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We introduce the numerical spectrum σ n (A) ⊆ C of an (unbounded) linear operator A on a Banach space X and study its properties. Our definition is closely related to the numerical range W (A) of A and always yields a superset of W (A). In the case of bounded operators on Hilbert spaces, the two notions coincide. However, unlike the numerical range, σ n (A) is always closed, convex and contains the spectrum of A. In the paper we strongly emphasise the connection of our approach to the theory of C 0 -semigroups.Mathematics Subject Classification (2010). 47A12, 47A10, 47D06.

Function distinguishable languages were introduced as a new methodology of defining characterizable subclasses of the regular languages which are learnable from text. Here, we give details on the implementation and the analysis of the corresponding learning algorithms. We also discuss problems which might occur in practical applications