A note on the connectedness problem for nest algebras

It has been conjectured that a certain operator T T belonging to the group G \mathcal {G} of invertible elements of the algebra Alg ⁡ Z \operatorname {Alg} {\mathbf {Z}} of doubly infinite upper-triangular bounded matrices lies outside the connected component of the identity in G \mathcal {G} . In this note we show that T T actually lies inside the connected component of the identity of G \mathcal {G} .