Spectral characterizations for Hermitian centrosymmetric K-matrices and Hermitian skew-centrosymmetric K-matrices

Let A and K be real symmetric matrices with K-2 = I. In the article "A spectral characterization of generalized real symmetric centrosymmetric and generalized real symmetric skew-centrosymmetric matrices" [D. Tao and M. Yasuda, SIAM J. Matrix Anal. Appl., 23 (2002), pp. 885-895], it was shown that (1) AK = KA if and only if the spectrum of A equals the spectrum of KA up to sign and (2) AK = -KA if and only if the spectrum of A equals the spectrum of KA multiplied by i. This paper extends these spectral characterizations from the case of real symmetric matrices to that of self-adjoint compact linear operators in a complex Hilbert space. Some consequences of these results are mentioned, including an application that describes the correspondence between the spectrum of a real symmetric Toeplitz matrix T and its associated Hankel matrix JT, where J is the so-called exchange matrix.