This question is based on a very similar [question][1] posted by me yesterday. A very nice solution was provided by Aleksei Kulikov. Here I modify my question slightly.

Let $Q_t$ be the heat semigroup on $L^2(\mathbb R)$. Is there a direct sum decomposition
$$\oplus_{k=1}^\infty \mathscr{H}_k\subseteq L^2(\mathbb R)$$ such that 

 1. Each $\mathscr H_k$ is an infinite dimensional Hilbert space.
 2. $Q_t(\mathscr H_k)\subseteq \mathscr H_k$ for all $k$ and $t>0$.
 3. If $Q^{(k)}_t$ denotes the restriction of $Q_t$ on $\mathscr H_k$ then $\sigma(Q^{(k)}_t)=\sigma(Q_t)=[0,1]$ for all $k$. Here $\sigma(A)$ means the spectrum of $A$.

For example, if $\mathscr{H}_1$ and $\mathscr{H}_2$ denote the Hilbert spaces of all $L^2(\mathbb R)$ functions that are even and odd respectively, it can be easily checked that $L^2(\mathbb R)=\mathscr{H}_1\oplus \mathscr{H}_2$, $Q_t(\mathscr H_k)\subseteq\mathscr{H}_k$ for $k=1,2$, and $\sigma(Q^{(k)}_t)=[0,1]$ for $k=1,2$. The above question asks whether we can have infinitely many such Hilbert spaces.

  [1]: https://mathoverflow.net/questions/479460/infinite-direct-sum-decomposition-of-the-heat-semigroup-on-mathbbrn