Given a C*-algebra with unit $1\in\mathcal{A}$.
For normal elements one has: $$A^*=A^{-1}\iff\sigma(A)\subseteq\mathbb{S}$$ $$A^*=A\iff\sigma(A)\subseteq\mathbb{R}$$ $$A\geq0\iff\sigma(A)\subseteq\mathbb{R}^+$$ $$A^2=A=A^*\iff\sigma(A)\subseteq\{0,1\}$$
For nonnormal elements this characterization breaks: $$N:=\begin{pmatrix}0&1\\0&0\end{pmatrix}:\quad\sigma(N)=\{0\}$$ But I'm still missing an example for a nonunitary element: $$\sigma(N)\subseteq\mathbb{S}:\quad N^*\neq N^{-1}$$ Do you have one at hand, please?
You already found a non-normal operator $N$ with spectrum $\{0\}$. Now simply note that $1+N$ will be another non-normal operator with spectrum $\{1\}$.