We propose a continuous approach for computing the pseudospectra of linear operators following a 'solve-then-discretize' strategy. Instead of taking a finite section approach or using a finite-dimensional matrix to approximate the operator of interest, the new method employs an operator analogue of the Lanczos process to work directly with operators and functions. The method is shown to be free of spectral pollution and spectral invisibility, fully adaptive, nearly optimal in accuracy, and well-conditioned. The advantages of the method are demonstrated by extensive numerical examples and comparison with the traditional method.

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