zbMATH — the first resource for mathematics

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

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.)
Full Text: EuDML