Let

$V_i \subset \mathbb{R}^n$ and $V_0 \supset V_1 \supset ... \supset V_i \supset ...$,

$A_i, B_i: V_i \rightarrow V_i$ be square non-symmetric positive definite matrices,

$Q_i:V_{i-1}\rightarrow V_i$ (respectively $Q_i^ T:V_i\rightarrow V_{i-1}$) be rectangular matrices representing $L^2$-projections, $I_i$ be the identity in $V_i$ and

$A_i=Q_i A_{i-1} Q_i^T$.

Is there a way to represent the following matrix series as a vector-of-matrices times bloc-matrix times vector-of-matrices multiplication of the form $\overline{v}^ T\overline{M} \overline{v}$? Does this matrix series sound familiar to anyone?

\begin{align} (I_0-B_0 A_0)&(I_0-Q_1^T A_1^{-1} Q_1 A_0)(I_0-B_0 A_0) + \\ +(I_0-B_0 A_0)Q_1^T(I_1-B_1 A_1)&(I_1-Q_2^T A_2^{-1} Q_2 A_1)(I_1-B_1 A_1)A_1^{-1}Q_1 A_0(I_0-B_0 A_0)+ \\ +(I_0-B_0 A_0)Q_1^T(I_1-B_1 A_1)Q_2^T(I_2-B_2 A_2)&(I_2-Q_3^T A_3^{-1} Q_3 A_2)(I_2-B_2 A_2)A_2^{-1}Q_2 A_1(I_1-B_1 A_1) A_1^{-1} Q_1 A_0(I_0-B_0 A_0) \\ &... \end{align}

Should there be several representations, I am particularly interested in $\overline{M}$ to be as full as possible.

Thanks!