Analysis On Resolvent Set Of Operators And The Invariant Subspace Problem

The spectral theory is an important part of the operator theory,and analysis on resolvent set of operators is an effective method in the spectral theory.In 2016,Douglas and Yang established and developed a new method on resolvent set to analyze the structure of the operator.They defined a non-Euclidean metric on resolvent set,associated with the operator and the action of the operator on the elements in the space.This metric is used to reflect the structures and algebraic properties of operators.They and other scholars have got rich achievements on the study of this method.This implies that the method is highly effective in analyzing the structures of operators.This paper uses the method to study the structures of the quasinilpotent operator and the Bergman shift in depth.Both of them are tightly connected with the invariant subspace problem.For the quasinilpotent operator,the power set is an important invariant defined by this nonEuclidean metric on resolvent set.The second chapter of this paper computes the power set of cyclic quasinilpotent operators and studies the connection with their invariant subspaces.The first section proves that the power set of the strictly cyclic quasinilpotent unilateral weighted shift only contains 1.Then the second section defines the strongly strictly cyclic quasinilpotent operator,characterizes its invariant subspaces,and shows that its power set also only contains 1.The third section shows that the Volterra integral operator V on L2[0,1],whose power set contains more than one point,is cyclic but not strictly cyclic.The third chapter generalizes Douglas and Yang’s method of analysis on resolvent set,and applies this to the study of the structure of Bergman shift Tz.The first section constructs two invariants on the resolvent set of Tz,to characterize the Bergman inner function and the inclusion relationship of two invariant subspaces for Tz.Let IA be the invariant subspace for Tz determined by a Bergman zero set A,and Sz be the compression of Tz on La2(D)?IA.It has been proved that σ(Sz)equals the closure of A.The second section uses the growth condition of the resolvent to give another proof to this result.The third section generalizes the definition of the power set to any boundary point of spectrum,and computes the power set of Sz*at different boundary points of its spectrum.