Let ${H}$ be a separable Hilbert space over the complex field. The class $S := \lbrace N|_{M} : N$ is normal on ${H}$ and ${M}$ is an invariant subspace for $N \rbrace$ of operators was introduced by Halmos and consists of subnormal operators. Each subnormal operator possesses a unique minimal normal extension $\hat{N}$ as shown by Halmos. Halmos proved that $\sigma(\hat{N}) \subseteq \sigma(S)$ and then Bram proved that $\sigma(S)$ is obtained by filling certain number of holes in the spectrum $\sigma(\hat{N})$ of the minimal normal extension $\hat{N}$ of a subnormal operator in ${S}$.