normal operator

The aim of this paper is to investigate the closed-ness of the range of the modular operators in Hilbert C*-modules. We present the conditions that, the reverse order law for the closed range modular operators and modular projections be holds. Also, we prove that, for modular operators A and B with closed ranges, if BA=0 then A†B† =0. Moreover, we give a novel characterization of the normal modular operators in Hilbert C*-modules.

Let B(H) denote the algebra of all bounded linear opera- tors acting on a complex Hilbert space H. For A, B ∈ B(H), define the bimultiplication operator M2,A,B on the class of Hilbert-Schmidt oper- ators by M2,A,B (X) = AXB. In this paper, we show that if B∗, the adjoint operator of B, is hyponormal, then co(W0(A)W0(B)) ⊆ W0(M2,A,B ), where co stands for the convex hull and W0(.) denotes the maximal numerical range. If in addition, A is hyponormal, we show that co(W0(A)W0(B)) = W0(M2,A,B ).

Let $T$ be a bounded linear operator on a Hilbert space $mathscr{H}$. We say that $T$ has the hyponormal property if there exists a function $f$, continuous on an appropriate set so that $f(|T|)geq f(|T^ast|)$. We investigate the properties of such operators considering certain classes of functions on which our definition is constructed. For such a function $f$ we introduce the $f$-Aluthge transform, $tilde{T}_{f}$. Given two continuous functions $f$ and $g$ with the property  $f(t)g(t)=t$, we also introduce the $(f,g)$-Aluthge transform, $tilde{T}_{(f,g)}$. The features of these transforms are discussed as well.