Real Linear Operators on Complex Hilbert Spaces with Applications

This dissertation studies real linear operators on complex Hilbert spaces. The focus is particularly on questions concerning spectral properties. Real linear operators arise naturally in applications such as the mathematical description of planar elasticity or the inverse conductivity problem utilized in electrical impedance tomography. Challenges in real linear operator theory stem from the observation that some foundational properties for complex linear operators may not hold for real linear operators in general. The findings in this dissertation include basic spectral properties of real linear operators. An analogue of the Weyl-von Neumann theorem is proved concerning the diagonalizability of self-adjoint antilinear operators. Properties of the characteristic polynomial of a finite rank real linear operator are studied. Addressing invariant subspaces, an analogue of Lomonosov's theorem is proved for compact antilinear operators. With respect to the Beltrami equation, real linear multiplication operators are discussed. A factorization of symplectic and metaplectic operators is presented in connection to optics.