Hilbert–Schmidt operator

In mathematics, a Hilbert–Schmidt operator, named after David Hilbert and Erhard Schmidt, is a bounded operator <math> A \colon H \to H </math> that acts on a Hilbert space <math> H </math> and has finite Hilbert–Schmidt norm

<math display="block">\|A\|^2_{\operatorname{HS}} \ \stackrel{\text{def}}{=}\ \sum_{i \in I} \|Ae_i\|^2_H,</math>

where <math>\{e_i: i \in I\}</math> is an orthonormal basis.<ref name="MathWorld">{{#invoke:citation/CS1|citation |CitationClass=web }}</ref><ref name="EOM">Template:Eom</ref> The index set <math> I </math> need not be countable. However, the sum on the right must contain at most countably many non-zero terms, to have meaning.Template:Sfn This definition is independent of the choice of the orthonormal basis. In finite-dimensional Euclidean space, the Hilbert–Schmidt norm <math>\|\cdot\|_\text{HS}</math> is identical to the Frobenius norm.

‖·‖_HS is well defined

The Hilbert–Schmidt norm does not depend on the choice of orthonormal basis. Indeed, if <math>\{e_i\}_{i\in I}</math> and <math>\{f_j\}_{j\in I}</math> are such bases, then <math display="block"> \sum_i \|Ae_i\|^2 = \sum_{i,j} \left| \langle Ae_i, f_j\rangle \right|^2 = \sum_{i,j} \left| \langle e_i, A^*f_j\rangle \right|^2 = \sum_j\|A^* f_j\|^2. </math> If <math>e_i = f_i, </math> then <math display="inline"> \sum_i \|Ae_i\|^2 = \sum_i\|A^* e_i\|^2. </math> As for any bounded operator, <math> A = A^{**}. </math> Replacing <math> A </math> with <math> A^* </math> in the first formula, obtain <math display="inline"> \sum_i \|A^* e_i\|^2 = \sum_j\|A f_j\|^2. </math> 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 on a Hilbert space is a Hilbert–Schmidt operator if and only if the Hilbert space is finite-dimensional. Given any <math>x</math> and <math>y</math> in <math>H</math>, define <math>x \otimes y : H \to H</math> by <math>(x \otimes y)(z) = \langle z, y \rangle x</math>, which is a continuous linear operator of rank 1 and thus a Hilbert–Schmidt operator; moreover, for any bounded linear operator <math>A</math> on <math>H</math> (and into <math>H</math>), <math>\operatorname{Tr}\left( A\left( x \otimes y \right) \right) = \left\langle A x, y \right\rangle</math>.Template:Sfn

If <math>T: H \to H</math> is a bounded compact operator with eigenvalues <math>\ell_1, \ell_2, \dots</math> of <math>|T| := \sqrt{T^*T}</math>, where each eigenvalue is repeated as often as its multiplicity, then <math>T</math> is Hilbert–Schmidt if and only if <math display="inline">\sum_{i=1}^{\infty} \ell_i^2 < \infty</math>, in which case the Hilbert–Schmidt norm of <math>T</math> is <math display="inline">\left\| T \right\|_{\operatorname{HS}} = \sqrt{\sum_{i=1}^{\infty} \ell_i^2}</math>.Template:Sfn

If <math>k \in L^2\left( \mu \times \mu \right)</math>, where <math>\left( X, \Omega, \mu \right)</math> is a measure space, then the integral operator <math>K : L^2\left( \mu \right) \to L^2\left( \mu \right)</math> with kernel <math>k</math> is a Hilbert–Schmidt operator and <math>\left\| K \right\|_{\operatorname{HS}} = \left\| k \right\|_2</math>.Template:Sfn

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

<math display="block">\langle A, B \rangle_\text{HS} = \operatorname{Tr}(B^* A) = \sum_i \langle Ae_i, Be_i \rangle.</math>

The Hilbert–Schmidt operators form a two-sided *-ideal in the Banach algebra of bounded operators on Template:Math. They also form a Hilbert space, denoted by Template:Math or Template:Math, which can be shown to be naturally isometrically isomorphic to the tensor product of Hilbert spaces

<math display="block">H^* \otimes H,</math>

where Template:Math is the dual space of Template:Math. 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).Template:Sfn 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).Template:Sfn

The set of Hilbert–Schmidt operators is closed in the norm topology if, and only if, Template:Math is finite-dimensional.

Properties

• Every Hilbert–Schmidt operator Template:Math is a compact operator.Template:Sfn
• A bounded linear operator Template:Math is Hilbert–Schmidt if and only if the same is true of the operator <math display="inline">\left| T \right| := \sqrt{T^* T}</math>, in which case the Hilbert–Schmidt norms of T and |T| are equal.Template:Sfn
• Hilbert–Schmidt operators are nuclear operators of order 2, and are therefore compact operators.Template:Sfn
• If <math>S : H_1 \to H_2</math> and <math>T : H_2 \to H_3</math> are Hilbert–Schmidt operators between Hilbert spaces then the composition <math>T \circ S : H_1 \to H_3</math> is a nuclear operator.Template:Sfn
• If Template:Math is a bounded linear operator then we have <math>\left\| T \right\| \leq \left\| T \right\|_{\operatorname{HS}}</math>.Template:Sfn
• Template:Math is a Hilbert–Schmidt operator if and only if the trace <math>\operatorname{Tr}</math> of the nonnegative self-adjoint operator <math>T^{*} T</math> is finite, in which case <math>\|T\|^2_\text{HS} = \operatorname{Tr}(T^* T)</math>.<ref name="MathWorld"/><ref name="EOM"/>
• If Template:Math is a bounded linear operator on Template:Math and Template:Math is a Hilbert–Schmidt operator on Template:Math then <math>\left\| S^* \right\|_{\operatorname{HS}} = \left\| S \right\|_{\operatorname{HS}}</math>, <math>\left\| T S \right\|_{\operatorname{HS}} \leq \left\| T \right\| \left\| S \right\|_{\operatorname{HS}}</math>, and <math>\left\| S T \right\|_{\operatorname{HS}} \leq \left\| S \right\|_{\operatorname{HS}} \left\| T \right\|</math>.Template:Sfn In particular, the composition of two Hilbert–Schmidt operators is again Hilbert–Schmidt (and even a trace class operator).Template:Sfn
• The space of Hilbert–Schmidt operators on Template:Math is an ideal of the space of bounded operators <math>B\left( H \right)</math> that contains the operators of finite-rank.Template:Sfn
• If Template:Math is a Hilbert–Schmidt operator on Template:Math then <math display="block">\|A\|^2_\text{HS} = \sum_{i,j} |\langle e_i, Ae_j \rangle|^2 = \|A\|^2_2</math> where <math>\{e_i: i \in I\}</math> is an orthonormal basis of H, and <math>\|A\|_2</math> is the Schatten norm of <math>A</math> for Template:Math. In Euclidean space, <math>\|\cdot\|_\text{HS}</math> is also called the Frobenius norm.

See also

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References

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