The author gives a clean characterization theorem which stems from the work of \textit{D. W. Stroock} and \textit{S. R. S. Varadhan} [Multidimensional diffusion processes (1979; Zbl 0426.60069)] on martingale problems in multidimensional diffusion. Let \(E^ \Delta\) be the one point compactification of the locally compact Hausdorff space \(E\). Let \(\Omega=D([0,\infty],E^ \Delta)\) be the usual space of right continuous paths in \(E^ \Delta\) with left limits (such that \(w(s)=\Delta\) implies \(w(t)=\Delta\) for all \(t\geq s)\). Main Theorem: Let \(L:D(L)\subseteq C_ 0(E)\to C_ 0(E)\) be a linear operator. The following three conditions are equivalent: (I) \(L\) is closable and \(\overline L\) generates a Feller semigroup on \(C_ 0(E)\). (II) \(D(L)\) is dense in \(C_ 0(E)\) and \(L\) solves the martingale problem maximally. (III) \(L\) satisfies the maximum principle, and both \(D(L)\) and Range \((\lambda I-L)\) are dense in \(C_ 0(E)\) for some \(\lambda>0\). Explanations: Solving the martingale problem means for each \(x\in E\) there is a unique probability measure \(P_ x\) such that for each \(f\in D(L)\), the stochastic process \(f(X(t))-f(X(0))-\int^ t_ 0Lf(X(s))ds\) is a \(P_ x\)-martingale and \(P_ x(X(0)=x)=1\). ``Maximally'' means if \(M\) is any operator for which the above is valid and if \(M\) is an extension of \(\overline L\), then \(M=\overline L\). (III) means that if \(f\in D(L)\) and if \(\text{Re} f(x_ 0)=\max\{\text{Re} f(x):x\in E\}>0\), then \(\text{Re} Lf(x_ 0)\leq 0\).