A family of operators with a specific property

Hi! I'm stuck trying to solve the below stated problem. (Any suggestions and help is more than welcome). Thanks!

PROBLEM: Define $\displaystyle T_b$ by $\displaystyle T_b f(t)=bf(t)$ (where $\displaystyle f(t)\in L^p([0,1])$ and $\displaystyle 1\leq p$ fixed). The task is to find all bounded linear maps $\displaystyle T$ from $\displaystyle L^p([0,1])$ to $\displaystyle L^p([0,1])$ with the property $\displaystyle TT_b=T_bT$.

Re: A family of operators with a specific property

Is $\displaystyle b$ a function?

Re: A family of operators with a specific property

No, b is just an independent variable.