Differentiation in the branges spaces and embedding theorems

The boundedness conditions for the differentiation operator in Hilbert spaces of entire functions (Branges spaces) and conditions under which the embedding K_и⊂L₂(μ) holds in spaces K_и associated with the Branges spacesH(E) are studied. Measure μ such that the above embedding is isometric are of special interest. It turns out that the condition E'/E∈H^∞(C⁺) is sufficient for the boundedness of the differentiation operator inH(E). Under certain restrictions on E, this condition is also necessary. However, this fact fails in the general case, which is demonstrated by the counterexamples constructed in this paper. The convex structure of the set of measures μ such that the embedding K_E ^* _/E⊂L²(μ) is isometric (the set of such measures was described by de Brages) is considered. Some classes of measures that are extreme points in the set of Branges measures are distinguished. Examples of measures that are not extreme points are also given. Bibliography: 7 titles. Translated fromProblemy Matematicheskogo Analiza, No. 19, 1999, pp. 27–68.