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 Î B(H), letΔ (T) =│T│½U│T│½, Γ (T) = │T│U, C(T)= (T – iI)(T+ iI)-1 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)½ and U is the identity
element.
For the Aluthge transform, we look at some properties on the range
R(Δ)={Δ (T): T Î B(H)},of Δ and we prove that R(Δ) 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 have the polar decomposition for binormal operators and examine some complete
contractivity of maps associated with the Duggal transform by exploring some relations between
the operator T,the Aluthge transform of T and the Duggal 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 relation
directly by algebraic formula, its normal extension and the Quaternionic Cayley transform for
bounded and unbounded operators and their inverses.
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