In
mathematics, a Hilbert–Schmidt operator, named after
David Hilbert and
Erhard Schmidt
Erhard Schmidt (13 January 1876 – 6 December 1959) was a Baltic German mathematician whose work significantly influenced the direction of mathematics in the twentieth century. Schmidt was born in Tartu (german: link=no, Dorpat), in the Govern ...
, is a
bounded 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 vector ...
that acts on a
Hilbert space and has finite Hilbert–Schmidt norm
where
is an
orthonormal basis
In mathematics, particularly linear algebra, an orthonormal basis for an inner product space ''V'' with finite dimension is a basis for V whose vectors are orthonormal, that is, they are all unit vectors and orthogonal to each other. For examp ...
.
The index set
need not be countable. However, the sum on the right must contain at most countably many non-zero terms, to have meaning. This definition is independent of the choice of the orthonormal basis.
In finite-dimensional
Euclidean space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean ...
, the Hilbert–Schmidt norm
is identical to the
Frobenius norm
In mathematics, a matrix norm is a vector norm in a vector space whose elements (vectors) are matrices (of given dimensions).
Preliminaries
Given a field K of either real or complex numbers, let K^ be the -vector space of matrices with m ro ...
.
, , ·, , is well defined
The Hilbert–Schmidt norm does not depend on the choice of orthonormal basis. Indeed, if
and
are such bases, then
If
then
As for any bounded operator,
Replacing
with
in the first formula, obtain
The independence follows.
Examples
An important class of examples is provided by
Hilbert–Schmidt integral operators.
Every bounded operator with a finite-dimensional range (these are called operators of finite rank) is a Hilbert–Schmidt operator.
The
identity operator
Identity may refer to:
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Arts and entertainment Film and television
* ''Identity'' (1987 film), an Iranian film
* ''Identity'' (2003 film) ...
on a Hilbert space is a Hilbert–Schmidt operator if and only if the Hilbert space is finite-dimensional.
Given any
and
in
, define
by
, which is a continuous linear operator of rank 1 and thus a Hilbert–Schmidt operator;
moreover, for any bounded linear operator ''
'' on
(and into
),
.
If
is a bounded compact operator with eigenvalues
of
, where each eigenvalue is repeated as often as its multiplicity, then
is Hilbert–Schmidt if and only if
, in which case the Hilbert–Schmidt norm of
is
.
If
, where
is a measure space, then the integral operator
with kernel
is a Hilbert–Schmidt operator and
.
Space of Hilbert–Schmidt operators
The product of two Hilbert–Schmidt operators has finite
trace-class norm; therefore, if ''A'' and ''B'' are two Hilbert–Schmidt operators, the Hilbert–Schmidt inner product can be defined as
The Hilbert–Schmidt operators form a two-sided
*-ideal in the
Banach algebra
In mathematics, especially functional analysis, a Banach algebra, named after Stefan Banach, is an associative algebra A over the real or complex numbers (or over a non-Archimedean complete normed field) that at the same time is also a Banach ...
of bounded operators on .
They also form a Hilbert space, denoted by or , which can be shown to be
naturally isometrically isomorphic to the
tensor product of Hilbert spaces In mathematics, and in particular functional analysis, the tensor product of Hilbert spaces is a way to extend the tensor product construction so that the result of taking a tensor product of two Hilbert spaces is another Hilbert space. Roughly spea ...
where is the
dual space of .
The norm induced by this inner product is the Hilbert–Schmidt norm under which the space of Hilbert–Schmidt operators is complete (thus making it into a Hilbert space).
The space of all bounded linear operators of finite rank (i.e. that have a finite-dimensional range) is a dense subset of the space of Hilbert–Schmidt operators (with the Hilbert–Schmidt norm).
The set of Hilbert–Schmidt operators is closed in the
norm topology if, and only if, is finite-dimensional.
Properties
* Every Hilbert–Schmidt 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 ...
.
* A bounded linear operator is Hilbert–Schmidt if and only if the same is true of the operator
, in which case the Hilbert–Schmidt norms of ''T'' and , ''T'', are equal.
* Hilbert–Schmidt operators are
nuclear operator
In mathematics, nuclear operators are an important class of linear operators introduced by Alexander Grothendieck in his doctoral dissertation. Nuclear operators are intimately tied to the projective tensor product of two topological vector spac ...
s of order 2, and are therefore
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 ...
s.
* If
and
are Hilbert–Schmidt operators between Hilbert spaces then the composition
is a
nuclear operator
In mathematics, nuclear operators are an important class of linear operators introduced by Alexander Grothendieck in his doctoral dissertation. Nuclear operators are intimately tied to the projective tensor product of two topological vector spac ...
.
* If is a bounded linear operator then we have
.
* is a Hilbert–Schmidt operator if and only if the
trace of the nonnegative self-adjoint operator
is finite, in which case
.
* If is a bounded linear operator on and is a Hilbert–Schmidt operator on then
,
, and
. In particular, the composition of two Hilbert–Schmidt operators is again Hilbert–Schmidt (and even a
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 ...
).
* The space of Hilbert–Schmidt operators on is an
ideal of the space of bounded operators
that contains the operators of finite-rank.
* If is a Hilbert–Schmidt operator on then
where
is an
orthonormal basis
In mathematics, particularly linear algebra, an orthonormal basis for an inner product space ''V'' with finite dimension is a basis for V whose vectors are orthonormal, that is, they are all unit vectors and orthogonal to each other. For examp ...
of ''H'', and
is the
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 H ...
of
for . In
Euclidean space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean ...
,
is also called the
Frobenius norm
In mathematics, a matrix norm is a vector norm in a vector space whose elements (vectors) are matrices (of given dimensions).
Preliminaries
Given a field K of either real or complex numbers, let K^ be the -vector space of matrices with m ro ...
.
See also
*
Frobenius inner product
In mathematics, the Frobenius inner product is a binary operation that takes two matrices and returns a scalar. It is often denoted \langle \mathbf,\mathbf \rangle_\mathrm. The operation is a component-wise inner product of two matrices as though ...
*
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 ...
References
*
*
{{DEFAULTSORT:Hilbert-Schmidt Operator
Operator theory