Determine all $T \in B(C[0,1])$ that commute with the multiplication operator $Mf=xf(x)$ Let $Mf=xf(x)$  and for any polynomial let $T$ be a "polynomial operator" $Tf=(a_{x}x^{n}+..+a_{0})f$
Clearly $M$ commutes with all such  "polynomial operators" and therefore since the weak limit of these operators in W.O.T are the continous functions we know that $T$ being pointwise multiplication by such functions also coummute since they are in the closed linear span of the polynomials in W.O.T. I cant seem to manage to prove the reverse inclusion i.e that this is all of them, maybe since it inst true but this is my approch so far. Hints?

Given such an operator $T $, note that for any polynomial $p $, you have
$$
T (p) = T (p \cdot 1) = p \cdot T (1) 
$$
Now use density of the polynomials to conclude $Tf = f \cdot T (1) $ for all $f \in C ([0,1]) $, so that $T $ is a multiplication operator.
