  We analyze when an arbitrary matrix pencil is equivalent to a dissipative
Hamiltonian pencil and show that this heavily restricts the spectral
properties. In order to relax the spectral properties, we introduce matrix
pencils with coefficients that have positive semidefinite Hermitian parts. We
will make a detailed analysis of their spectral properties and their numerical
range. In particular, we relate the Kronecker structure of these pencils to
that of an underlying skew-Hermitian pencil and discuss their regularity,
index, numerical range, and location of eigenvalues. Further, we study matrix
polynomials with positive semidefinite Hermitian coefficients and use
linearizations with positive semidefinite Hermitian parts to derive sufficient
conditions for a spectrum in the left half plane and derive bounds on the
index.
