OPERATORS WITH COMPACT IMAGINARY PART

In this note we initiate a study of the old unsolved problem whether every T ∈ L(H) of the form T = H + iK with K compact has a nontrivial invariant subspace, using [6] as our main tool. In case K ≥ 0 we obtain some positive results. 1. Let H be a separable, infinite dimensional, complex Hilbert space. We write L(H) for the algebra bounded linear operators on H, K for the ideal of compact operators in L(H), and pi for the quotient (Calkin) map L(H) → L(H)/K. For T in L(H) we will write, as usual, σ(T), σe(T), σle(T), and ‖T‖e, for the spectrum, essential (Calkin) spectrum, left essential spectrum, and essential (Calkin) norm of T, respectively, and C∗(S) for the unital C∗-algebra generated by a collection S ⊂ L(H). We remind the reader that the question whether every operator T in L(H) with compact imaginary part (i.e., T = H + iK with H, K selfadjoint and K ∈ K) has a nontrivial invariant subspace has been around for over fifty years, but, nevertheless, is still open. In fact, the major results in the area, by Livsic [9], Macaev [10], Schwartz [12] and Gohberg-Krein [7], are over forty-five years old. The purpose of this note is to use the little known but useful main theorem from [6] to make some progress on this old outstanding problem. For the reader’s convenience, we first state the main result from [6], which characterizes the set of all (positive semidefinite) Q that arise from the Aronszajn-Smith “invariant subspace procedure ” applied to an arbitrary quasitriangular operator (cf. [11, Chapter 4]), in the form this theorem from [6] was used in [5] and [4], where it was also the main tool. Theorem 1.1. Let A be a unital, norm-separable, norm-closed subalgebra of L(H) and let Q ∈ L(H) be such that 0 Q 1. Then the following statements are equivalent: a) there exists a sequence {Qn}n∈N of projections in L(H) such tha