Approximation rates are analyzed for deep surrogates of maps between
infinite-dimensional function spaces, arising e.g. as data-to-solution maps of
linear and nonlinear partial differential equations. Specifically, we study
approximation rates for Deep Neural Operator and Generalized Polynomial Chaos
(gpc) Operator surrogates for nonlinear, holomorphic maps between
infinite-dimensional, separable Hilbert spaces. Operator in- and outputs from
function spaces are assumed to be parametrized by stable, affine representation
systems. Admissible representation systems comprise orthonormal bases, Riesz
bases or suitable tight frames of the spaces under consideration. Algebraic
expression rate bounds are established for both, deep neural and spectral
operator surrogates acting in scales of separable Hilbert spaces containing
domain and range of the map to be expressed, with finite Sobolev or Besov
regularity. We illustrate the abstract concepts by expression rate bounds for
the coefficient-to-solution map for a linear elliptic PDE on the torus