Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Absolutely continuous component of a class of integral operators


Author: Richard E. Sours
Journal: Proc. Amer. Math. Soc. 37 (1973), 521-524
MSC: Primary 47G05
DOI: https://doi.org/10.1090/S0002-9939-1973-0312333-1
MathSciNet review: 0312333
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: The operator $ T:{L^2}(0,\infty ) \to {L^2}(0,\infty )$ defined by

$\displaystyle Tf(x) = \int_0^\infty {\frac{{k(x){{(k(t))}^ - }}}{{x + t}}} \;f(t)\;dt,$

where $ {(k(t))^ - }$ is the complex conjugate, is studied and conditions are given which are sufficient to characterize the absolutely continuous component.

References [Enhancements On Off] (What's this?)

  • [1] A. Erdélyi et al., Higher transcendental functions. Vol. 1. The hypergeometric function, Legendre functions, McGraw-Hill, New York, 1953. MR 15, 419.
  • [2] J. S. Howland, Trace class Hankel operators, Quart. J. Math. Oxford Ser. (2) 22 (1971), 147-159. MR 0288630 (44:5826)
  • [3] T. Kato, Perturbation theory for linear operators, Die Grundlehren der math. Wissenschaften, Band 132, Springer-Verlag, New York, 1966. MR 34 #3324. MR 0203473 (34:3324)
  • [4] W. Koppleman, Spectral multiplicity theory for a class of singular integral operators, Trans. Amer. Math. Soc. 113 (1964), 87-100. MR 29 #1555a. MR 0164256 (29:1555a)
  • [5] J. D. Pincus, On the spectral theory of singular integral operators, Trans. Amer. Math. Soc. 113 (1964), 101-128. MR 29 #1555b. MR 0164257 (29:1555b)
  • [6] -, Commutators, generalized eigenfunction expansions and singular integral operators, Trans. Amer. Math. Soc. 121 (1966), 358-377. MR 32 #6228. MR 0188796 (32:6228)
  • [7] -, Commutators and systems of singular integral equations. I, Acta Math. 121 (1968), 219-249. MR 39 #2026. MR 0240680 (39:2026)
  • [8] M. Rosenblum, On the Hilbert matrix. II, Proc. Amer. Math. Soc 9 (1958), 581-585. MR 20 #6038. MR 0099599 (20:6038)
  • [9] -, A spectral theory for self-ādjoint singular integral operators, Amer. J. Math. 88 (1966), 314-328. MR 33 #6453. MR 0198294 (33:6453)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 47G05

Retrieve articles in all journals with MSC: 47G05


Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1973-0312333-1
Keywords: Integral operator, commutator systems, absolutely continuous component
Article copyright: © Copyright 1973 American Mathematical Society

American Mathematical Society