# Injectivity of integral operators

Let $$K:L^2[0,1]^{d_1}\to L^2[0,1]^{d_2}$$ be integral operator $$(Kf)(y) = \int f(x)k(x,y)d x.$$ If $$d_1>d_2$$ is it possible for $$K$$ to be injective?, e.g. let's take $$d_1=2,d_2=1$$.

More generally, does the injectivity of $$K$$ impose any restrictions on $$d_1,d_2$$.

• Is there any information on $k(x,y)$? For instance, in the trivial case when $k(x,y)$ does not depend on $y$ and vanishes for all $x \in (0,0.5)$, then $Kf(y) =0 \ \forall y$ for some non-zero function $f$, so the operator is not injective. – Yuxin Wang Nov 29 '18 at 1:43
• Thank you. I don't have any conditions. I need an example of injective $K$ when $d_1>d_2$ (or at least in knowing whether this is possible). I'm not interested in counterexamples. – Lionville Nov 29 '18 at 1:45
• Well, what are your thoughts on the question so far? – jgon Nov 29 '18 at 2:30