Accessible Operators on Ultraproducts of Banach Spaces
An operator between ultraproducts (X_i)_U -> (Y_i)_U is 'accessible' if it is induced pointwise by a family of possibly nonlinear maps f_i: X_i -> Y_i via f((x_i)_U) = (f_i(x_i))_U. Such a family induces a bounded linear operator exactly when the quasilinearity constants Q[f_i] tend to zero along the ultrafilter, and the operator is an ultraproduct of linear operators iff the f_i are asymptotically close to linear maps. This links accessible operators to non-split short exact sequences (twisted sums) of quasi-Banach spaces and to Kalton's notion of K-spaces and K_0-spaces.
The final version will appear in Extracta Mathematicæ, https://revista-em.unex.es/index.php/EM
Cabello Sanchez studies 'accessible' operators between ultraproducts of Banach spaces -- operators of the form f((x_i)_U) = (f_i(x_i))_U induced by a family of (possibly nonlinear) maps f_i: X_i -> Y…