fa.functional analysis – “Weakly” nuclear operators (terminology)

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 $$T:Vto W$$ 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 the composition of $$T$$ induces a nuclear operator $$widehat{V}_qto widehat{W}_p$$ where $$widehat{V}_q$$, $$widehat{W}_p$$ denote the Banach completions of $$V$$ and $$W$$ with respect to $$q$$ and $$p$$ respectively.

Note that by definition a locally convex topological vector space $$V$$ is nuclear if and only if the identity endomorphism of $$V$$ is weakly linear in the above sense.

So my first question is: Does there already exist a notion equivalent to that of “weakly nuclear operators” above ? 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 $$T:Vto W$$ is weakly nuclear then for any summable family $$(x_i)_{iin I}$$ in $$V$$, $$(T(x_i))_{iin I}$$ 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 ?