Aluthge Transforms of Complex Symmetric Operators

. If denotes the polar decomposition of a bounded linear operator T , then the Aluthge transform of T is defined to be the operator . In this note we study the relationship between the Aluthge transform and the class of complex symmetric operators ( T is complex symmetric if there exists a conjugate-linear, isometric involution 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 and are unitarily equivalent, (3) if T is complex symmetric, then if and only if T is normal, (4) if and only if T 2 = 0, and (5) every operator which satisfies T 2 = 0 is necessarily complex symmetric.