Seminormal operators with compact self-commutators

Putnam’s inequality $\pi ||{T^ \ast }T - T{T^ \ast }|| \leqq \operatorname {meas}_2({\text {sp}}(T))$ is verified for seminormal operators $T$ when ${T^ \ast }T - T{T^ \ast }$ is compact. The spectrum of the seminormal singular integral operator defined for $f \in {L^2}(0,1)$ by \[ Tf(s) = sf(s) + i\left ( {\frac {{\phi (s)}}{{\pi i}}\int _0^1 {\frac {{f(t)\bar \phi (t)}}{{s - t}}dt} } \right ),\quad s \in [0,1],\] where $\phi$ is any fixed essentially bounded measurable function on $[0,1]$, is computed.