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Zhu, Sen

Approximation of complex symmetric operators. (English) Zbl 1336.47015

Math. Ann. 364, No. 1-2, 373-399 (2016).
Summary: An operator \(T\) on a complex Hilbert space \(\mathcal H\) is called a complex symmetric operator if there exists a conjugate-linear, isometric involution \(C:\mathcal H\to\mathcal H\) so that \(CTC=T^\ast\). In this paper, we study the approximation of complex symmetric operators. By virtue of an intensive analysis of compact operators in singly generated \(C^\ast\)-algebras, we obtain a complete characterization of norm limits of complex symmetric operators and provide a classification of complex symmetric operators up to approximate unitary equivalence. This gives a general solution to the norm closure problem for complex symmetric operators. As an application, we provide a concrete description of partial isometries which are norm limits of complex symmetric operators.

Cited in 15 Documents

MSC:

47A58 Linear operator approximation theory
47A55 Perturbation theory of linear operators
47C10 Linear operators in \({}^*\)-algebras

Keywords:

approximation of complex symmetric operators; compact operators in singly generated \(C^\ast\)-algebras; norm closure problem; partial isometries
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References:

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