I am concerned with the question if one could treat certain linear operators in a separable Banach space as multiplication operatos on a closed subspace of $C[0,1]$. In a Hilbert space it is know by spectral theorem that any linear operator is unitarily equivalent to a multiplication operator in L^2. Sure that does not hold for a more general Banach spaces. But is it possible for certain operators, for example the Laplace operator?
There is a famous result of Banach and Mazur that a separable Banach space is isometric isomorph to a closed subspace of $C[0,1]$(Sadly, at least in the internet of the Country I am in right now temporarily, I could not find a full public and free accessible proof of this theorem, does any one have a link, maybe an open book or public lecture note?). So I lack a bit of material to investigate my question.
Has anyone already encountered this question and can say that this is in general not possible or if there are certain (nonlinear) operators, i.e. Laplace, in seperable Banach spaces(which are not a Hilbert space) which can be seen as multiplication operators?
thank you in advance for your answer!