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 In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise natu ...

and Calkin sequence spaces (also called rearrangement invariant sequence spaces). The correspondence is implemented by mapping an operator to its

singular value In mathematics, in particular functional analysis, the singular values, or ''s''-numbers of a compact operator T: X \rightarrow Y acting between Hilbert spaces X and Y, are the square roots of the (necessarily non-negative) eigenvalues of the se ...

sequence. It originated from

John von Neumann John von Neumann (; hu, Neumann János Lajos, ; December 28, 1903 – February 8, 1957) was a Hungarian-American mathematician, physicist, computer scientist, engineer and polymath. He was regarded as having perhaps the widest cove ...

's study of symmetric norms on matrix algebras. It provides a fundamental classification and tool for the study of two-sided ideals of

compact operator In functional analysis, a branch of mathematics, a compact operator is a linear operator T: X \to Y, where X,Y are normed vector spaces, with the property that T maps bounded subsets of X to relatively compact subsets of Y (subsets with compact ...

s 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 In functional analysis and related areas of mathematics, a sequence space is a vector space whose elements are infinite sequences of real or complex numbers. Equivalently, it is a function space whose elements are functions from the natural ...

''j'' within ''l''_∞ can be embedded in ''B''(''H'') using an arbitrary orthonormal basis _''n''=0^∞. Associate to a sequence ''a'' from ''j'' the bounded operator ::::

(a) = \sum_^\infty a_n ,  e_n \rangle \langle e_n , ,

where

bra–ket notation In quantum mechanics, bra–ket notation, or Dirac notation, is used ubiquitously to denote quantum states. The notation uses angle brackets, and , and a vertical bar , to construct "bras" and "kets". A ket is of the form , v \rangle. Mathem ...

has been used for the one-dimensional projections onto the subspaces spanned by individual basis vectors. The sequence of absolute values of the entries 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, ... Note that it is identical to the

s of the operator diag(''a''). Another notation for the decreasing rearrangement is ''a''*. A ''Calkin (or rearrangement invariant) sequence space'' is a linear subspace ''j'' of the bounded sequences ''l''_∞ such that if ''a'' is a bounded sequence and μ(''n'',''a'') ≤ μ(''n'',''b''), ''n'' 0, 1, 2, ..., for some ''b'' in ''j'', then ''a'' belongs to ''j''.

Correspondence

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

j = \ .

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

J = \ .

Here μ(''A'') and μ(''a'') are the

s of the operators ''A'' and diag(''a''), respectively. Calkin's Theorem states that the two maps are inverse to each other. We obtain, :Calkin correspondence: The two-sided ideals of

bounded operators In functional analysis and operator theory, a bounded linear operator is a linear transformation L : X \to Y between topological vector spaces (TVSs) X and Y that maps bounded subsets of X to bounded subsets of Y. If X and Y are normed vector sp ...

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

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

Examples

Suppose ''H'' is a separable infinite-dimensional Hilbert space. *

Bounded operators In functional analysis and operator theory, a bounded linear operator is a linear transformation L : X \to Y between topological vector spaces (TVSs) X and Y that maps bounded subsets of X to bounded subsets of Y. If X and Y are normed vector sp ...

. The improper two-sided ideal ''B''(''H'') corresponds to ''l''_∞. *

Compact operator In functional analysis, a branch of mathematics, a compact operator is a linear operator T: X \to Y, where X,Y are normed vector spaces, with the property that T maps bounded subsets of X to relatively compact subsets of Y (subsets with compact ...

s. The proper and norm closed two-sided ideal ''K''(''H'') corresponds to ''c''₀, the space of sequences converging to zero. * Finite rank operators. The smallest two-sided ideal ''F''(''H'') of finite rank operators corresponds to ''c''₀₀, 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. In particular, the trace class operators correspond to ''l''_''1'' and the Hilbert-Schmidt operators correspond to ''l''_''2'' . * 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 : Lorentz sequence spaces.

Notes

References

* * {{cite book , isbn=978-3-11-026255-1 , author= S. Lord, F. A. Sukochev. D. Zanin , year=2012 , url=http://www.degruyter.com/view/product/177778 , title=Singular traces: theory and applications , publisher=De Gruyter , location=Berlin Operator algebras Hilbert spaces Von Neumann algebras