# “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 $$T:V\to W$$ 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 ?