Hilbert–Schmidt operator explained

A \colon H \to H that acts on a Hilbert space H and has finite Hilbert–Schmidt norm

$$\backslash |A\backslash |^2\_\; \backslash \; \backslash stackrel\backslash \; \backslash sum\_\; \backslash |Ae\_i\backslash |^2\_H,$$

where

\{e _{i:} i \in I\} is an orthonormal basis.^[1] The index set 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} is identical to the Frobenius norm.

||·|| is well defined

The Hilbert–Schmidt norm does not depend on the choice of orthonormal basis. Indeed, if

\{e _{i\}} _{i\in} and \{f _{j\}} _{j\in} are such bases, then$$\backslash sum\_i\; \backslash |Ae\_i\backslash |^2\; =\; \backslash sum\_\; \backslash left|\; \backslash langle\; Ae\_i,\; f\_j\backslash rangle\; \backslash right|^2\; =\; \backslash sum\_\; \backslash left|\; \backslash langle\; e\_i,\; A^*f\_j\backslash rangle\; \backslash right|^2\; =\; \backslash sum\_j\backslash |A^*\; f\_j\backslash |^2.$$If e _{i} = f _{i,} then $\backslash sum\_i\; \backslash |Ae\_i\backslash |^2\; =\; \backslash sum\_i\backslash |A^*\; e\_i\backslash |^2.$ As for any bounded operator, A = A ^{**} . Replacing A with A ^{*} in the first formula, obtain $\backslash sum\_i\; \backslash |A^*\; e\_i\backslash |^2\; =\; \backslash sum\_j\backslash |A\; f\_j\backslash |^2.$ 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 ⊗ y : H \to H by (x ⊗ 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 ⊗ 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,} ... of |T| = \sqrt{T ^{*T}} , where each eigenvalue is repeated as often as its multiplicity, then T is Hilbert–Schmidt if and only if , in which case the Hilbert–Schmidt norm of T is $\backslash left\backslash |\; T\; \backslash right\backslash |\_\; =\; \backslash sqrt$.

If

k \in L ^{2\left(} \mu x \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

$$\backslash langle\; A,\; B\; \backslash rangle\_\backslash text\; =\; \backslash operatorname(A^*\; B)\; =\; \backslash sum\_i\; \backslash langle\; Ae\_i,\; Be\_i\; \backslash rangle.$$

The Hilbert–Schmidt operators form a two-sided

• -ideal

in the Banach algebra 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

$$H^*\; \backslash otimes\; H,$$

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.
• A bounded linear operator is Hilbert–Schmidt if and only if the same is true of the operator $\backslash left|\; T\; \backslash right|\; :=\; \backslash sqrt$, 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} \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 is a bounded linear operator then we have

\left\| T \right\| \leq \left\| T \right\| _{\operatorname{HS} }. \operatorname{Tr} of the nonnegative self-adjoint operator T ^{*} T is finite, in which case \|T\| ^{2} _{HS} = \operatorname{Tr}(T ^{*} T) .^[1]

• If is a bounded linear operator on and is a Hilbert–Schmidt operator on then

\left\| S ^{*} \right\| _{\operatorname{HS} } = \left\| S \right\|_, \left\| T S \right\| _{\operatorname{HS} } \leq \left\| T \right\| \left\| S \right\|_, and \left\| S T \right\| _{\operatorname{HS} } \leq \left\| S \right\|_ \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 is an ideal of the space of bounded operators

B\left( H \right) that contains the operators of finite-rank.

• If is a Hilbert–Schmidt operator on then $$\backslash |A\backslash |^2\_\backslash text\; =\; \backslash sum\_\; |\backslash langle\; e\_i,\; Ae\_j\; \backslash rangle|^2\; =\; \backslash |A\backslash |^2\_2$$ 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, \| ⋅ \| _{HS} is also called the Frobenius norm.

References

• Book: Conway, John B.. John B. Conway

. John B. Conway. A course in functional analysis . Springer-Verlag . New York . 1990 . 978-0-387-97245-9 . 21195908.

• Book: Schaefer, Helmut H.. Helmut H. Schaefer

. Helmut H. Schaefer. Topological Vector Spaces . Springer New York Imprint Springer . . 3 . New York, NY . 1999 . 978-1-4612-7155-0 . 840278135 .

Notes and References

1. Web site: Moslehian . M. S. . Hilbert–Schmidt Operator (From MathWorld) .