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It is well known that two hermitian n x n matrices K, H, where H is positive definite, H > 0, can be simultaneously diagonalized. The key to the proof is to consider C[superscript]n, where C is the complex number field, as a Hilbert space [Fraktur capital]H [subscript]H with the inner product given by (f,g) = g*Hf, where f,g [lowercase Greek]Epsilon C[superscript]n, considered as a space of column vectors. Then the operator A = H⁻¹K is selfadjoint in [Fraktur capital]H [subscript]H, and the spectral theorem readily yields the result. Of course such A, when K is not hermitian, can also be investigated in [Fraktur capital]H [subscript]H. We consider a similar problem where K, H are replaced by a pair of ordinary differential expressions L and M, where M > 0 in some sense. Two difficulties arise: (1) there are many natural choices for a selfadjoint H > 0 generated by M, and hence many choices for [Fraktur capital]H [subscript]H, and (2), once a choice for H has been made, there are many choices for the analogue of A. In our work we consider all possible choices for H > 0 and the analogue of A.

It is well known that two hermitian n x n matrices K, H, where H is positive definite, H > 0, can be simultaneously diagonalized. The key to the proof is to consider C[superscript]n, where C is the complex number field, as a Hilbert space [Fraktur capital]H [subscript]H with the inner product given by (f,g) = g*Hf, where f,g [lowercase Greek]Epsilon C[superscript]n, considered as a space of column vectors. Then the operator A = H⁻¹K is selfadjoint in [Fraktur capital]H [subscript]H, and the spectral theorem readily yields the result. Of course such A, when K is not hermitian, can also be investigated in [Fraktur capital]H [subscript]H. We consider a similar problem where K, H are replaced by a pair of ordinary differential expressions L and M, where M > 0 in some sense. Two difficulties arise: (1) there are many natural choices for a selfadjoint H > 0 generated by M, and hence many choices for [Fraktur capital]H [subscript]H, and (2), once a choice for H has been made, there are many choices for the analogue of A. In our work we consider all possible choices for H > 0 and the analogue of A.