Title

Aluthge Transforms of Complex Symmetric Operators

Document Type

Article

Department

Mathematics (Pomona)

Publication Date

2008

Keywords

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

Abstract

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.

Rights Information

© 2008 Springer-Verlag

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