The spectrum of a class of singular integral operators (Q796039)

The author proves that the spectrum of \((Sf)(s)=sf(s)+B^*(s)\pi^{- 1}\int_{E}B(t)f(t)(s-t)^{-1}dt,\) where E is a bounded measurable subset of \(R^ 1\), coincides with the set \(\{z=x+iy\quad | y| \leq ess\lim \sup_{t\to x}\| B(t)\|^ 2,\quad x\in essential\quad closure\quad of\quad E\}.\) Here \(f\in L^ 2\!_ n(R)\), which is the space of \(C^ n-valued\) functions on \(R^ 1(n>1)\). For the case \(n=1\) see \textit{K. F. Clancey} and \textit{C. R. Putnam} [Comment. Math. Helv. 46, 451-456 (1971; Zbl 0228.47035)].