Vol. 39, No. 1, 1971

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Spectral theory for a first-order symmetric system of ordinary differential operators

Sorrell Berman

Vol. 39 (1971), No. 1, 13–30

For a symmetric differential expression associated with a first order system

A0(t)x′ + A (t)x,a < t < b

where A0 and A are n × n matrices and x is an n × 1 vector, a spectral decomposition will be developed. That is, if S is a closed symmetric differential operator determined by the differential system, the explicit nature of the generalized resolutions of the identity for all the self-adjoint extensions of S in any Hilbert space will be determined in terms of a fundamental matrix and spectral matrices associated with these extensions. An important aspect is that these self-adjoint extensions may be defined in Hilbert spaces larger than the natural one in which the operator S is defined.

Mathematical Subject Classification 2000
Primary: 47E05
Secondary: 34B25
Received: 4 September 1970
Revised: 19 June 1971
Published: 1 October 1971
Sorrell Berman