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Let $(H_0, \langle \,,\,\rangle_0)$ be a real separable Hilbert space, and let $(H_1, \langle \,,\,\rangle_1)$ be a Hilbert space such that $H_1 \subset H_0$ is dense and such that the inclusion $(H_1, \|\,\|_1) \to (H_0,\|\,\|_0)$ is compact.

For us it suffices to think of the sequence spaces \begin{eqnarray*} \ell^2 &=& \{ (x_1, x_2, \dots ) \mid \sum_{k=1}^\infty x_k^2 < \infty \} \quad \;\; \mbox{ with inner product } \langle x,y \rangle_0 = \sum_{k=1}^\infty x_k\, y_k \\ \ell^2_1 &=& \{ (x_1, x_2, \dots ) \mid \sum_{k=1}^\infty k^2\, x_k^2 < \infty \} \quad \mbox{ with inner product } \langle x,y \rangle_1 = \sum_{k=1}^\infty k^2\,x_k\, y_k \end{eqnarray*}

For $j =0,1$ denote by $L (H_j)$ the Banach space of bounded linear operators $H_j \to H_j$ with the norm topology, and by $GL (H_j)$ and $O(H_j)$ the groups of invertible resp. orthogonal transformations in $L(H_j)$.

By a result of Putnam and Wintner in PNAS 1951, $GL (H_j)$ and $O(H_j)$ are connected, and
by Kuiper's theorem they are even contractible.

We wonder about a relative version of these results: We consider $O (H_0) \cap GL (H_1)$, namely the set of orthogonal transformations of $H_0$ that restrict to continuous maps $H_1 \to H_1$, endowed with the norm topology $\|A\|_{L(H_0,H_0)}+\|A\|_{L(H_1,H_1)}$.

Question. Is $O (H_0) \cap GL (H_1)$ path-connected, or even contractible?

We studied the proofs of Putnam and Wintner, and of Kuiper, but were not able to adapt them to our situation. The problem is that the homotopies $A(t)$ in their proofs from $A$ to $id$ are not such that $A(t)$ restrict to continuous maps $H_1 \to H_1$. Is there something known about the above question, or about the topology of $O (H_0) \cap GL (H_1)$ ?

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You might look at the treatment of Kuiper's theorem in Kronheimer-Mrowka's book Monopoles and 3-manifolds. They consider a similar situation in Chapter 33 (Coupled Morse homology).

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