relation between orthogonal projections?

Question

Let $\mathbb{H}$ be a Hilbert space with respect to two different inner products $[\cdot,\cdot]$ and $[\cdot,\cdot]^1$. Also let $W$ be a closed subspace of $\mathbb{H}$. Also let us assume that $\pi_W$ and $\pi^{1}_{W}$ respectively be the orthogonal projections onto $W$ with respect to the inner products $[\cdot,\cdot]$ and $[\cdot,\cdot]^1$. Is there exist any relation between $\pi_W$ and $\pi^{1}_{W}$?

Answer 1

Well, as they are both projections onto $W$, they both are idempotent, they both have a range of $W$ (which is the eigenspace corresponding to eigenvalue $1$), and their kernels complement $W$, in that they direct sum with $W$ to give all of $\mathbb{H}$.

Other than that, there's not a lot. If you can find a closed subspace $U$ that complements $W$, then you can define an inner product that that make $U = W^\perp$ (or at least you can for separable Hilbert spaces, and I think it holds more generally), and hence $U = \operatorname{ker} \pi_W$.