At each iterative step of the generalised Lanczos process for the pair of matrices (A,B), we obtain the following factorisation: $$ A Q_k = B Q_{k+1} \widehat{T}_k, $$ where $Q_k^T B Q_k = I_k$ and $\widehat{T}_k$ denotes a tridiagonal matrix $T_k$ augmented with a additional row containing a non zero entry in the position $t_{k+1,k}$. Under this notation, we also have $Q_k^T A Q_k = T_k$.
Using this factorisation, i'm looking to describe an expression for fractional powers of matrices. In particular, i'd like to write down $\left(B^{-1}A\right)^{1-\alpha}$ for $\alpha \in [0,1]$ in terms of both $Q_k$ and $T_k$.
So far, i'm at the following point. Multiply both sides by $Q_{k}^T$, so that $$ A(Q_kQ_k^T) = BQ_kTQ_k^T \quad \implies \quad B^{-1}A = Q_kTQ_k^T (Q_kQ_k^T)^{-1}, $$ which I believe to be perfectly valid and correct. Can I then say: $$ \left(B^{-1}A\right)^{1-\alpha} = \left(Q_kTQ_k^T (Q_kQ_k^T)^{-1}\right)^{1-\alpha} = (Q_kQ_k^T)^{-1} Q_kT^{1-\alpha}Q_k^T. $$ I believe (and am pretty sure) that this step is wrong. If so, please could you point me in the right direction for obtaining such a relation? This isn't for any kind of assessment, purely based on personal interest, as i'd like to understand how the algorithm can be used in the evaluation of fractional matrix powers. Therefore, i'm happy with either a short or thorough explanation, would would probably prefer the latter in order to improve my understanding.