The paper introduces operator deep smoothing, a novel approach for implied volatility smoothing that leverages neural operators to directly map observed option data to smooth, no-arbitrage surfaces. Unlike traditional methods that require per-instance optimization (e.g., SVI calibration), this method trains a single graph neural operator (GNO) offline on historical data, enabling fast online inference. The key innovation is the use of neural operators, which are discretization-invariant and can handle the varying number and spatial arrangement of option quotes that occur naturally in markets. The authors adapt the GNO architecture to accept point clouds of implied volatilities with their coordinates (time-to-expiry and log-moneyness) and output a continuous surface. Training is performed on ten years of intraday S&P 500 options data, with a loss function that combines mean squared error against observed quotes and penalty terms for calendar and butterfly arbitrage violations. The trained model (with ~100k parameters) achieves state-of-the-art accuracy, outperforming the industry-standard SVI parametrization and matching the performance of per-surface neural networks, while being significantly faster at inference. The method also demonstrates robustness to input subsampling and generalizes well to other US indices without retraining. This work opens the door for using neural operators in financial engineering, particularly for tasks involving irregularly sampled spatial data, and provides a practical tool for practitioners needing fast, reliable implied volatility surfaces.