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Additivity properties of operator ranges. (English) Zbl 1339.47001
Let \(L(\mathcal{H})\) be the algebra of all bounded linear operators on a Hilbert space \(\mathcal{H}\) and for \(A\in L(\mathcal{H})\) let \(R(A)\) stand for the range of \(A\). In this paper, the authors consider conditions on operators \(A,B\in L(\mathcal{H})\) that ensure the additivity property of operator ranges \(R(A+B)=R(A)+R(B)\). In particular, they examine cases when \( R(A)+R(B)\) is closed or dense in \(\mathcal{H}\), when \( R(A)+R(B)= \mathcal{H}\). Some generalizations and refinements of known results are obtained (see in this connection [P. A. Fillmore and J. P. Williams, Adv. Math. 7, 254–281 (1971; Zbl 0224.47009); C.-Y. Deng et al., Linear Algebra Appl. 437, No. 9, 2366–2385 (2012; Zbl 1276.47004)] and the literature therein).

MSC:
47A05 General (adjoints, conjugates, products, inverses, domains, ranges, etc.)
46C07 Hilbert subspaces (= operator ranges); complementation (Aronszajn, de Branges, etc.)
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