Take a Hilbert space $(\mathcal H,(\cdot,\cdot)_{\mathcal H})$ and two equivalent inner products $(\cdot,\cdot)_1$ and $(\cdot,\cdot)_2$ on $\mathcal H$, i.e. such that there are $a,b \in \mathbb R$ with $0 < a \leq b$ satisfying $ a (f,f)_1 \leq (f,f)_2 \leq b(f,f)_1 \quad \forall f \in \mathcal H $ Suppose that the metric induced by $(\cdot,\cdot)_i$ makes $(\mathcal H,(\cdot,\cdot)_i)$ a Hilbert space. In this way we get an equivalence relation $\sim$ on the set $X$ of the inner products on $\mathcal H$.
- If $A : \mathcal H \to \mathcal H$ is, with respect to the inner product of $\mathcal H$, a positive, invertible self-adjoint operator then $ \mathcal H \times \mathcal H \ni (f,g) \mapsto (Af,g) \in\mathbb C $ is an inner product on $\mathcal H$ equivalent to $(\cdot,\cdot)_{\mathcal H}$.
- If $H,K \in X$ are such that $H \sim K$, is it true that $H$ and $K$ are related by an operator $A : (\mathcal H,H) \to (\mathcal H,H)$ as in the previous point?
- What can be said on $\tilde X := X/\sim$? How many inequivalent inner products on $\mathcal H$ are there? Is there a way to classify them?
For the second point, assuming that $(\mathcal H,H)$ and $(\mathcal H,K)$ are separable, I thought to take two complete orthonormal systems $\{a_i\}_{i \in I}$ on $(\mathcal H,H)$ and $\{b_i\}_{i \in I}$ on $(\mathcal H,K)$ and define $A := T^* T$, where $T : (\mathcal H,H) \to (\mathcal H,H)$ is defined by $ f \mapsto Tf := \sum_{i \in I} K(f,b_i) a_i $ In this way, using $H(a_i,a_j) = \delta_{ij} = K(b_i,b_j)$ , $ \begin{align*} H(f,Ag) = H(Tf,Tg) &= \sum_{i,j,k \in I} K(f,b_i) H(a_i,a_j) K(b_j,g) \\ &= \sum_{i,j,k \in I} K(f,b_i) K(b_i,b_j) K(b_j,g) = K(f,g) \end{align*} $ Is this reasoning valid? What can be said in general?