Pologrupy operátorů a jejich orbity

Title: Semigroups of operators and its orbits Author: Jan Vršovský Department: Institute of Mathematics of the Academy of Sciences of the Czech Republic Supervisor: prof. RNDr. Vladimír Müller, DrSc., Institute of Mathematics of the AS CR Abstract: The orbit of a bounded linear operator T on a Banach space is a se- quence T n x, n = 0, 1, 2, . . ., where x is a fixed vector. The orbits are closely connected to the dynamics of operator semigroups and to the invariant sub- spaces and subsets. The thesis studies the relation between the operator and its orbits. The subject of the first part is the relation between sequences T n x and T n , stability and orbits tending to infinity. The second part deals with dense orbits - hypercyclicity and related notions. In the third part, an ana- logue of reflexive algebras of operators, orbit reflexive operators are defined and studied. Apart from "normal" orbits of a single operator, the weak orbits and orbits of C0-semigroups are also touched. Keywords: operator, semigroup, orbit, hypercyclic, orbit reflexiv