L^2 kernel operators and Hilbert-Schmidt operators on R^3

Assume the Axiom of Countable Choice (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)). Write $L^2=L^2({\mathbb{R}}^3;\mathbb{C})$ for the complex Hilbert space of Lebesgue (equivalence classes of) square-integrable functions with inner product $⟨f,g⟩={\int }_{{\mathbb{R}}^3}f\bar{g}$ linear in the first argument (The space $L^p(\mu )$ as the quotient by null functions, Complex Lp classes and Euclidean test-function conventions, Hilbert space), and write $L^2({\mathbb{R}}^3\times {\mathbb{R}}^3;\mathbb{C})$ for the same construction over the product with Lebesgue measure; identify Lebesgue measure on ${\mathbb{R}}^6$ with the completion of the product of Lebesgue measures on the two ${\mathbb{R}}^3$ factors.

Kernel data and sections. Let $k\in L^2({\mathbb{R}}^3\times {\mathbb{R}}^3;\mathbb{C})$, an almost-everywhere equivalence class, and choose a finite-valued Lebesgue-measurable representative $k_0:{\mathbb{R}}^3\times {\mathbb{R}}^3\to \mathbb{C}$. When $f\in L^2$ is used, choose likewise a finite-valued Lebesgue-measurable representative $f_0:{\mathbb{R}}^3\to \mathbb{C}$. For almost every $x\in {\mathbb{R}}^3$, the section $k_x(y):=k_0(x,y)$ is measurable, by applying the completed-product Fubini theorem to the positive and negative parts of the real and imaginary components of $k_0$ (Tonelli and Fubini for the completed product, with only almost-everywhere section measurability).

Domain and action. For $f\in L^2$ say that the integral defining $T_{k}f$ exists at $x\in {\mathbb{R}}^3$ if the section product $y\mapsto k_0(x,y)f_0(y)$ is measurable and lies in $L^1({\mathbb{R}}^3;\mathbb{C})$, and at such points put $(T_{k}f)(x):={\int }_{{\mathbb{R}}^3}{k}_0(x,y)f_0(y)dy\mathrm{.}$ Define $D(T_k):=\{f\in L^2:x\mapsto (T_{k}f)(x)\text{ exists for almost every }x\text{ and agrees a.e. with a class in }{L}^2\},$ and for $f\in D(T_k)$ let $T_{k}f$ denote that class. The assignment $f\mapsto T_{k}f$ is called the $L^2$-kernel operator with kernel $k$.

This definition is independent of both representative choices. If $k_1$ is another representative of $k$, its disagreement set with $k_0$ is null for the completed product Lebesgue measure. Fubini applied to the indicator of that set gives $k_0(x,\cdot )=k_1(x,\cdot )$ almost everywhere for almost every $x$. For each such $x$ and any representative of $f$, the two section products therefore have the same $L^1$ status and, when integrable, the same integral. Replacing $f_0$ by another representative changes the section product only on a null set of $y$, so it too preserves integrability and the integral. Consequently the almost-everywhere sections, the subset $D(T_k)\subseteq L^2$, and the resulting output class are unchanged by either representative choice.

Hilbert-Schmidt operators. A linear operator $T$ on $L^2$ is called a Hilbert-Schmidt operator if there is $k\in L^2({\mathbb{R}}^3\times {\mathbb{R}}^3;\mathbb{C})$ with $D(T)=D(T_k)$ and $T=T_k$; such a $k$ is called an $L^2$ kernel of $T$.

This definition asserts nothing about whether $D(T_k)$ is all of $L^2$, whether $T_k$ is bounded, or whether $T_k$ is compact; those properties are proved in L^2 kernel operators are bounded and compact. A finite product kernel ${\sum }_{i=1}^n{f}_i(x)\overline{g_i(y)}$ with $f_i,g_i\in L^2$ is also a class in $L^2({\mathbb{R}}^3\times {\mathbb{R}}^3;\mathbb{C})$, as the same lemma records, so $T_k$ is defined for it whenever that membership is used.