Abstract :
Let A = (Aij) be an n × n matrix of operators acting on the Bannach space X = X1 circled plus X2 circled plus…circled plusXn endowed with the double vertical bar·double vertical barx norm. Gershgorin circle theorem extends to this setting:image then imageMoreover, assume that J is a proper nonempty subset of {1.2.…n}, if union operatorinegated set membershipjGj and union operatorinegated set membershipjGj are disjoint, then there exist invariant subspaces Y1 and Y2 for A such thatimagewhere Y1 similar, equals circled plusiset membership, variantjXiand Y2 similar, equals circled plusinegated set membershipjXi. 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)Icircled pluscdots, three dots, centeredcircled plus(1/xn)I)A(x1Icircled pluscdots, three dots, centeredcircled plusxnI)=Ax is similar to A. Let image. The minimal Gershgorin set G(A) is defined to be image. As in the scalar case it has the property that image where ΩA={B=(Bii):Bii=Aiifori=1.…nandshort parallelBijshort parallel=short parallelAijshort parallelifi≠j} It is proved that if each Xi is a Hilbert space and each Aii is normal, i = 1.…n, thenimage where ∂G(A) denotes the boundary of G(A). It is worth remarking that the closure of σ(ΩA) is necessary.
