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Complex matrices that are structured with respect to a possibly degenerate indefinite inner product are studied. Based on the theory of linear relations, the notion of an adjoint is introduced: the adjoint of a matrix is defined as a linear relation which is a matrix if and only if the inner product is nondegenerate. This notion is then used to give alternative definitions of selfadjoint and unitary matrices in degenerate inner product spaces and it is shown that those coincide with the definitions that have been used in the literature before. Finally, a new definition for normal matrices is given which allows the generalization of an extension result for positive invariant subspaces from the case of nondegenerate inner products to the case of degenerate inner products.

The polynomial ω = P ( λ ) + iQ ( λ ) with real P ( λ ) and Q ( λ ) which belongs to Hermite-Biehler class (all its zeros lie in the open upper half-plane) and is symmetric $$ (\omega ( - \bar \lambda ) = \overline {\omega (\lambda )} ) $$ is modified as follows $$ (\omega _c (\lambda ) = \tilde P(\lambda ^2 + c) + i\lambda \tilde \hat Q(\lambda ^2 + c), c > 0. $$ Here $$ \tilde P(\lambda ^2 ) = P(\lambda ),\tilde \hat Q(\lambda ^2 ) = \lambda ^{ - 1} Q(\lambda ) $$ and $$ P(\lambda ) = \frac{{\omega (\lambda ) + \omega ( - \lambda )}} {2},Q(\lambda ) = \frac{{\omega (\lambda ) - \omega ( - \lambda )}} {{2i}} $$ The conditions are obtained necessary and sufficient for a set of complex numbers to be the zeros of a polynomial of the form ω _c( λ ).

The Douglas lemma on majorization and factorization of Hilbert space operators is extended to the setting of Krein space operators.

The behavior of the domain, the range, the kernel and the multivalued part of a polynomial in a linear relation is analyzed, respectively.