= Solution
The <adjoint operator> $\alpha^*:W\to V$ is uniquely characterized by
$$
\langle\alpha v,w\rangle_W=\langle v,\alpha^*w\rangle_V\quad\text{for all }v,w.
$$
Let $A=\beta\beta^*+\gamma^*\gamma:V\to V$. Its quadratic form is nonnegative:
$$
\langle Av,v\rangle=\|\beta^*v\|^2+\|\gamma v\|^2.
$$
If $Av=0$, both norms vanish. In particular $v\in\ker\gamma=\operatorname{im}\beta$, while $\beta^*v=0$ means $v$ is orthogonal to $\operatorname{im}\beta$, by the defining <adjoint operator> identity. A vector belonging to a <vector subspace> and its <orthogonal complement> must be zero. Hence $A$ is injective, and finite dimensionality makes it surjective.
Thus
$$
\boxed{\beta\beta^*+\gamma^*\gamma\text{ is an isomorphism of }V.}
$$
The same calculation shows it is a <positive-definite operator> when $V\ne\{0\}$.
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