Hilbert-Schmidt Integral operators are usually defined from $H=L_2[a,b]$ into $H=L_2[a,b]$ as
$$(Tf)(x) = \int_a^b K(x,y)f(y) dy,$$
provided that $K(x,y)$ is a Hilbert Schmidt kernel, namely
$$\int_a^b \int_a^b |K(x,y)|^2dx dy < \infty.$$
I was wondering if the following extension is used with the same name. Letting $H_1 = L_2[a,b]$, $H_2=L_2[c,d]$, and
$$\int_a^b \int_c^d |K(x,y)|^2dx dy < \infty,$$
consider the integral operators $T_1:H_2 \rightarrow H_1$ and $T_2:H_1 \rightarrow H_2$
$$(T_1 f)(x) = \int_c^d K(x,y)f(y) dy,$$
$$(T_2 f)(y) = \int_a^b K(x,y)g(x) dx.$$
Can one still call these operators and corresponding kernel of Hilbert-Schmidt type?
Best Answer
The abstract definition of Hilbert-Schmidt operators on a Hilbert space $H$ is that for a complete orthonormal set $A$ a bounded operator $$ T: H \to H $$ is a Hilbert-Schmidt operator if $$ \sum_{a \in A} \| T a \|^2 \lt \infty $$ We can, of course, without problems say that we actually have two different, but isomorphic, Hilbert spaces $H_1$ and $H_2$ and reformulate the definition for $$ T: H_1 \to H_2 $$ which would not change anything (definitions nor theorems) besides additional indexes. So, yes, I think one can and should talk about Hilbert-Schmidt operators in your example.