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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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