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

AbstractLet 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(Hi) of operators A for which σ(A)={0} implies A is nilpotent (possibly, the 0 operator) and A○=(λX0B)((A○−λ)−1(0){(A○−λ)−1(0)}⊥) at every non-zero λ∈σp(A○) for some operators X and B such that λ∉σp(B) and σ(A○)={λ}∪σ(B). If CS(m) denotes the set of upper triangular operator matrices A=(Aij)i,j=1m∈B(⊕i=1nHi), where Aii∈C(i) and Aii has SVEP for all 1⩽i⩽m, then σ is continuous on CS(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