Hilbert–Schmidt operator

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

‖ A ‖ HS 2   = def   ∑ i ∈ I ‖ A e i ‖ H 2 , {\displaystyle \|A\|_{\operatorname {HS} }^{2}\ {\stackrel {\text{def}}{=}}\ \sum _{i\in I}\|Ae_{i}\|_{H}^{2},}

where { e i : i ∈ I } {\displaystyle \{e_{i}:i\in I\}} is an orthonormal basis. The index set I {\displaystyle I} 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 ‖ ⋅ ‖ HS {\displaystyle \|\cdot \|_{\text{HS}}} 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 { e i } i ∈ I {\displaystyle \{e_{i}\}_{i\in I}} and { f j } j ∈ I {\displaystyle \{f_{j}\}_{j\in I}} are such bases, then ∑ i ‖ A e i ‖ 2 = ∑ i , j | ⟨ A e i , f j ⟩ | 2 = ∑ i , j | ⟨ e i , A ∗ f j ⟩ | 2 = ∑ j ‖ A ∗ f j ‖ 2 . {\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}.} If e i = f i , {\displaystyle e_{i}=f_{i},} then ∑ i ‖ A e i ‖ 2 = ∑ i ‖ A ∗ e i ‖ 2 . {\textstyle \sum _{i}\|Ae_{i}\|^{2}=\sum _{i}\|A^{*}e_{i}\|^{2}.} As for any bounded operator, A = A ∗ ∗ . {\displaystyle A=A^{**}.} Replacing A {\displaystyle A} with A ∗ {\displaystyle A^{*}} in the first formula, obtain ∑ i ‖ A ∗ e i ‖ 2 = ∑ j ‖ A f j ‖ 2 . {\textstyle \sum _{i}\|A^{*}e_{i}\|^{2}=\sum _{j}\|Af_{j}\|^{2}.} 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 x {\displaystyle x} and y {\displaystyle y} in H {\displaystyle H} , define x ⊗ y : H → H {\displaystyle x\otimes y:H\to H} by ( x ⊗ y ) ( z ) = ⟨ z , y ⟩ x {\displaystyle (x\otimes y)(z)=\langle z,y\rangle x} , which is a continuous linear operator of rank 1 and thus a Hilbert–Schmidt operator; moreover, for any bounded linear operator A {\displaystyle A} on H {\displaystyle H} (and into H {\displaystyle H} ), Tr ⁡ ( A ( x ⊗ y ) ) = ⟨ A x , y ⟩ {\displaystyle \operatorname {Tr} \left(A\left(x\otimes y\right)\right)=\left\langle Ax,y\right\rangle } .

If T : H → H {\displaystyle T:H\to H} is a bounded compact operator with eigenvalues ℓ 1 , ℓ 2 , … {\displaystyle \ell _{1},\ell _{2},\dots } of | T | = T ∗ T {\displaystyle |T|={\sqrt {T^{*}T}}} , where each eigenvalue is repeated as often as its multiplicity, then T {\displaystyle T} is Hilbert–Schmidt if and only if ∑ i = 1 ∞ ℓ i 2 < ∞ {\textstyle \sum _{i=1}^{\infty }\ell _{i}^{2}<\infty } , in which case the Hilbert–Schmidt norm of T {\displaystyle T} is ‖ T ‖ HS = ∑ i = 1 ∞ ℓ i 2 {\textstyle \left\|T\right\|_{\operatorname {HS} }={\sqrt {\sum _{i=1}^{\infty }\ell _{i}^{2}}}} .

If k ∈ L 2 ( μ × μ ) {\displaystyle k\in L^{2}\left(\mu \times \mu \right)} , where ( X , Ω , μ ) {\displaystyle \left(X,\Omega ,\mu \right)} is a measure space, then the integral operator K : L 2 ( μ ) → L 2 ( μ ) {\displaystyle K:L^{2}\left(\mu \right)\to L^{2}\left(\mu \right)} with kernel k {\displaystyle k} is a Hilbert–Schmidt operator and ‖ K ‖ HS = ‖ k ‖ 2 {\displaystyle \left\|K\right\|_{\operatorname {HS} }=\left\|k\right\|_{2}} .

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

⟨ A , B ⟩ HS = Tr ⁡ ( A ∗ B ) = ∑ i ⟨ A e i , B e i ⟩ . {\displaystyle \langle A,B\rangle _{\text{HS}}=\operatorname {Tr} (A^{*}B)=\sum _{i}\langle Ae_{i},Be_{i}\rangle .}

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 BHS(H) or B2(H), which can be shown to be naturally isometrically isomorphic to the tensor product of Hilbert spaces

H ∗ ⊗ H , {\displaystyle H^{*}\otimes H,}

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 | T | := T ∗ T {\textstyle \left|T\right|:={\sqrt {T^{*}T}}} , 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 S : H 1 → H 2 {\displaystyle S:H_{1}\to H_{2}} and T : H 2 → H 3 {\displaystyle T:H_{2}\to H_{3}} are Hilbert–Schmidt operators between Hilbert spaces then the composition T ∘ S : H 1 → H 3 {\displaystyle T\circ S:H_{1}\to H_{3}} is a nuclear operator.
• If T : H → H is a bounded linear operator then we have ‖ T ‖ ≤ ‖ T ‖ HS {\displaystyle \left\|T\right\|\leq \left\|T\right\|_{\operatorname {HS} }} .
• T is a Hilbert–Schmidt operator if and only if the trace Tr {\displaystyle \operatorname {Tr} } of the nonnegative self-adjoint operator T ∗ T {\displaystyle T^{*}T} is finite, in which case ‖ T ‖ HS 2 = Tr ⁡ ( T ∗ T ) {\displaystyle \|T\|_{\text{HS}}^{2}=\operatorname {Tr} (T^{*}T)} .
• If T : H → H is a bounded linear operator on H and S : H → H is a Hilbert–Schmidt operator on H then ‖ S ∗ ‖ HS = ‖ S ‖ HS {\displaystyle \left\|S^{*}\right\|_{\operatorname {HS} }=\left\|S\right\|_{\operatorname {HS} }} , ‖ T S ‖ HS ≤ ‖ T ‖ ‖ S ‖ HS {\displaystyle \left\|TS\right\|_{\operatorname {HS} }\leq \left\|T\right\|\left\|S\right\|_{\operatorname {HS} }} , and ‖ S T ‖ HS ≤ ‖ S ‖ HS ‖ T ‖ {\displaystyle \left\|ST\right\|_{\operatorname {HS} }\leq \left\|S\right\|_{\operatorname {HS} }\left\|T\right\|} . 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 B ( H ) {\displaystyle B\left(H\right)} that contains the operators of finite-rank.
• If A is a Hilbert–Schmidt operator on H then ‖ A ‖ HS 2 = ∑ i , j | ⟨ e i , A e j ⟩ | 2 = ‖ A ‖ 2 2 {\displaystyle \|A\|_{\text{HS}}^{2}=\sum _{i,j}|\langle e_{i},Ae_{j}\rangle |^{2}=\|A\|_{2}^{2}} where { e i : i ∈ I } {\displaystyle \{e_{i}:i\in I\}} is an orthonormal basis of H, and ‖ A ‖ 2 {\displaystyle \|A\|_{2}} is the Schatten norm of A {\displaystyle A} for p = 2. In Euclidean space, ‖ ⋅ ‖ HS {\displaystyle \|\cdot \|_{\text{HS}}} 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

1. ^ a b Moslehian, M. S. "Hilbert–Schmidt Operator (From MathWorld)".
2. ^ a b Voitsekhovskii, M. I. (2001) , "Hilbert-Schmidt operator", Encyclopedia of Mathematics, EMS Press
3. ^ a b Schaefer 1999, p. 177.
4. ^ a b c Conway 1990, p. 268.
5. ^ a b c d e f g h i Conway 1990, p. 267.

• 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. Vol. 3. New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135.

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