spectrum of a convolution operator is connected

Question

Let $k \in L^1(\mathbb{R}; \mathbb{C})$ and define a linear operator $T$ acting on $L^2(\mathbb{R}; \mathbb{C})$ by $$ Tf(x) := [k \ast f] (x) \ldotp$$ Linearity is obvious, while the fact that $T$ is well-defined and continuous is a result of Young's inequality. I'm supposed to show that the spectrum of $T$ is connected but I don't really know how to start - the definition of being connected seems to me to be fairly distant from any properties of spectrum I've seen. For a moment I was hoping that perhaps it's convex, but I don't really think it's true, so any hints on how to approach the problem are most welcome.