Real linear operators arise in a range of applications of mathematical physics. In this paper, basic properties of real linear operators are studied and their spectral theory is developed. Suitable extensions of classical operator theoretic concepts are introduced. Providing a concrete class, real linear multiplication operators are investigated and, motivated by the Beltrami equation, related problems of unitary approximation are addressed. Mathematics Subject Classification (2010). Primary 47A10; Secondary 47B38.

Abstract. Antilinear operators on a complex Hilbert space arise in various contexts in mathematical physics. In this paper, an analogue of the Weyl-von Neumann theorem for antilinear self-adjoint operators is proved, i.e. that an antilinear self-adjoint operator is the sum of a diagonalizable operator and of a compact operator with arbitrarily small Schatten p-norm. In doing so, we discuss conjugations and their properties. A spectral integral representation for antilinear self-adjoint operators is constructed.Mathematics Subject Classification (2010). Primary 47A10; Secondary 47B38.

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Abstract. The metaplectic group is generated by the Fourier transform and multiplications by functions of particular exponential type. Based on the use of the metaplectic representation and a factorization of symplectic matrices, in this paper a bound on the number of terms needed to factor an arbitrary metaplectic operator is derived. The approach is constructive and numerically stable, leading to a reliable factorization algorithm in practice. The problem is partially motivated by the task of constructing lens systems in diffractive optics.Mathematics subject classification (2010): 42B10, 78M25, 65F30.

Real linear operators emerge in a range of mathematical physics applications. In this paper spectral questions of compact real linear operators are addressed. A Lomonosov-type invariant subspace theorem for antilinear compact operators is proved. Properties of the characteristic polynomial of a finite rank real linear operator are investigated. A related numerical function, defined as a normalization of the characteristic polynomial, is studied. An extension to trace-class operators is discussed.