Hilbert–Schmidt operator

‖·‖{{sub|HS}} is well defined

The Hilbert–Schmidt norm does not depend on the choice of orthonormal basis. Indeed, if {e_i}{i\in I} and {f_j}{j\in I} are such bases, then \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, then \sum_i |Ae_i|^2 = \sum_i|A^* e_i|^2. As for any bounded operator, A = A^{**}. Replacing A with A^* in the first formula, obtain \sum_i |A^* e_i|^2 = \sum_j|A f_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 and y in H, define x \otimes y : H \to H by (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 on H (and into H), \operatorname{tr}\left( A\left( x \otimes y \right) \right) = \left\langle A x, y \right\rangle.

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

If k \in L^2\left( \mu \times \mu \right), where \left( X, \Omega, \mu \right) is a measure space, then the integral operator K : L^2\left( \mu \right) \to L^2\left( \mu \right) with kernel k is a Hilbert–Schmidt operator and \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

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

References

References

1. Moslehian, M. S.. "Hilbert–Schmidt Operator (From MathWorld)".
2. Voitsekhovskii, M. I.. "Hilbert-Schmidt operator".