We study the approximation rates of a class of deep neural network approximations of operators, which arise as data-to-solution maps G † of linear elliptic partial differential equations (PDEs), and act between pairs X, Y of suitable infinite-dimensional spaces. We prove expression rate bounds for approximate neural operators G with the structure G = R ◦ A ◦ E, with linear encoders E and decoders R. The constructive proofs are via a ...