This book summarizes the study of similarity and reducing subspaces for a class of multiplication operators on the Bergman space, Dirichlet space, and Sobolev disc algebra.It also investigates the compactness of certain radial and Hankel operators on the weighted Bergman space. The research methods combine operator theory techniques with infinite matrix manipulations.The book consists of five chapters. In the first chapter, we characterize the similarity and commutant of certain analytic Toeplitz operators. The second chapter focuses on the similarity of multiplication operator induced by finite Blaschke product and characterize the compactness of a class of Hankel operators. The third chapter discusses the quasi-similarity and reducing subspaces of a class of multiplication operator with monomial symbol in the Fock space. The fourth chapter explores the similarity and reducing subspaces of a class of multiplication operator induced by monomial symbol in the Sobolev disc algebra. The fifth chapter addresses the algebra matrix and similarity classification of certain Cowen-Douglas operators.The theme of the book is straightforward. Its contents can be comfortably understood by readers with a background in single-variable complex analysis and basic knowledge of operator theory.