Abstract. It is becoming more and more apparent that the class of well-bounded operators of type {\rm(B)} is naturally suited for the investigation of many problems in analysis. In this note a complete description is given of the structure and spectral properties of this family of operators in the class of Banach spaces called hereditarily indecomposable. The special properties of this class of spaces (intensively studied in [9]) forces such operators to have a particularly simple form.

AMS Subject Classification (1991): 47B40, 47B06

Received April 22, 1994. (Registered under 5600/2009.)