On some topics in operator theory : An unfinished story about mathematical control

Abstract: This thesis considers differentiation of non-negative, fractional order, composed with Hardy spacetypeHankel operators. H2-boundedness is characterized in terms of a reproducing kernel thesis.The setting of operator-valued symbols is considered, in which H2-boundedness is characterized interms of Carleson embeddings, provided that the order of differentiation is strictly positive. Somenew results are deduced for the zeroth order. The complexity of the Carleson embedding conditionsis demonstrated by means of examples. Natural corresponding factorization theorems are proved.Some results are phrased in terms of control theory. An attempt is made at describing Hilbert spacecontraction semigroups which can be modeled by a weighted backward shift.