Spectra of 3 × 3 upper triangular operator matrices

Abstract

Let H 1, H 2, and H 3 be complex separable Hilbert spaces. Given A ∈ B(H 1), B ∈ B(H 2), and C ∈ B(H 3), write $${M_{D,E,F}} = \left( {\begin{array}{*{20}{c}} A&D&E 0&B&F 0&0&C \end{array}} \right)$$ , where D ∈ B(H 2,H 1), E ∈ B(H 3,H 1), and F ∈ B(H 3,H 2) are unknown operators. This paper gives a complete description of the intersection ∩ D,E,F σ(M D,E,F ), where D, E, and F range over the respective sets of bounded linear operators. Further, we show that σ(A) ∪ σ(B) ∪ σ(C) = σ(M D,E,F ) ∪ W, where W is the union of certain gaps in σ(M D,E,F ), which are subsets of (σ(A) ∩ σ(B)) ∪ (σ(B) ∩ σ(C)) ∪ (σ(A) ∩ σ(C)). Finally, we obtain a necessary and sufficient condition for the relation σ(M D,E,F ) = σ(A)∪σ(B)∪σ(C) to hold for any D, E, and F.