The paper “Convexity of Decentralized Controller Synthesis” by me and Sanjay Lall was submitted to Transactions on Automatic Control. This paper concerns the general problem of optimal structured controller synthesis. Such problems are hard in general, but can sometimes be expressed as convex optimization problems, which makes them much more tractable. This paper shows that under suitable assumptions, the simple algebraic condition of quadratic invariance is necessary and sufficient for the set of achievable closed-loop maps to be convex. This work is a generalization of some results that first appeared in my thesis and applied only to bounded linear operators. The new results also cover extended spaces, which means that they can be applied in cases when the plant or controller is unstable. Our paper is also available on arXiv.