Recently, I'd come across the following kind of operators and I wonder if they have been considered before and given a name.

Let's say that a linear map $T:V\to W$ between locally convex topological space is "weakly nuclear" if for every continuous semi-norm $p$ on $W$, there exists a continuous semi-norm $q$ on $V$ such that $T$ induces a nuclear operator $\widehat{V}_q\to \widehat{W}_p$ where $\widehat{V}_q$, $\widehat{W}_p$ denote the Banach completions of $V$ and $W$ with respect to $q$ and $p$ respectively.

The property of $T$ being "weakly nuclear" is easily seen to be indeed weaker than that of $T$ being "nuclear". For Banach spaces the two notions are equivalent but otherwise seem genuinely different e.g. the identity endomorphism of $V$ is weakly nuclear if the space $V$ is nuclear (and this is actually an if and only if).

So my first question is: Has a notion equivalent to that of "weakly nuclear operators" above already been considered ? If so, I would be happy to have a reference.

One reason for me to consider this kind of operators is that they enjoy the following nice property with respect to summable families: if $T:V\to W$ is weakly nuclear then for any summable family $(x_i)_{i\in I}$ in $V$, $(T(x_i))_{i\in I}$ is absolutely summable in $W$. This is actually not hard to prove directly using the definition of nuclear operators between Banach spaces but I would prefer to have a reference (e.g. because this is not my main area of expertise).

So my second question would be: in case the answer to the first question is yes, is there any reference for the above result ? Otherwise, is it a particular case of a more general theorem/result with a reference ?