In mathematics, a singular trace is a trace on a space of linear operators of a separable Hilbert space that vanishes
on operators of finite rank. Singular traces are a feature of infinite-dimensional Hilbert spaces such as the space of square-summable sequences and spaces of square-integrable functions. Linear operators on a finite-dimensional Hilbert space have only the zero functional as a singular trace since all operators have finite rank. For example, matrix algebras have no non-trivial singular traces and the matrix trace is the unique trace up to scaling.
American mathematician Gary Weiss and, later, British mathematician Nigel Kalton observed in the infinite-dimensional case that there are non-trivial singular traces on the ideal of trace class operators.[1][2]
Therefore, in distinction to the finite-dimensional case, in infinite dimensions the canonical operator trace is not the unique trace up to scaling. The operator trace is the continuous extension of the matrix trace from finite rank operators to all trace class operators, and the term singular derives from the fact that a singular trace vanishes where the matrix trace is supported, analogous to a singular measure vanishing where Lebesgue measure is supported.
Singular traces measure the asymptotic spectral behaviour of operators and have found applications in the noncommutative geometry of French mathematician Alain Connes.[3][4]
In heuristic terms, a singular trace corresponds to a way of summing
numbers a1, a2, a3, ... that is completely orthogonal or 'singular' with respect to the usual sum a1 + a2 + a3 + ... .
This allows mathematicians to sum sequences like the harmonic sequence (and operators with similar spectral behaviour) that are divergent for the usual sum. In similar terms a (noncommutative) measure theory or probability theory can be built for distributions like the Cauchy distribution (and operators with similar spectral behaviour) that do not have finite expectation in the usual sense.
^Weiss, Gary (1980). "Commutators of Hilbert-Schmidt Operators, II". Integral Equations and Operator Theory. 3 (4): 574–600. doi:10.1007/BF01702316. S2CID 189875793.
^N. J. Kalton (1989). "Trace-class operators and commutators". Journal of Functional Analysis. 86: 41–74. doi:10.1016/0022-1236(89)90064-5.
^
Connes, Alain (1988). "The action functional in noncommutative geometry" (PDF). Communications in Mathematical Physics. 117 (4): 673–683. Bibcode:1988CMaPh.117..673C. doi:10.1007/bf01218391. S2CID 14261310.
^
A. Connes (1995). Noncommutative Geometry(PDF). New York: Academic Press. ISBN 978-0-08-057175-1.
a singulartrace is a trace on a space of linear operators of a separable Hilbert space that vanishes on operators of finite rank. Singulartraces are...
defined using singular values. For example, the Ky Fan-k-norm is the sum of first k singular values, the trace norm is the sum of all singular values, and...
the largest singular value of M . {\displaystyle \mathbf {M} .} The last of the Ky Fan norms, the sum of all singular values, is the trace norm (also known...
singular values of A {\displaystyle A} . The second equality is proven by explicit computation of t r a c e ( A ∗ A ) {\displaystyle \mathrm {trace}...
The technological singularity—or simply the singularity—is a hypothetical future point in time at which technological growth becomes uncontrollable and...
larger than the space of trace class operators. Dixmier traces are examples of singulartraces. Some applications of Dixmier traces to noncommutative geometry...
Godzilla Singular Point (Japanese: ゴジラ S.P ＜シンギュラポイント＞, Hepburn: Gojira Shingyura Pointo) is a Japanese anime television series directed by Atsushi Takahashi...
In general relativity, a naked singularity is a hypothetical gravitational singularity without an event horizon. When there exists at least one causal...
used to compute the trace. This trace of trace-class operators generalizes the trace of matrices studied in linear algebra. All trace-class operators are...
invertible is called singular or degenerate. A square matrix with entries in a field is singular if and only if its determinant is zero. Singular matrices are...
"three or more"). English and other languages present number categories of singular or plural, both of which are cited by using the hash sign (#) or by the...
In time series analysis, singular spectrum analysis (SSA) is a nonparametric spectral estimation method. It combines elements of classical time series...
especially quantum information and the study of open quantum systems, the trace distance T is a metric on the space of density matrices and gives a measure...
ISBN 978-0-8176-4775-9. Lord, Steven; Sukochev, Fedor; Zanin, Dmitriy (2012). SingularTraces: Theory and Applications (1 ed.). De Gruyter. doi:10.1515/9783110262551...
operators on an infinite dimensional Hilbert space we require that they be trace class and self-adjoint, in which case similar definitions apply, but we...
The Singularities is a novel by the Irish author John Banville, published in 2022. It is based on characters and themes from the author's earlier novels...
be regular; otherwise, the statistical model is said to be singular. Examples of singular statistical models include the following: normal mixtures, binomial...
extremely complex grammatical number systems, with singular, dual, paucal, greater paucal, and plural. Traces of the dual and paucal can be found in some Slavic...
ISBN 978-3-11-019805-8 S. Lord, F. A. Sukochev. D. Zanin (2012). Singulartraces: theory and applications. Berlin: De Gruyter. doi:10.1515/9783110262551...
with a 2 + b c = a {\displaystyle a^{2}+bc=a} is idempotent. The only non-singular idempotent matrix is the identity matrix; that is, if a non-identity matrix...
showed that the singularity disappeared after a change of coordinates. In 1933, Georges Lemaître realized that this meant the singularity at the Schwarzschild...
changes its trace with respect to the metric unless f = 0 {\displaystyle f=0} . In Riemannian geometry and pseudo-Riemannian geometry, the trace-free Ricci...
Global Information