Definition: inverse of an unbounded operator.

I’m looking at two different definitions of an inverse and having difficulty finding an authoritative reference. I’d be pleased to get assistance.

Let BB be a Banach space and TT an unbounded linear operator with dense domain D(T)D(T) and range R(T)R(T).

Both definitions agree that if an inverse T−1T^{-1} exists then T−1(T(x))=xT^{-1}(T(x)) = x for all x∈D(T)x \in D(T).

They differ in requiring T(T−1(y))=yT(T^{-1}(y)) = y either for all y∈R(T)y \in R(T), or all y∈By \in B. So is it R(T)R(T) or BB ? I am particularly interested in the context of the resolvent set of an operator, i.e. the values λ∈C\lambda \in \Bbb C such that (T−λI)(T – \lambda I) is invertible.

A second question I have is that I can see no reason why this would not generalize to unbounded linear transformations between two different Banach spaces: am I correct in this assumption ?

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I found a reference: p.307 in faculty.jacobs-university.de/poswald/teaching/FunctAnal/…
– Tom Collinge
yesterday

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