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Journal of Operator Theory

Volume 72, Issue 2, Fall 2014  pp. 451-473.

Function theory and spectral mapping theorems for antilinear operators

Authors:  Marko Huhtanen (1) and Allan Peramaki (2)
Author institution: (1) Division of Mathematics, Department of Electrical and Information Engineering, University of Oulu, 90570 Oulu 57, Finland
(2) Department of Mathematics and Systems Analysis, Aalto University, Box 1100, FIN-02015, Finland

Summary:  Unlike in complex linear operator theory, polynomials or, more generally, Laurent series of antilinear operators cannot be modelled with complex analysis. There exists a corresponding function space, though, surfacing in spectral mapping theorems. These spectral mapping theorems are inclusive in general. Equality can be established in the self-adjoint case. The arising functions are shown to possess a biradial character. It is shown that to any given set of Jacobi parameters corresponds a biradial measure yielding these parameters in an iterative orthogonalization process in this function space, once equipped with the corresponding $L^2$ structure.

Keywords:  antilinear operator, Laurent series, spectral mapping, biradial function, biradial measure, Jacobi operator, Hankel operator

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