A Note on the Connectedness of the Invertible Group of a Nest Algebra (Communications in Mathematical Research, 2021)
The connectedness of the invertibles question for arbitrary nest has beenreduced to the case of the lower triangular operators with respect to a fixed orthonormal basis $e_n$ for $n ≥ 1$. For each $f ∈ H^∞$, let $T_f$ be the Toeplitz operator. Inthis paper we prove that $T_f$ can be connected to the identity through a path in theinvertible group of the lower triangular operators if $f$ satisfies certain conditions.