compute the norm of a linear operator

Question

good morning everyone, I am preparing my functional analysis exam and I can't resolve this exercise:

Let $$ H = L^2([0, 3])$$ and let T : H → H be defined by

$$Tu(t) = (1 + t^2)u(t)$$

i) Prove that T is linear and continuous and compute its norm.

I find out that the norm of T is lower than 10 but I cannot show that It is greater or equal than 10 to conclude that it is actually equal to 10. can anyone help me?

thank you in advance

Answer 1

Let $u_n(t)=\sqrt nI_{(3-\frac 1 n,3)}$. Then $\|u_n\|=1$ for all $n$. Now $\|Tu_n\|^{2}=n\int _{3-1/n} ^{3} (1+t^{2})^{2} \, dt \to 100$ so $\|T\| \geq 10$. As you have already noted the bound $(1+t^{2})^{2} \leq 100$ gives the opposite inequality.