Sufficient criteria for the spectrum of a bounded operator to be convex.

Question

Let $A$ be an operator on some (separable) Hilbert space. If the Hilbert space is for example $l^2(\mathbb{Z}^d)$ and I consider the discrete Laplacian $\Delta$. Then $\sigma(\Delta) = \lbrack 0, 2d \rbrack$. This is a convex set. Similarly, if I take the Hilbert space $L^2 \left( \lbrack 0, 1 \rbrack \right)$ I can look at the operator that takes $(M_x f)(x) = x f(x)$. Then $\sigma(M_x) = \lbrack 0,1 \rbrack$ which is also convex. These are just some example, but is there some theory which investigates whether the spectrum of a (not necessarily normal) operator $A$ is convex?