Suppose we have the space $L^p(R,R^n)$ where $1 \leq p < \infty$ (i.e the space of functions that take values in $R^n$ and are $L^p$ integrable) and suppose $T_m: L^p(R,R^n) \to L^p(R,R^n) $ is a linear operator given by $T_mf(x)=m(x)f(x)$ where $m(x)$ is a an $n \times n$ matrix with coefficients that are functions of $x$. It is claimed that $T_m$ is bounded if and only if $m$ is essentially bounded and $||T_m||=||m||_\infty$. My question is what does it mean for $m$ to be essentially bounded and what is $||m||_\infty$.
I understand this in one dimension, i.e $m$ is essentially bounded means that the essential supremum of $m$ which is the infimum of the set $ \{M: m(x) \leq M \mbox{ almost everywhere } \}$ and the $L_\infty$ norm is precisely the infimum of this set.
What would the $L_\infty$ norm be in the matrix valued case?
Thank you.