Let \(\overline{M_{\Lambda }}\) be the closed span of the system \(\{t^{\lambda _n}\}_{n=1}^{\infty }\) in \(L^2 (0,1)\) where \(\Lambda =\{\lambda _n\}_{n=1}^{\infty }\) is a strictly increasing sequence of positive real numbers such that \(\sum _{n=1}^{\infty }\lambda _n^{-1}<\infty \) and \(\inf (\lambda _{n+1}-\lambda _n)>0\) . We refer to \(\overline{M_{\Lambda }}\) as the Müntz space of \(\Lambda \) . We investigate properties of \(\overline{M_{\Lambda }}\) and of the unique biorthogonal family \(\{r_n (t)\}_{n=1}^{\infty }\) to the system \(\{t^{\lambda _n}\}_{n=1}^{\infty }\) in \(\overline{M_{\Lambda }}\) . In the spirit of the “Clarkson-Erdős-Schwartz Phenomenon”, we obtain a series representation for functions in \(\overline{M_{\Lambda }}\) and then show that the system \(\{t^{\lambda _n}\}_{n=1}^{\infty }\) is a strong Markushevich basis for \(\overline{M_{\Lambda }}\) . As a result, we construct a general class of compact operators on \(\overline{M_{\Lambda }}\) that admit spectral synthesis: one of them is the operator \(T_{\rho }(f)=f(\rho x)\) for \(\rho \in (0,1)\) . Moreover, we show that the restriction of the Hardy-Cesàro operator H on \(\overline{M_{\Lambda }}\) , with \(H: \overline{M_{\Lambda }}\rightarrow \overline{M_{\Lambda }}\) , admits spectral synthesis, where \( Hf(x) : = \frac{1}{x} \int _{0}^{x} f(t)\, dt, \qquad x\in (0,1]. \)
