Continuity of the spectrum on a class of upper triangular operator matrices

Let B ( H ) denote the algebra of operators on an infinite dimensional complex Hilbert space H , and let A ○ ∈ B ( K ) denote the Berberian extension of an operator A ∈ B ( H ) . It is proved that the set theoretic function σ, the spectrum, is continuous on the set C ( i ) ⊂ B ( H i ) of operators A for which σ ( A ) = { 0 } implies A is nilpotent (possibly, the 0 operator) and A ○ = ( λ X 0 B ) ( ( A ○ − λ ) − 1 ( 0 ) { ( A ○ − λ ) − 1 ( 0 ) } ⊥ ) at every non-zero λ ∈ σ p ( A ○ ) for some operators X and B such that λ ∉ σ p ( B ) and σ ( A ○ ) = { λ } ∪ σ ( B ) . If C S ( m ) denotes the set of upper triangular operator matrices A = ( A i j ) i , j = 1 m ∈ B ( ⊕ i = 1 n H i ) , where A i i ∈ C ( i ) and A i i has SVEP for all 1 ⩽ i ⩽ m , then σ is continuous on C S ( m ) . It is observed that a considerably large number of the more commonly considered classes of Hilbert space operators constitute sets C ( i ) and have SVEP.