Aluthge Transforms of Complex Symmetric Operators

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Mathematics (Pomona)

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Complex symmetric operator, Aluthge transform, generalized Aluthge transform, Duggal transform, nilpotent operator, unitary equivalence, hyponormal operator, p-hyponormal operator, Hankel operator, Toeplitz operator, compressed shift, symmetric matrix


If T=U∣T∣ denotes the polar decomposition of a bounded linear operator T, then the Aluthge transform of T is defined to be the operator T˜=∣T∣12U∣T∣12 . 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:H→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 (T˜)∗ and (T∗)˜ are unitarily equivalent, (3) if T is complex symmetric, then T˜=T if and only if T is normal, (4) T˜=0 if and only if T 2 = 0, and (5) every operator which satisfies T 2 = 0 is necessarily complex symmetric.

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