Measurable enumeration of eigenelements

Abstract

We prove that for eigenelements of a measurable family of self-adjoint linear operators in separable Hilbert space there exists a measurable enumeration. We also prove a similar result for measurable families of bounded linear operators having at most countably many eigenvalues (under certain restrictions on the parameter space). The proof of the latter result is based on descriptive set theory, while in the case of self-adjoint (and some more general) operators the proof is constructive.

Keywords:

• Measurable space
• eigenvalue
• eigenvector
• measurable enumeration

^*Department of Mathematics, University of North Carolina at Charlotte, Charlotte, NC 28223; Email:[email protected].

^†International Institute of Earthquake Prediction Theory and Mathematical Geophysics of Russian Academy of Sciences, Warshavskoe shosse 79, korpus 2, Moscow 113556, Russia.

^‡Department of Mathematics, Caltech, Pasadena, CA 91125; [email protected].

^ξResearch partially supported by NSF Grant DMS 96-19880.

^*Department of Mathematics, University of North Carolina at Charlotte, Charlotte, NC 28223; Email:[email protected].

^†International Institute of Earthquake Prediction Theory and Mathematical Geophysics of Russian Academy of Sciences, Warshavskoe shosse 79, korpus 2, Moscow 113556, Russia.

^‡Department of Mathematics, Caltech, Pasadena, CA 91125; [email protected].

^ξResearch partially supported by NSF Grant DMS 96-19880.

Notes

^*Department of Mathematics, University of North Carolina at Charlotte, Charlotte, NC 28223; Email:[email protected].

^†International Institute of Earthquake Prediction Theory and Mathematical Geophysics of Russian Academy of Sciences, Warshavskoe shosse 79, korpus 2, Moscow 113556, Russia.

^‡Department of Mathematics, Caltech, Pasadena, CA 91125; [email protected].

^ξResearch partially supported by NSF Grant DMS 96-19880.

Additional information

Notes on contributors

Alexander Y. Gordon

* †

Alexander S. Kechris

‡ ξ