Let $a\in \mathcal{L}(L^2([0, 1], \mathbb{R}))$ be a multiplication operator. I wonder whether there is any work on calculating its [essential spectrum][1]. Is there any way to explicitly compute its essential growth bound and elements of its discrete spectrum? 

What about the $n$-dimensional case, i.e., $a\in \mathcal{L}(L^2([0, 1], \mathbb{R}^{n\times n}))$? (Note that the $n$-dim multiplication operator may not be self-adjoint.)


  [1]: http://math.mit.edu/~eyjaffe/Short%20Notes/Functional%20Analysis/Essential%20Spectrum.pdf