Complex aspects of gravity

This paper presents reflections on the validity of a series of mathematical methods and technical assumptions that are encrusted in macrophysics (related to gravitational interaction), that seem to have little or no physical significance. It is interesting to inquire what a change can occur if one removes some of the traditional assumptions.Comment: 10 page

Nehari problems and equalizing vectors for infinite-dimensional systems

For a class of infinite-dimensional systems we obtain a simple frequency domain solution for the suboptimal Nehari extension problem. The approach is via $J$-spectral factorization, and it uses the concept of equalizing vectors. Moreover, the connection between the equalizing vectors and the Nehari extension problem is given. \u

J-spectral factorization and equalizing vectors

For the Wiener class of matrix-valued functions we provide necessary and sufficient conditions for the existence of a $J$-spectral factorization. One of these conditions is in terms of equalizing vectors. A second one states that the existence of a $J$-spectral factorization is equivalent to the invertibility of the Toeplitz operator associated to the matrix to be factorized. Our proofs are simple and only use standard results of general factorization theory. Note that we do not use a state space representation of the system. However, we make the connection with the known results for the Pritchard-Salamon class of systems where an equivalent condition with the solvability of an algebraic Riccati equation is given. For Riesz-spectral systems another necessary and sufficient conditions for the existence of a $J$-spectral factorization in terms of the Hamiltonian is added