Hilbert-Schmidt Operators

Abstract Hilbert-Schmidt Operators

Hilbert-Schmidt operators are a special class of compact operators that play an important role in functional analysis and quantum mechanics. They can be thought of as infinite-dimensional analogues of matrices with finite Frobenius norm.

Hilbert-Schmidt Operator

An operator $T$ on a Hilbert space \newcommand{\H}{\mathcal{H}} is called Hilbert-Schmidt if its Hilbert-Schmidt norm $\Vert T{\Vert }_{\text{HS}}^2:={\sum }_j\Vert T{e}_j{\Vert }^2<\infty .$

e.g.

• In ${\mathbb{R}}^n$, the Hilbert–Schmidt norm is identical to the Frobenius norm.

Proposition

The product of two Hilbert-Schmidt operators is a trace class operator. Thus we can define the Hilbert–Schmidt inner product of two Hilbert-Schmidt operators as $⟨A,B{⟩}_{\text{HS}}:=\mathrm{Tr}(B^{\ast }A)={\sum }_j⟨A{e}_j,B{e}_j⟩.$

Proposition

Suppose \H is a complex Hilbert space. Then the space of all Hilbert-Schmidt operators, denoted $B_{\text{HS}}(\stackrel{˝}{)}$ or $S_2(\stackrel{˝}{)}$, is an two-sided $\ast$-ideal in the Banach algebra $B(\stackrel{˝}{)}$. Moreover, $B_{\text{HS}}(\stackrel{˝}{)}$ is isometrically isomorphic to the Hilbert space tensor product \overline{\H} \otimes \H, where \overline{\H} is the conjugate Hilbert space of \H.
This means for each Hilbert-Schmidt operator $T$, we can define a bilinear form \beta_{T}\colon \H \times \H \to \C,\quad (x,y) \mapsto \langle x, Ty \rangle_{\H}.

Proof

Integral Operators

Integral Operator

On \newcommand{\R}{\mathbb{R}}L^{2}(\R^d), we can define an operator $T_K:L^2({\mathbb{R}}^d)\to L^2({\mathbb{R}}^d)$ by the formula $T_K(f):={\int }_{{\mathbb{R}}^d}K(x,y)f(y)\text{dd}y,$ we call it an integral operator with associated kernel $K:{\mathbb{R}}^d\times {\mathbb{R}}^d\to \mathbb{R}$, where $K$ is a measurable function such that ${\int }_{{\mathbb{R}}^d}|K(x,y){|}^2\text{dd}y<\infty$ for almost every $x\in {\mathbb{R}}^d$.

Proposition

An integral operator $T_K$ is Hilbert-Schmidt if the associated kernel $K$ is an element of $L^2({\mathbb{R}}^d\times {\mathbb{R}}^d)$.

Proposition

Let $T$ be a Hilbert-Schmidt operator on $L^2({\mathbb{R}}^d)$ with kernel $K$. Then

• for every $f\in L^2({\mathbb{R}}^d)$, and almost every $x\in {\mathbb{R}}^d$, the function $y\mapsto K(x,y)f(y)$ is integrable.
• $T$ is bounded, and $\Vert T\Vert \le \Vert K{\Vert }_{L^2({\mathbb{R}}^d\times {\mathbb{R}}^d)}$.
• the adjoint of $T$ is also a Hilbert-Schmidt operator with kernel $\overline{K(y,x)}$.