# 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 $\setR$-linear operator $\oper{A}$ the resolvent operator is defined outside the spectrum of $\oper{A}$ while the cosolvent operator is defined outside the proper values of $\oper{A}$. 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 $\oper{A}$. An application of this leads to a decomposition of $\setR$-linear operators. Classes of structured $\setR$-linear operators are considered.

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

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