On the holomorphism of subordinate semigroups

The concept of a subordinate semigroup was introduced by S. Bochner in his research on probability theory. Subordination in the Bochner sense is a method of obtaining new strongly continuous semigroups from the original one by integrating over the so—called subordinator, which is a convolutional semigroup of sub-probability measures on [0,\infty). The report will consider the holomorphic conditions of semigroups subordinate to a given strongly continuous semigroup of operators in a Banach space. In this case, we will discuss both conditions related to the case of one-parameter semigroups (the case first considered by A. Carasso and T. Kato) and to the case of multiparametric strongly continuous semigroups of operators in a Banach space. These results express the so-called "improving property" of the Bochner-Phillips functional calculus.