Aluthge Operator Field and Its Numerical Range and Spectral Properties

For an arbitrary operator T acting on a Hilbert space we consider a field of operators ( Δ _z(T)) called the Aluthge operator field associated with T . After giving preliminary results, we establish that two fields (left and right), canonically linked to the Altuthge field ( Δ _z(T)) and a support subspace, are constant on each horizontal segment where they are defined. This result leads to a positive solution of a conjecture stated by Jung-Ko-Pearcy in 2000. Then we do a detailed spectral study of ( Δ _z(T)) and we give a Yamazaki type formula in this context.