On the Hilbert Space $L^2([0,\infty))$, define the linear operator $\Lambda$ by $$(\Lambda f)(x)=2f(x+1),~~~~~~x\ge0.$$
Describe the Ker$(\Lambda)$ and Ker$(\Lambda ^*)$.
My attempt: Let $2f(x+1)=0 ~~for~~ x\ge0$
Then I have Ker$(\Lambda)$=$\{f\in L^2([0,\infty)):f(x)=0~, x\in[1,\infty)\}$.
But I found the solution is Ker$(\Lambda)$=$\{f\in L^2([0,\infty)):f(x)=0~~a.e. x\in[0,1]\}$
Why? How to compute the Kernel?