted μ(n,a), n = 0, 1, 2, ..., since it is identical to the singular values of the operator diag(a). Another notation for the decreasing rearrangement is a*.

A Calkin (or rearrangement invariant) sequence space j is a linear subspace of the bounded sequences l_∞ such that μ(n,a) ≤ μ(n,b), n = 0, 1, 2, ..., for some b from j implies that the bounded sequence a belongs to j.

Correspondence

Associate to a two-sided ideal J the sequence space j given by

$j = \{ a \in l_\infty : {\rm diag}(\mu(a)) \in J \} .$

Associate to a sequence space j the two-sided ideal J given by

$J = \{ A \in B(H) : \mu(A) \in j \} .$

Here μ(A) and μ(a) are the singular values of the operators A and diag(a), respectively. Calkin's Theorem^[2] states that the two maps are inverse to each other. We obtain,

Calkin correspondence: The two-sided ideals of bounded operators on an infinite dimensional separable Hilbert space and the Calkin sequence spaces are in bijective correspondence.

It is sufficient to know the association only between positive operators and positive sequences, hence the map μ: J₊ → j₊ from a positive operator to its singular values implements the Calkin correspondence.

Another way of interpreting the Calkin correspondence, since the sequence space j is equivalent to the diagonal of the operator ideal J with respect to an arbitrary orthonormal basis, is that two-sided ideals are completely determined by their diagonals.

Examples

Suppose H is a separable infinite-dimensional Hilbert space.

• Bounded operators. The improper two-sided ideal B(H) corresponds to l_∞.

• Compact operators. The proper and norm closed two-sided ideal K(H) corresponds to c₀, the space of sequences converging to zero.

• Finite rank operators. The smallest two-sided ideal F(H) of finite rank operators corresponds to c₀₀, the space of sequences with finite non-zero terms.

• Schatten p-ideals. The Schatten p-ideals L_p, p ≥ 1, correspond to the l_p sequence spaces.

• Weak-L_p ideals. The weak-L_p ideals L_p,∞, p ≥ 1, correspond to the weak-l_p sequence spaces.

• Lorentz ψ-ideals. The Lorentz ψ-ideals for an increasing concave function ψ : [0,∞) → [0,∞) correspond to the Lorentz sequence spaces.

Notes

1. J. von Neumann (1937). "Some matrix inequalities and metrization of matrix space". Tomsk. University Review 1: 286–300.
2. J. W. Calkin (1941). "Two-sided ideals and congruences in the ring of bounded operators in Hiulbert space". Ann. Math. (2) 42 (4): 839–873. doi:10.2307/1968771.

References

• B. Simon (2005). Trace ideals and their applications. Providence, Rhode Island: Amer. Math. Soc. ISBN 978-0-8218-3581-4.

• S. Lord, F. A. Sukochev. D. Zanin (2012). Singular traces: theory and applications. Berlin: De Gruyter. ISBN 978-3-11-026255-1.