  Operator learning problems arise in many key areas of scientific computing
where Partial Differential Equations (PDEs) are used to model physical systems.
In such scenarios, the operators map between Banach or Hilbert spaces. In this
work, we tackle the problem of learning operators between Banach spaces, in
contrast to the vast majority of past works considering only Hilbert spaces. We
focus on learning holomorphic operators - an important class of problems with
many applications. We combine arbitrary approximate encoders and decoders with
standard feedforward Deep Neural Network (DNN) architectures - specifically,
those with constant width exceeding the depth - under standard $\ell^2$-loss
minimization. We first identify a family of DNNs such that the resulting Deep
Learning (DL) procedure achieves optimal generalization bounds for such
operators. For standard fully-connected architectures, we then show that there
are uncountably many minimizers of the training problem that yield equivalent
optimal performance. The DNN architectures we consider are `problem agnostic',
with width and depth only depending on the amount of training data $m$ and not
on regularity assumptions of the target operator. Next, we show that DL is
optimal for this problem: no recovery procedure can surpass these
generalization bounds up to log terms. Finally, we present numerical results
demonstrating the practical performance on challenging problems including the
parametric diffusion, Navier-Stokes-Brinkman and Boussinesq PDEs.
