#### 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* is*complex 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

#### Terms of Use & License Information

#### DOI

10.1007/s00020-008-1564-y

#### Recommended Citation

Garcia, S.R., Aluthge transforms of complex symmetric operators, Integral Equations Operator Theory 60, no. 3, (2008), 357-367. MR2392831 (2008m:47052). doi: 10.1007/s00020-008-1564-y