CONTINUITY OF SPECTRA AND COMPACT PERTURBATIONS

In this note we give conditions for continuity of spectrum, approximative point spectrum and defect spectrum on the set fTg+K(X), where T 2 B(X) and K(X) is the set of compact operators. X into itself. For T 2 B ( X ), let � ( T ), � e( T ), � le( T ), � re( T ), � a( T ) ands( T ) denote respectively the spectrum, the essential spec- trum, the left essential spectrum, the right essential spectrum, the approximate point spectrum and the surjective spectrum. We set � lre( T ) = � le( T ) \ � re( T ). Let S denote the collection of all non-empty compact subsets of C. Equip- ping S with the Hausdorff metric, the spectrum can be viewed as a function � : B ( X ) ! S mapping operators T 2 B ( X ) into their spectrum � ( T ). In general the spectrum is not continuous, but it is always upper semi-continuous (see (8)). Given an operator T 2 B ( X ), we write N ( T ) and R ( T ) for null space and range of T. Denote � ( T ) = dim N ( T ) and � ( T ) = dim X=R ( T ). An operator T is called upper semi-Fredholm ( T 2 � +( X )), respectively lower semi-Fredholm ( T 2 � ( X )), if R ( T ) is closed and � ( T ) < 1 , respectively � ( T ) < 1 . If T is either upper or lower semi-Fredholm, then T is called a semi-Fredholm operator ( T 2 � ( X )). When T is both upper and lower semi-Fredholm, T is a Fredholm operator ( T 2 �( X )). The index of a semi-Fredholm operator T is dened as i ( T ) = � ( T ) � ( T ). We set � + ( X ) = f T 2 B ( X ) j T 2 � +( X ) and i ( T ) � 0 g , � + ( X ) = f T 2 B ( X ) j T 2 � ( X ) and i ( T ) � 0 g and � 0( X ) = f T 2 �( X ) j i ( T ) = 0 g . The ascent of T 2 B ( X ), denoted by asc(