Aluthge Transforms of Complex Symmetric Operators

Abstract.

If $T = U\mid T\mid$ denotes the polar decomposition of a bounded linear operator T, then the Aluthge transform of T is defined to be the operator $\tilde{T} = {\mid}T{\mid}^\frac{1}{2}U {\mid}T{\mid}^\frac{1}{2}$. In this note we study the relationship between the Aluthge transform and the class of complex symmetric operators (T iscomplex symmetric if there exists a conjugate-linear, isometric involution $C : {\mathcal{H}} \rightarrow {\mathcal{H}}$ so that T = CT*C). In this note we prove that: (1) the Aluthge transform of a complex symmetric operator is complex symmetric, (2) if T is complex symmetric, then $(\tilde{T})^*$ and $\widetilde{(T^*)}$ are unitarily equivalent, (3) if T is complex symmetric, then $\tilde{T} = T$ if and only if T is normal, (4) $\tilde{T} = 0$ if and only if T ² = 0, and (5) every operator which satisfies T ² = 0 is necessarily complex symmetric.