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Aluthge Transforms of Complex Symmetric Operators


If \(T = U\mid T\mid\) denotes the polar decomposition of a bounded linear operator T, then the Aluthge transform of T is defined to be the operator \(\tilde{T} = {\mid}T{\mid}^\frac{1}{2}U {\mid}T{\mid}^\frac{1}{2}\). 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 \(C : {\mathcal{H}} \rightarrow {\mathcal{H}}\) 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 \((\tilde{T})^*\) and \(\widetilde{(T^*)}\) are unitarily equivalent, (3) if T is complex symmetric, then \(\tilde{T} = T\) if and only if T is normal, (4) \(\tilde{T} = 0\) if and only if T 2 = 0, and (5) every operator which satisfies T 2 = 0 is necessarily complex symmetric.

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Correspondence to Stephan Ramon Garcia.

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This work partially supported by National Science Foundation Grant DMS 0638789.

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Garcia, S.R. Aluthge Transforms of Complex Symmetric Operators. Integr. equ. oper. theory 60, 357–367 (2008).

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Mathematics Subject Classification (2000).