A note on universal operators between separable Banach spaces

We compare two types of universal operators constructed relatively recently by Cabello Sanchez, and the authors. The first operator Omega acts on the Gurari.i space, while the second one P-S has values in a fixed separable Banach space S. We show that if S is the Gurarii space, then both operators are isometric. We also prove that, for a fixed space S, the operator P-S is isometrically unique. Finally, we show that Omega is generic in the sense of a natural infinite game.