Spectral Properties and Fixed Point Theory for Block Operator Matrix

There are three questions that this work tries to answer. First, we investigate some spectral of fine properties of a 3 3 block operator matrix with unbounded entries and with domain consisting of vectors which satisfy certain relations between their components. An application to transport equations that describes the neutron transport in a plane-parallel domain with width 2a, or the transfer of unpolarized light in a plane-parallel atmosphere of optical thickness 2a is given. Second, we discuss under which conditions a 2 2 operator matrix with nonlinear entries, acting on a product of convex closed subsets of Banach spaces have a fixed point. As application, we give some existence results for a mixed stationary problem on Lp-spaces (1 < p < 1) inspired from the rotenberg's model. finally, we study some algebraic and topological properties of a new set defined by ghyp(t) := { a : t ai is hypercyclic} for a given bounded linear operator acting on separable banach space.