On spectral resolutions connected with self-adjoint differential vector-operators in a Hilbert space

Abstract

We show that spectral resolutions of differential vector-operators may be represented as a specific direct sum integral operator with a kernel written in terms of generalized vector-operator eigenfunctions. Then we prove that a generalized eigenfunction measurable with respect to the spectral parameter may be decomposed using a set of analytical defining systems of coordinate operators.

Keywords:

• Vector-operators
• Cyclic vector
• Spectral representation
• Ordered representation
• Multiplicity
• Unitary transformation
• 1999 Mathematics Subject Classifications: Primary: 34L05
• 47B25
• 47B37
• 47A16

Acknowledgements

The first author would like to express his thanks to the Abdus Salam International Center for Theoretical Physics for the hospitality during his visit in April, 2004. The second author was supported by the Abdus Salam ICTP (grants AC-84 and MISC-03/14). The authors would also like to express their thanks to Professor Everitt, who had drawn the attention of the authors to the problem.