Journal of Operator Theory

Volume 95, Issue 1, Winter 2026 pp. 275-302.

A functional model and tridiagonalisation for symmetric anti-linear operators

Authors: Alexander Pushnitski (1), Frantisek Stampach (2)
Author institution: (1) Department of Mathematics, King's College London, Strand, London, WC2R 2LS, U.K.
(2) Department of Mathematics, Czech Technical University in Prague, Trojanova 13, 12000 Prague 2, Czech Republic

Summary: We consider the class of bounded symmetric anti-linear operators $B$ with a cyclic vector. We associate with $B$ the spectral data consisting of a~probability measure and a~function. In terms of the spectral data of $B$, we introduce a functional model operator $\mathcal{B}$ acting on a model space. We prove an anti-linear variant of the spectral theorem demonstrating that $B$ is unitarily equivalent to $\mathcal{B}$. Next, we show that $B$ is also unitarily equivalent to an anti-linear tridiagonal operator and discuss connection with orthogonal polynomials in the anti-linear setting.

DOI: http://dx.doi.org/10.7900/jot.2024jul19.2461
Keywords: anti-linear operators, functional model, spectral theorem, Jacobi matrix, anti-orthogonal polynomials

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