The Gantmakher–Krein theorem for completely indecomposable operators in spaces of functions

O. Y. Kushel

Belarusian State University

Abstract: For a completely continuous non-negative operator $A$ acting in the space $L_p(\Omega)$ or $C(\Omega)$ the existence of $k$ positive eigenvalues is proved under some additional conditions on its $j$-th $(1<j\le k)$ exterior power $\wedge^jA$. For the case where the operator $A$ is completely indecomposable, the simplicity of all non-zero eigenvalues is proved and the connection between the imprimitivity indices of $A$ and $\wedge^jA$ is examined.

UDC: 517.948:330.105

Received: 05.12.2005