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Complex symmetric operators and applications. (English) Zbl 1087.30031
Let \(C\) be an antilinear isometric involution on a complex Hilbert space \(H\). A bounded linear operator \(T\) on \(H\) is called \(C\)-symmetric if \(C T= T^{\ast} C\). This is equivalent to the symmetry of \(T\) with respect to the bilinear form \([f, g]= \langle f, Cg\rangle\). If \((e_{i})_{i\in I}\) is an orthonormal basis in \(H\) which is left invariant by \(C\), i.e. \(Ce_{i}=e_{i}\), then \(C\)-symmetry of \(T\) is simple complex symmetry of the associated matrix.
In this paper, the authors study a few classes of Hilbert space operators whose matrix representations are complex symmetric with respect to a preferred orthonormal basis. They explore applications of this symmetry to Jordan canonical models, self-adjoint extensions of symmetric operators, rank-one unitary perturbations of the compressed shift, Darlington synthesis and matrix-valued inner functions, and free bounded analytic interpolation in the disk.

MSC:
30D55 \(H^p\)-classes (MSC2000)
47A15 Invariant subspaces of linear operators
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