Norm of operator between two $L^p$ spaces?

Question

In my reading, I've come across notation like $ ||T||_{2 \to \infty} $, where $T$ is an operator defined on every $L^p$ space. What does this mean? Is it simply the norm of $T$ viewed as an operator from $L^2$ to $L^\infty$? Can this norm be expressed in terms of the norms $||T||_2$ and $||T||_{\infty}$ ?

Answer 1

Just as you suspect, $\left\|\cdot\right\|_{p,q}$ is the norm $L_p \rightarrow L_q$. In this context $\left\|\cdot\right\|_{2}$ is just $\left\|\cdot\right\|_{2,2}$ and so on.