Hyponormal operators having real parts with simple spectra

Abstract:

Let ${T^ \ast }T - T{T^ \ast } = D \geq 0$ and suppose that the real part of $T$ has a simple spectrum. Then $D$ is of trace class and $\pi$ trace$(D)$ is a lower bound for the measure of the spectrum of $T$. This latter set is specified in terms of the real and imaginary parts of $T$. In addition, the spectra are determined of self-adjoint singular integral operators on ${L^2}(E)$ of the form $A(x)f(x) + \Sigma {b_j}(x)H[f{\bar b_j}](x)$, where $E \ne ( - \infty ,\infty ),A(x)$ is real and bounded, $\Sigma |{b_j}(x){|^2}$ is positive and bounded, and $H$ denotes the Hilbert transform.