A common property of \(R(E,F)\) and \(B(\mathbb R^n, \mathbb R^m)\) and a new method for seeking a path to connect two operators (Q625892)

For real Banach spaces \(E\) and \(F\), let \(B(E, F)\) be the space of all bounded linear operators from \(E\) to \(F\), and, for each nonnegative integer \(r\), let \(\Sigma_r\) be the subset of \(B(E, F)\) consisting of operators of rank \(r\). It is shown that (1) if \(F\) is infinite-dimensional, then \(\Sigma_r\) is pathwise connected for every \(r\), and (2) if \(E= \mathbb R^n\) and \(F= \mathbb R^m\) (resp., and, in addition, \(m\neq n\)), then \(\Sigma_r\) is pathwise connected with the dimension (of the associated tangent space at each operator in (\(\Sigma_r\)) equal to \((m+ n)r- r^2\) for \(0\leq r<\min\{n,m\}\) (resp., \(0\leq r\leq\min\{n,m\}\)). These are all proved by geometrically constructing specific paths connecting two arbitrary operators in \(\Sigma_r\).