The homotopy type of the linear group of Lebesgue–Bochner and Besov spaces

In this article we study the homotopical properties of linear groups of some Banach spaces. Our first main result asserts that for $1<p,q<\infty$ the linear group $GL\left(L_p\right(L_q\left)\right)$ of the Lebesgue–Bochner space $L_p\left(L_q\right)$ is contractible to a point, where $L_p$ and $L_q$ are both considered on $[0,1]$ equipped with the standard Lebesgue measure. The proof of this result is based on techniques drawn from the geometry of UMD-spaces. In addition, we establish the contractibility to a point of the general linear groups of $L_1\left(L_p\right)$ and $L_{\infty }\left(L_q\right)$, $1<p,q<\infty$. The proof is based on the techniques drawn from the theory of vector-valued Köthe spaces. We also prove that for $1<p<\infty$ and for a reflexive symmetric sequence space E the linear group $GL\left({\ell }_p\right(E\left)\right)$ is contractible to a point, where ${\ell }_p$ is the space of p-summable sequences and ${\ell }_p\left(E\right)$ is the ${\ell }_p$-sum of E spaces. As a consequence of the latter result we deduce the contractibility to a point of the linear group of a Besov space $B_p^{s,q}$, $1<p,q<\infty$, $s>0$. We conclude with few open problems.