Weighted PLB-spaces of continuous functions arising as tensor products of a Fréchet and a DF-space

Countable projective limits of countable inductive limits, so-called PLB-spaces, of weighted Banach spaces of continuous functions have recently been investigated by Agethen et al., who analyzed locally convex properties in terms of the defining double sequence of weights. We complement their results by considering a defining sequence which is the product of two single sequences. By associating these two sequences with a weighted Fréchet, resp. LB-space of continuous functions or with two weighted Fréchet spaces (by taking the reciprocal of one of the sequences) we derive a representation of the PLB-space as the tensor product of a Fréchet and a DF-space and exhibit a connection between the invariants (DN) and (Ω) for Fréchet spaces and locally convex properties of the PLB-space resp. of the forementioned tensor product.