Schatten Class Operator

In mathematics, specifically

functional analysis Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (e.g. inner product, norm, topology, etc.) and the linear functions defined ...

, a ''p''th Schatten-class operator is a

bounded linear operator 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 vect ...

on a

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 ...

with finite ''p''th

Schatten norm In mathematics, specifically functional analysis, the Schatten norm (or Schatten–von-Neumann norm) arises as a generalization of ''p''-integrability similar to the trace class norm and the Hilbert–Schmidt norm. Definition Let H_1, H_2 be ...

. The space of ''p''th Schatten-class operators is a

Banach space In mathematics, more specifically in functional analysis, a Banach space (pronounced ) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between ve ...

with respect to the Schatten norm. Via

polar decomposition In mathematics, the polar decomposition of a square real or complex matrix A is a factorization of the form A = U P, where U is an orthogonal matrix and P is a positive semi-definite symmetric matrix (U is a unitary matrix and P is a positive ...

, one can prove that the space of ''p''th Schatten class operators is an ideal in ''B(H)''. Furthermore, the

satisfies a type of Hölder inequality: :

\,  S T\,  _ \leq \,  S\,  _ \,  T\,  _ \ \mbox \ S \in S_p , \  T\in S_q \mbox  1/p+1/q=1.

If we denote by

S_\infty

the Banach space 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 on ''H'' with respect to the

operator norm In mathematics, the operator norm measures the "size" of certain linear operators by assigning each a real number called its . Formally, it is a norm defined on the space of bounded linear operators between two given normed vector spaces. Intr ...

, the above Hölder-type inequality even holds for

p \in,\infty

. From this it follows that

\phi : S_p \rightarrow S_q '

T \mapsto \mathrm{tr}(T\cdot )

is a well-defined contraction. (Here the prime denotes (topological) dual.) Observe that the ''2''nd Schatten class is in fact the Hilbert space of

Hilbert–Schmidt operator In mathematics, a Hilbert–Schmidt operator, named after David Hilbert and Erhard Schmidt, is a bounded operator A \colon H \to H that acts on a Hilbert space H and has finite Hilbert–Schmidt norm \, A\, ^2_ \ \stackrel\ \sum_ \, Ae_i\, ^2_ ...

s. Moreover, the ''1''st Schatten class is the space of

trace class 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- ...

operators. Operator theory