Gershgorin's theorem for matrices of operators

Abstract

Let A = (Aij) be an n × n matrix of operators acting on the Bannach space X = X1 ⊕ X2 ⊕…⊕Xn endowed with the ‖·‖x norm. Gershgorin circle theorem extends to this setting:Gi=σ(Aii)∪{λ:λ∉σ(Aii)and∥(λ−Aii)−1∥−1≤∑j=Li/jn∥Aij∥} then σ(A)⊂∪i=1nGiMoreover, assume that J is a proper nonempty subset of {1.2.…n}, if ⋃i∉jGj and ⋃i∉jGj are disjoint, then there exist invariant subspaces Y1 and Y2 for A such thatσ(A|Y1)⊂∪i∈JGiandσ(A|Y2)⊂∪i∉JGiwhere Y1 ≃ ⊕i∈jXiand Y2 ≃ ⊕i∉jXi. The notion of minimal Gershgorin sets that follows is one possible generalization of what Varga studied in the scalar case. (For partitioned matrices he used a more refined generalization.) For any positive n-dimensional vector x = (x1.…, xn) the operator ((1/x1)I⊕⋯⊕(1/xn)I)A(x1I⊕⋯⊕xnI)=Ax is similar to A. Let Gix(A)=σ(Aii)∪{z:z∉σ(Aii)and∥(Aii−z)−1∥−1≤∑j=Lj/in(xj/xi)∥Aij∥}. The minimal Gershgorin set G(A) is defined to be G(A)=∩x>0∪i−AnGix(A). As in the scalar case it has the property that σ(ΩA)=∪B∈ΩAσ(B)⊂G(A) where ΩA={B=(Bii):Bii=Aiifori=1.....nand∥Bij∥=∥Aij∥ifi≠j} It is proved that if each Xi is a Hilbert space and each Aii is normal, i = 1.…n, then∂G(A)⊂σ(ΩA)¯ where ∂G(A) denotes the boundary of G(A). It is worth remarking that the closure of σ(ΩA) is necessary.