Article PDF
Avoid common mistakes on your manuscript.
Literatur
F. Rellich, Spektraltheorie in nicht-separablen Räumen, Math. Annalen 110 (1934), p. 342–356.
J. Ernest Wilkins, Definitely self-conjugate adjoint integral equations, Duke Math. Journal 11 (1944), p. 155–166.
G. A. Bliss, Definitely self-adjoint boundary value problems, Transactions Am. Math. Soc. 44 (1938), p. 413–428.
W. T. Reid, Expansion problems associated with a system of linear integral equations, Transactions Am. Math. Soc. 33 (1931), p. 475–485.
A. C. Zaanen, On the theory of linear integral equations VIII, Proc. Kon. Ned. Akad. v. Wetensch. (Amsterdam) 50 (1947), p. 465–473 and p. 612–617 (=Indagationes Math. 9 (1947), p. 271–279 and p. 320–325).
A. C. Zaanen, Ueber vollstetige symmetrische und symmetrisierbare Operatoren, Nieuw Arch. v. Wisk. (2), 22 (1943), p. 57–80.
A. C. Zaanen, On the theory of linear integral equations I, Proc. Kon. Ned. Akad. v. Wetensch. (Amsterdam) 49 (1946), p. 194–204 (=Indagationes Math. 8 (1946), p. 91–101).
A. C. Zaanen, On the theory of linear integral equations II–VI, Proc. Kon. Ned. Akad. v. Wetensch. (Amsterdam) 49 (1946), p. 205–212, 202–301, 409–423, 571–585, 608–621 (=Indagationes Math. 8 (1946), p. 102–109, 161–170, 264–278, 352–366, 367–380).
Symmetrisable transformationsK such that bothH andK are of integral type with bounded kernelsH(x, y) andK(x, y) were introduced for the first time byJ. Marty, Valeurs singulières d'une équation de Fredholm, Comptes Rendus de l'Acad. des sc. (Paris) 150 (1910), p. 1499–1502.
Some authors use the name of singular value for the reciprocal value of λ.
Part of the idea of this proof is derived from the proof that an integral equation with a non-vanishing Hermitian kernel has at least one characteristic value ≠ 0, as given inO. D. Kellogg, On the existence and closure of sets of characteristic functions, Math. Annalen 86 (1922), p. 14–17.
Some authors call λ an eigenvalue ofK, and reserve the name of characteristic value for the reciprocal value of λ.
A bounded normal transformation in a Hilbert space of infinite dimension is completely continuous if and only if its spectrum converges to o.
Cf. p. 198, footnote 1.
Cf., p. 198. footnote 4.
F. Smithies, The Fredholm theory of integral equations, Duke Math. Journal 8 (1941), p. 107–130, Lemma 2. 6.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Zaanen, A.C. Normalisable transformations in Hilbert space and systems of linear integral equations. Acta Math. 83, 197–248 (1950). https://doi.org/10.1007/BF02392637
Published:
Issue Date:
DOI: https://doi.org/10.1007/BF02392637