Complex-self-adjointness. (English) Zbl 07640518
Let \(H\) be a linear operator in a complex separable Hilbert space \(\mathcal{H}\). Then \(H\) is said to be complex-symmetric with respect to \(C\) if the adjoint \(H^*\) is an extension of \(CHC^{-}\) for some antiunitary operator \(C\) in \(\mathcal{H}\). Furthermore, if \(H^*=CHC^{-1}\), it is said to be complex-self-adjoint with respect to \(C\).
There have been substantial works on this class of operators subject to \(C^2=I\). Particularly, Garcia and Putinar’s work on the subject stands out. In this paper, the authors extend some of S. R. Garcia and M. Putinar’s results in [Trans. Am. Math. Soc. 359, No. 8, 3913–3931 (2007; doi:10.1090/S0002-9947-07-04213-4)] to the general case. This paper also includes examples of complex-self-adjoint operators in different Hilbert spaces that are not necessarily subject to \(C^2=I\).
There have been substantial works on this class of operators subject to \(C^2=I\). Particularly, Garcia and Putinar’s work on the subject stands out. In this paper, the authors extend some of S. R. Garcia and M. Putinar’s results in [Trans. Am. Math. Soc. 359, No. 8, 3913–3931 (2007; doi:10.1090/S0002-9947-07-04213-4)] to the general case. This paper also includes examples of complex-self-adjoint operators in different Hilbert spaces that are not necessarily subject to \(C^2=I\).
Reviewer: Miyeon Kwon (Platteville)
MSC:
| 47A15 | Invariant subspaces of linear operators |
| 47B28 | Nonselfadjoint operators |
| 47B35 | Toeplitz operators, Hankel operators, Wiener-Hopf operators |
| 81Q12 | Nonselfadjoint operator theory in quantum theory including creation and destruction operators |
Keywords:
complex symmetric operator; complex self adjoint operator; singular value decomposition; polar decomposition; antilinear eigenfunction expansionReferences:
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