On some transforms of linear operators in a hilbert space

Abstract

Let B (H) be algebra of all bounded linear operator in a complex separable Hilbert space. For an operator T E B (H). let II en = I T I ''u I T I ''. r en = I T I u, C (T) = (T _ iI) (T+ iI)-! be the Aluthge transform, Duggal transform and Cayley transform respectively with U being a partial isometry, T= U , T' is the polar decomposition where , T' = (T*T)1h and U is the identity element.

For the Aluthge transform, we look at some properties on the range R (A) = {A en: TE B (H)}. of II and we prove that R(II) neither closed nor dense in B(H). We shall also discuss the properties of the spectrum and numerical range of Aluthge transform and their relationships and extend its iterated convergences of the Aluthge transform.

For the Duggal transform, we shall obtain the results about the polar decomposition of Duggal transform by giving the necessary and sufficient condition for the Duggal transform of T to bave the polar decomposition for binormal operators and examine some complete contractivity of maps associated with the Duggal transform by expioring some relations between the operator T. the Aluthge transform ofT and the Ouggai transform of T by studying maps between the Riesz-Dunford algebras associated with the operators. Finally under the cayley transform we define the cayley transform of a linear

Finally under the Cayley transform, we define the Cayley transform of a linear relation directly by algebraic formula. its normal extension and the Quatemionic Cayley transform for bounded and unbounded operators and their inverses.