In mathematics, a weak trace class operator is a
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 c ...
on a
separable 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 natural ...
''H'' with
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 self ...
s the same order as the
harmonic sequence.
When the dimension of ''H'' is infinite, the ideal of weak trace-class operators is strictly larger than the ideal of
trace class operator In mathematics, specifically functional analysis, a trace-class operator is a linear operator for which a trace may be defined, such that the trace is a finite number independent of the choice of basis used to compute the trace. This trace of trace ...
s, and has fundamentally different properties. The usual
operator trace on the trace-class operators does not extend to the weak trace class. Instead the ideal of weak trace-class operators admits an infinite number of linearly independent quasi-continuous traces, and it is the smallest two-sided ideal for which all traces on it are
singular trace
In mathematics, a singular trace is a Von Neumann algebra#Weights, states, and traces, trace on a space of linear operators of a separable Hilbert space that vanishes
on operators of finite-rank operator, finite rank. Singular traces are a feature ...
s.
Weak trace-class operators feature in the
noncommutative geometry
Noncommutative geometry (NCG) is a branch of mathematics concerned with a geometric approach to noncommutative algebras, and with the construction of ''spaces'' that are locally presented by noncommutative algebras of functions (possibly in some ge ...
of French mathematician
Alain Connes
Alain Connes (; born 1 April 1947) is a French mathematician, and a theoretical physicist, known for his contributions to the study of operator algebras and noncommutative geometry. He is a professor at the , , Ohio State University and Vande ...
.
Definition
A
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 c ...
''A'' on an infinite dimensional
separable 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 natural ...
''H'' is ''weak trace class'' if μ(''n'',''A'') O(''n''
−1), where μ(''A'') is the sequence of
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 self ...
s. In mathematical notation the two-sided
ideal
Ideal may refer to:
Philosophy
* Ideal (ethics), values that one actively pursues as goals
* Platonic ideal, a philosophical idea of trueness of form, associated with Plato
Mathematics
* Ideal (ring theory), special subsets of a ring considere ...
of all weak trace-class operators is denoted,
::::
where
are the compact operators. The term weak trace-class, or weak-''L''
1, is used because the operator ideal corresponds, in J. W. Calkin's
correspondence between two-sided ideals of bounded linear operators and rearrangement invariant sequence spaces, to the
weak-''l''1 sequence space.
Properties
* the weak trace-class operators admit a
quasi-norm defined by
::::
:making ''L''
1,∞ a quasi-Banach operator ideal, that is an ideal that is also a
quasi-Banach space.
See also
*
Lp space
In mathematics, the spaces are function spaces defined using a natural generalization of the Norm (mathematics)#p-norm, -norm for finite-dimensional vector spaces. They are sometimes called Lebesgue spaces, named after Henri Lebesgue , although ...
*
Spectral triple In noncommutative geometry and related branches of mathematics and mathematical physics, a spectral triple is a set of data which encodes a geometric phenomenon in an analytic way. The definition typically involves a Hilbert space, an algebra of op ...
*
Singular trace
In mathematics, a singular trace is a Von Neumann algebra#Weights, states, and traces, trace on a space of linear operators of a separable Hilbert space that vanishes
on operators of finite-rank operator, finite rank. Singular traces are a feature ...
*
Dixmier trace
In mathematics, the Dixmier trace, introduced by , is a non-normal trace on a space of linear operators on a Hilbert space larger than the space of trace class operators. Dixmier traces are examples of singular traces.
Some applications of Dixmie ...
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