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
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
