## What is the intersection of the closures of left invertible operators and right invertible operators?

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?

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I think you need to clarify your last sentence. Do you mean: "does this equality hold when $B(X)$ is replaced by an arbitrary Banach algebra $A$"? – Yemon Choi Jun 17 at 6:26
Thanks Choi. Yes, the last sentence should be "does this equality hold when $B(X)$ is replaced by an arbitrary Banach algebra $A$ with an identity". – Qingping Zeng Jun 17 at 6:47
The tags on this question need editing. (As far as I know, I do not have enough reputation to do this myself.) – Philip Brooker Jun 17 at 11:06
are you talking about bounded operators? and hence taking the closure in the operator norm? – Delio Mugnolo Feb 7 at 9:31