Paranormal operators in semi-Hilbertian spaces

Abstract

The purpose of this paper is to obtain structural properties for a class of linear operators on semi-Hilbertian spaces i.e., spaces generated by positive semi-definite sesquilinear forms. This kind of spaces appears in many problems concerning linear and bounded operators on Hilbert spaces and is intensively studied in the present. We call the elements of this class A-paranormal operators. An operator $$T\in {{\mathcal {B}}}({{\mathcal {H}}})$$ is said to be A-paranormal if $$\Vert Tx\Vert _A^2\le \Vert T^2x\Vert _A ,\;\forall \; x\in {{\mathcal {H}}}: \Vert x\Vert _A=1.$$ Some of the basic properties of this class are studied. Moreover, the product, tensor product and the sum of finite numbers of these type are discussed.