J-Selfadjoint Krein Space Operators and Aluthge Transforms

In this paper, we give a refined polar decomposition of a J-selfadjoint operator on a Krein space by decomposing the partial isometry in the polar decomposition. Using this refined polar decomposition, we prove some relations between J-selfadjoint operators and their (s, t)-Aluthge transforms. We investigate various spectra of a J-adjoint T-# and a J-adjoint (T) over tilde (#) of its Aluthge transform for a bounded Krein space operator T. Using their spectra, we develop Fredholm theory and J-Fredholm theory for the products (TT)-T-# and TT# and their Aluthge transforms ((TT)-T-#) over tilde and (TT#) over tilde, which become Fredholm, J-Fredholm, (a-)Weyl, or (a-)Browder.