On the imaginary component of a dissipative operator with slowly increasing resolvent

We consider the class $\Lambda$ (RZhMat., 1970, 6B675) of bounded dissipative operators with real spectrum acting in the infinite-dimensional separable Hilbert space $\mathfrak H$ whose resolvents $R_A(\lambda)$ satisfy the following growth condition:
$$\varlimsup_{y\to+0}\int_{-\infty}^\infty(1+x^2)^{-1}\ln^+y\,\|R_A(x+iy)\|\,dx<\infty.$$
Principal results:
1. The operator $H\geqslant0$ is the imaginary component of an operator $A\in\Lambda$ (i.e., $H=(1/2i)(A-A^*)$) if and only if $0$ is either an eigenvalue of infinite multiplicity for $H$ or a limit point for the spectrum of $H$.
2. All linear operators with imaginary component $H\geqslant0$ and real spectrum belong to the class $\Lambda$ if and only if $H$ is nuclear: $\operatorname{tr}H<\infty$.
Bibliography: 10 titles.