Classical neural networks are known for their ability to approximate mappings between finite-dimensional spaces, but they fall short in capturing complex operator dynamics across infinite-dimensional function spaces. Neural operators, in contrast, have emerged as powerful tools in scientific machine learning for learning such mappings. However, standard neural operators typically lack mechanisms for mixing or attending to input information across space and time. In this work, we introduce the Banach neural operator (BNO) -- a novel framework that integrates Koopman operator theory with deep neural networks to predict nonlinear, spatiotemporal dynamics from partial observations. The BNO approximates a nonlinear operator between Banach spaces by combining spectral linearization (via Koopman theory) with deep feature learning (via convolutional neural networks and nonlinear activations). This sequence-to-sequence model captures dominant dynamic modes and allows for mesh-independent prediction. Numerical experiments on the Navier-Stokes equations demonstrate the method's accuracy and generalization capabilities. In particular, BNO achieves robust zero-shot super-resolution in unsteady flow prediction and consistently outperforms conventional Koopman-based methods and deep learning models.