Composition Operators and Generalized Paranormal Operators on Semi-Hilbert Spaces

In this paper, the interaction between composition operator and generalized paranormal operator in semi-Hilbert space is discussed. Semi-Hilbert spaces are a natural extension of classical Hilbert spaces, through a semi-inner product, which is associated to a positive bounded operator (A). Theories of operator theory and spectral analysis can be extended in this general framework in a flexible way. In the present study, some composition operators, (n,A)-paranormal operators and (k)-quasi-(A)-paranormal operators are focused. They are given in one treatise in the form of basic definitions, algebraic properties, and characteristics of their spectra. Furthermore, the concept of (A)-adjoint operators and the connection between (A)-adjoint and (A)-numerical range are explored, as well as its application in the characterisation of classes of operators. Several examples are presented to show the application of the ideas developed. The paper also points to other directions for future research such as weighted composition operators and reproducing kernel Hilbert spaces.