On the stability problem of a pair of adjoint operators

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Abstract
As an introduction, the eigenvalue problem for a linear operator T having a discrete point spectrum and a complete set of eigenfunctions is studied. The bivariational principle for T and its adjoint operator T° is derived, and the biorthogonal properties of their eigenfunctions are discussed. The main part of the paper is then concerned with the problem whether these features can be extended also to a general pair of adjoint operators, T and T°, in which case the eigenvalue problem is replaced by the more general stability problem. The stability problem for a pair of adjoint operators—T and T°—is first formulated in terms of nonorthogonal projectors—O and O°—which decompose these operators and satisfy the commutation relations TO=OT and T°O°=O°T°. In the case of a finite space, these skew-projectors may be explicitly expressed in product forms derived from the reduced Cayley–Hamilton equation for the operator T. It is shown that, if the stable subspaces defined by these projectors are properly classified by their Segre characteristics, one may explicitly derive the form of the projectors for the irreducible stable subspaces associated with the individual Jordan blocks of the so-called classical canonical forms of the matrix representations of T and T°. It is further shown that, in such a case, the biorthonormality property of the generalized eigenfunctions is still valid, and that a bivariational principle may be derived. The extension of these results to infinite spaces is finally briefly discussed.