For $s \in (0,1)$, we consider the *spectral fractional Laplacian*
\begin{align}
(-\Delta)^{-s}u = \sum_{k=1}^{\infty}\lambda_k^{-s}(\phi_k,u)_{L^2}\phi_k
\end{align}
where
\begin{align*}
\begin{cases}
-\Delta\phi_k = \lambda_k\phi_k &\mbox{in }\Omega,\\
\partial_\nu \phi_k = 0 &\mbox{on }\partial\Omega,\qquad k\geq 1.
\end{cases}
\end{align*}
Then, for $\alpha \in (0,1)$, we define the space
\begin{align*}
H^\alpha(\Omega) = \{ u\in L^2(\Omega):~ \nabla (-\Delta)^{(\alpha-1)/2}u\in L^2(\Omega) \},
\end{align*}
and the space
\begin{align*}
\tilde H^\alpha(\Omega) = \{ u\in L^2(\Omega)~:~ \sum_{j=1}^\infty \lambda_j^\alpha|(\phi_j,u)_{L^2(\Omega)}|^2<\infty\}
\end{align*}

**Questions:**

*Are these two spaces equivalent?*Do Sobolev embeddings, Poincaré inequalities, etc hold in them? If yes, where can I find a reference?

**UPDATE.**
After the discussion in the comments, let me reformulate the question in a way that avoids (hopefully) some issues:

Let $(\phi_k)\subset L^2(\Omega)$ be a complete orthonormal system for $\mathcal L^2(\Omega)= \{ u\in L^2(\Omega):\int_\Omega u d x =0 \}$ composed of eigenfunctions of $-\Delta$ with homogeneous Neumann boundary conditions with eigenvalues $(\lambda_k)\subset (0,\infty)$.

**Q1:** Is \begin{align}
(-\Delta)^{-s}u = \sum_{k=1}^{\infty}\lambda_k^{-s}(\phi_k,u)_{L^2}\phi_k
\end{align}
well-defined?

**Q2:** Define
\begin{align*}
\mathcal H^\alpha(\Omega) = \{ u\in \mathcal L^2(\Omega):~ \nabla (-\Delta)^{(\alpha-1)/2}u\in L^2(\Omega) \},
\end{align*}
and the space
\begin{align*}
\tilde{\mathcal H}^\alpha(\Omega) = \{ u\in \mathcal L^2(\Omega)~:~ \sum_{j=1}^\infty \lambda_j^\alpha|(\phi_j,u)_{L^2(\Omega)}|^2<\infty\}
\end{align*}
Are these spaces well-defined and are they equivalent?

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