On Similarity And Reducing Subspaces Of The N-Shift Plus Certain Weighted Volterra Operator

The invariant subspace and reducing subspace problems are interesting and impor-tant themes in operator theory.It is conjectured that every bounded linear operator does have a non-trivial closed invariant subspace.The invariant subspace and reducing subspace problems on the Hardy space and the Bergman space have been studied ex-tensively in the literature.Let g（z）be an n-degree polynomial（n ≥ 2）.Inspired by Sarason’s results（[13],[14]）,we introduce the operator Ti defined by the multiplication operator Mg plus the weighted Volterra operator Vg on the Bergman space.We show that the operator Ti is similar to Mg on some Hilbert space S_g²（D）.Then for g（z）= zn,by using matrix and the operator theory,the reducing subspaces of the corresponding operator T₂ on the Bergman space are characterized.We use the characterization of the reducing subspaces of T₂ to obtain a description of reducing subspaces of Mzn on Sn2（D）.The main structure of this article is:In the first section,we give out some concepts.We introduce the concept of the weighted Bergman space on the unit disk,similarity,invariant subspaces,reducing sub-spaces,commutant,projection operator and so on.In the second section,by defining Hilbert spaces S_g²（D）,we prove that the multipli-cation operator Mzn is similar to T₁ on the Hilbert spaces.In the third section,we consider the case of g（z）= zn with the help of projec-tion operators,reducing subspaces,matrix multiplication,we characterize the reducing subspaces of T₂,it is shown that T₂ has exactly 2 reducing subspaces.In the fourth section,we prove that the multiplication operator Mzn is similar to T₂,we obtain that the multiplication operator Mzn has 2ⁿ reducing subspaces with minimal reducing subspaces L₀,,L₁,…,L_n-1.