Range Inclusion and Diagonalization of Complex Symmetric Operators

We consider the range inclusion and the diagonalization in the Jordan algebra SC of C -symmetric operators, that are, bounded linear operators T satisfying CTC =T* , where C is a conjugation on a separable complex Hilbert space H . For T∈ SC , we aim to describe the set CR(T) of those operators A∈ SC satisfying the range inclusion R(A)⊂ R(T) . It is proved that (i) CR(T)=TSC T if and only if R(T) is closed, (ii) CR(T)=TSC T̄ , and (iii) CR(T)̄ is the closure of CR(T) in the strong operator topology. Also, we extend the classical Weyl–von Neumann Theorem to SC , showing that every self-adjoint operator in SC is the sum of a diagonal operator in SC and a compact operator with arbitrarily small Schatten p -norm for p∈ (1,∞ ) .