How to find out the following value?

Question

I am given with Hilbert space $$X = L_2(0, \infty) = \{ f: \int_0^{\infty} |f(x)|^2 dx < \infty\}$$ Now define the Shift operator $$T(t): X \to X \ : \ T(t)(f(x)) = f(t+x) \quad t \geq 0$$ and $$\|T(t)f(x)\|^2 = \int_0^{\infty} |f(x+t)|^2 dx $$ I am able to prove that $$\|T(t)\| \leq 1$$ for each $t \geq 0$. But i have to prove that $$\lim_{t \to \infty} \frac{\log \|T(t)\|}{t} = 0$$ How to do this?

Answer 1

You can just show that $ \|T(t)\| = 1 $ for all $ t $, which seems quite reasonable, since shifting a function doesn't really change its total weight, since the integration interval is unbounded. We just need to pick the right function $ f \not= 0$ such that $ \|T(t)(f)\| = \|f\| $.

Since $ \int_0^\infty{|f(x+t)|^2dx} = \int_0^\infty{|f(x)|^2dx}$ is equivalent to $ \int_0^t |f|^2 = 0 $, we just need to find a function in $ X $ which is zero on $ [0,t] $ but whose norm is non-zero. Surely a continuous stitching of $ 0 $ and a standard square-integrable function could achieve that, thus giving the norm of $ T(t) $.