Let $ X $ and $Y$ be Hilbert space, $A:X \to Y $ compact and injective, $(\sigma_n;v_n,u_n)$ be its singular value decomposition, that is, $$ Av_n = \sigma_n u_n \\ A^* u_n = \sigma_n v_n $$ Since $\mathcal{N}(A) = \{ 0 \}$, $ \{ v_n \} $ form a complete orthonormal basis of $X$. Now we define $X_0$ as the completion of $X$ in norm $$ \Vert x \Vert_{X_0} := (\sum_j \sigma^2_j\vert \langle x, v_j \rangle \vert^2)^{\frac{1}{2}} $$ Set $\theta = 1-2\mu$, $p=2$, $$ X_\mu :=[X_0,X]_{\theta,p}, \quad \mu \in (0,\frac{1}{2}) $$ where $[\cdot,\cdot]_{\theta,p}$ denotes the interpolation space.
Question: Can we determine that $X_\mu$ is the completion of $X$ in norm $$ \Vert x \Vert_{X_\mu} := (\sum_j \sigma^{4\mu}_j\vert \langle x, v_j \rangle \vert^2)^{\frac{1}{2}} $$ ?
PS: This was proposed in MSE before.