I am reading Foundations of Time-Frequency Analysis by K. Grochenig and I have come across the following property of the short-time Fourier transform, defined by $$ (V_g f)(p,q) := \int_{\mathbb{R}^d} f(t)e^{2 \pi i p \cdot t} g(t-q) \, dt $$ which is, for $f_1,f_2,g_1,g_2 \in L^2(\mathbb{R}^d)$, $$ \langle V_{g_1}f_1 , V_{g_2}f_2 \rangle_{L^2(\mathbb{R}^{2d})} = \langle f_1, f_2 \rangle_{L^2} \langle g_2 , g_1 \rangle_{L^2} $$ The part I am confused about is the case when $g_1 \perp g_2$. Wouldn't that then imply that $$V_{g_2}^*V_{g_1} = 0 $$
but then, due to the inversion formula, $$I = (V_{g_2}V_{g_2}^*)(V_{g_1}V_{g_1}^*) = 0$$ I am clearly missing something. Any help would be appreciated.
Edit: I think the inversion formula is only a left-inverse. $V_g^*V_g=I$ but not necessarily $V_gV_g^*$. This resolves the confusion above.