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We study various aspects of Schur analysis in the slice hyperholomorphicsetting. We present two sets of results: first, we give new results on thefunctional calculus for slice hyperholomorphic functions. In particular, weintroduce and study some properties of the Riesz projectors. Then we prove aBeurling-Lax type theorem, the so-called structure theorem. A crucial factwhich allows to prove our results, is the fact that the right spectrum of aquaternionic linear operator and the S-spectrum coincide. Finally, we study theKrein-Langer factorization for slice hyperholomorphic generalized Schurfunctions. Both the Beurling-Lax type theorem and the Krein-Langerfactorization are far reaching results which have not been proved in thequaternionic setting using notions of hyperholomorphy other than slicehyperholomorphy

Krein-Langer factorization and related topics in the slice hyperholomorphic setting