Mosic-Abyzov Inverse of the Anti-triangular Perturbation Matrix
This paper investigated the Mosic-Abyzov inverse of an anti-triangular perturbation matrix over Banach algebra.Let a,b∈A⊕,under the conditions of b⊕a=0 and babπ=0,it is proved that (a1b0)∈M2(A)⊕ using the Peirce decomposition.Based on the additive decomposition of matrices,it is also proved that if b2 a=0 and ababπ=0,then(a1b0)∈M2(A)⊕.Moreover,it is shown that(a1b0)∈ M2(A)⊕ if b⊕ a=0 and(ab)bπ=(ba)bπ by means of the(a1b0)∈M(A)⊕.Moreover,it is shown that (a1b0)∈M2(A)⊕ifb⊕a=0 and(ab)bπ=(ba)b by means of the solvability of the equation ax+1=xbx.
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