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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.)
##### Keywords:
operator ranges; Douglas’ theorem; oblique projections
Full Text:
##### References:
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