A Note on Aluthge Transforms of Complex Symmetric Operators and Applications

Abstract.

In this paper, we reprove that: (i) the Aluthge transform of a complex symmetric operator $\tilde{T} = |T|^{\frac{1}{2}} U|T|^{\frac{1}{2}}$ is complex symmetric, (ii) if T is a complex symmetric operator, then $(\tilde{T})^{*}$ and $\widetilde{T^{*}}$ are unitarily equivalent. And we also prove that: (iii) if T is a complex symmetric operator, then $\widetilde{(T^{*})}_{s,t}$ and $(\tilde{T}_{t,s})^{*}$ are unitarily equivalent for s, t > 0, (iv) if a complex symmetric operator T belongs to class wA(t, t), then T is normal.