Spectral Properties and Fixed Point Theory for Block Operator Matrix

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Fixed Point Theory, Spectral Properties for Block Operator Matrix and Applications to Transport Equation

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.

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