Question

From Douglas Zare's answer (see http://mathoverflow.net/questions/99777/does-x-embed-in-y-and-y-embed-in-x-always-imply-that-x-isomorphic-on), one know that $$ \overline{G_{l}(X,Y)} \bigcap \overline{G_{r}(X,Y) } = \overline{G(X,Y)}$$ does not hold in general, where $G_{l}((X,Y),G_{r}((X,Y)$ and $G(X,Y)$ denote the set of left invertible operators, right invertible operators and invertible operators. (We say an operator $T$ left invertible, if $ST=I$ for some operator $S$.) But does this equality holds when $X=Y$? If not, for what kinds of $X$, this equality holds? Furthermore, does this equality holds when $B(X)$ is replaced by a Banach algebra $A$ with an identity?