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On a criterion for solvability of Fredholm equations. (Russian) Zbl 0594.47009
Let A be a bounded linear operator in a Hilbert space H and let $$b\in H$$. The author proves: If the equation $$Ax=b$$ has a solution then $$r^ 2AA^*-\| b\|^ 2P$$ is a nonnegtive operator for some $$r>0$$, where P is the orthogonal projection on the subspace spanned by b; if the range of A is closed or A is compact the reverse implication is true. This results are applied to Fredholm integral equations of the first and the second kind with Hilbert-Schmidt kernels.
[Remark: The abstract results are special cases of theorem 1 in R. G. Douglas, Proc. Am. Math. Soc. 17, 413-415 (1966; Zbl 0146.125); from this theorem follows that no extra conditions are necessary for the reverse implication above.]
Reviewer: K.-H.Förster

##### MSC:
 47A50 Equations and inequalities involving linear operators, with vector unknowns 47A53 (Semi-) Fredholm operators; index theories 47B10 Linear operators belonging to operator ideals (nuclear, $$p$$-summing, in the Schatten-von Neumann classes, etc.)
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