Unbounded spectra and operators

Question

In a paper I have encountered the following non-autonomous infinite dimensional dynamical system: $$ \frac{d}{dt} \begin{pmatrix} f \\ g \end{pmatrix} = \underbrace{\begin{pmatrix} 0 & 1 \\ -t^{-2} \Delta_{S^2} & -2 t^{-1} \end{pmatrix}}_{A(t)} \begin{pmatrix} f \\ g \end{pmatrix}, $$ where $f,g \in C^\infty(S^2)$ for $t>0$, and $\Delta_{S^2}$ is the Laplace-Beltrami operator on the sphere. The operator $A(t)$ has unbounded spectrum in both directions, and therefore the above problem is said to be ill-posed, meaning one cannot expect a solution to exist forward or backward in time for generic initial data. I am not an expert in functional analysis, and the only thing I know is that if $A(t)$ is unbounded, it is not easy to prove that an abstract Cauchy problem is well posed. Does the fact that $A(t)$ have an unbounded spectrum automatically imply that it is an unbounded operator? And if not, why does the above property imply that the problem is ill-posed? Any reference would be highly appreciated.