Journal of Operator Theory

Volume 58, Issue 2, Fall 2007 pp. 229-250.

The real linear resolvent and cosolvent operators

Authors: Marko Huhtanen (1) and Olavi Nevanlinna (2)
Author institution: (1) Institute of Mathematics, Helsinki University of Technology, Box 1100, FIN-02015, Finland
(2) Institute of Mathematics, Helsinki University of Technology, Box 1100, FIN-02015, Finland

Summary: For an $\mathbb{R}$-linear operator $\mathcal{A}$ the resolvent operator is defined outside the spectrum of $\mathcal{A}$ while the cosolvent operator is defined outside the proper values of $\mathcal{A}$. In this paper these two functions are studied. Series expansions are given. A new characterization for the eigenvalues of real matrices is obtained. The cosolvent operator is used to define and analyze analytic functions of $\mathcal{A}$. An application of this leads to a decomposition of $\mathbb{R}$-linear operators. Classes of structured $\mathbb{R}$-linear operators are considered.

Keywords: Eigenvalues, proper values, resolvent operator, cosolvent operator, minimal polynomial

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