Hilbert–Schmidt operator

Short description: Nuclear operator of order 2; a bounded operator A on a Hilbert space H such that tr(A*A) is finite

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

[math]\displaystyle{ \|A\|^2_{\operatorname{HS}} \ \stackrel{\text{def}}{=}\ \sum_{i \in I} \|Ae_i\|^2_H, }[/math]

where [math]\displaystyle{ \{e_i: i \in I\} }[/math] is an orthonormal basis. The index set [math]\displaystyle{ I }[/math] 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, the Hilbert–Schmidt norm [math]\displaystyle{ \|\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]\displaystyle{ \{e_i\}_{i\in I} }[/math] and [math]\displaystyle{ \{f_j\}_{j\in I} }[/math] are such bases, then [math]\displaystyle{ \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]\displaystyle{ e_i = f_i, }[/math] then [math]\displaystyle{ \sum_i \|Ae_i\|^2 = \sum_i\|A^* e_i\|^2. }[/math] As for any bounded operator, [math]\displaystyle{ A = A^{**}. }[/math] Replacing [math]\displaystyle{ A }[/math] with [math]\displaystyle{ A^* }[/math] in the first formula, obtain [math]\displaystyle{ \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]\displaystyle{ x }[/math] and [math]\displaystyle{ y }[/math] in [math]\displaystyle{ H }[/math], define [math]\displaystyle{ x \otimes y : H \to H }[/math] by [math]\displaystyle{ (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]\displaystyle{ A }[/math] on [math]\displaystyle{ H }[/math] (and into [math]\displaystyle{ H }[/math]), [math]\displaystyle{ \operatorname{Tr}\left( A\left( x \otimes y \right) \right) = \left\langle A x, y \right\rangle }[/math].

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

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

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]\displaystyle{ \langle A, B \rangle_\text{HS} = \operatorname{Tr}(A^* B) = \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 H. They also form a Hilbert space, denoted by B_HS(H) or B₂(H), which can be shown to be naturally isometrically isomorphic to the tensor product of Hilbert spaces

[math]\displaystyle{ H^* \otimes H, }[/math]

where H^∗ is the dual space of H. 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, H is finite-dimensional.

Properties

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

See also

• Frobenius inner product – Binary operation, takes two matrices and returns a scalar
• Sazonov's theorem
• Trace class – Compact operator for which a finite trace can be defined

References

• Conway, John B. (1990). A course in functional analysis. New York: Springer-Verlag. ISBN 978-0-387-97245-9. OCLC 21195908. 
• Schaefer, Helmut H. (1999). Topological Vector Spaces. GTM. 3. New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135.