Spectra of 3 × 3 upper triangular operator matrices

Abstract

Let H₁, H₂, and H₃ be complex separable Hilbert spaces. Given A ∈ B(H₁), B ∈ B(H₂), and C ∈ B(H₃), 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(H₂,H₁), E ∈ B(H₃,H₁), and F ∈ B(H₃,H₂) are unknown operators. This paper gives a complete description of the intersection ∩_D,E,Fσ(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.