Similarities of subspace lattices in Banach spaces

A collineation of a subspace lattice $\text{fL}$ in a complex Banach space $\text{eX}$ is an invertible operator $S$ on $\text{eX}$ with the property that the image $S\text{eM}$ of a subspace $\text{eM}$ belongs to $\text{fL}$ if and and only if $\text{eM}$ belongs to it. Hence, $S$ is a collineation of $\text{fL}$ if and only if it implements an order automorphism of $\text{fL}$. We study the group $\text{Col}(\text{fL})$ of all collineations of $\text{fL}$ and its subgroup $\text{Grp}(\text{Alg}(\text{fL}))$ of all invertible operators that fix every subspace in $\text{fL}$. We show that $\text{Grp}(\text{Alg}(\text{fL}))$ is a normal subgroup of $\text{Col}(\text{fL})$; moreover, if $\text{fL}$ is a reflexive subspace lattice, then $\text{Col}(\text{fL})$ is the normalizer of $\text{Grp}(\text{Alg}(\text{fL}))$ in the group of all invertible operators on $\text{eX}$. One of the main questions that we consider is whether $\text{Grp}(\text{Alg}(\text{fL}))$ is a complemented subgroup in $\text{Col}(\text{fL})$. For certain subspace lattices $\text{fL}$, such as some realizations of the diamond or the double triangle, some nests in the space of continuous functions on $[0,1]$, and the classical Volterra nest in $L^1[0,1]$, we characterize the complement of $\text{Grp}(\text{Alg}(\text{fL}))$ in $\text{Col}(\text{fL})$. On the other hand, for the Volterra nests in $L^p[0,1]$, where $1<p<\mathrm{\infty }$, a further study is needed, and we prove only some partial results.