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)



Norms of positive operators on $ L\sp p$-spaces

Authors: Ralph Howard and Anton R. Schep
Journal: Proc. Amer. Math. Soc. 109 (1990), 135-146
MSC: Primary 47B38; Secondary 47A30
MathSciNet review: 1000156
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Let $ 0 \leq T:{L^p}(Y,\nu ) \to {L^q}(X,\mu )$ be a positive linear operator and let $ \vert\vert T\vert{\vert _{p,q}}$ denote its operator norm. In this paper a method is given to compute $ \vert\vert T\vert{\vert _{p,q}}$ exactly or to bound $ \vert\vert T\vert{\vert _{p,q}}$ from above. As an application the exact norm $ \vert\vert V\vert{\vert _{p,q}}$ of the Volterra operator $ Vf(x) = \int_0^x {f(t)dt} $ is computed.

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

  • [B] B. Beauzamy, Introduction to Banach spaces and their geometry, North-Holland, 1982. MR 670943 (84g:46017)
  • [G] E. Gagliardo, On integral transformations with positive kernel, Proc. Amer. Math. Soc. 16 (1965), 429-434. MR 0177314 (31:1577)
  • [Gr] R. Grzaślewics, On isometric domains of positive operators on $ {L^p}$-spaces, Colloq. Math LII (1987), 251-261. MR 893541 (88h:47041)
  • [H-L-P] G.H. Hardy, J.E. Littlewood, and G. Pólya, Inequalities, Cambridge University Press, 1959.
  • [H-S] P.R. Halmos and V.S. Sunder, Bounded integral operators on $ {L^2}$ Spaces, Springer-Verlag, 1978. MR 517709 (80g:47036)
  • [K] U. Krengel, Ergodic theorems, De Gruyter, 1985. MR 797411 (87i:28001)
  • [M] B. Maurey, Théorèmes de factorisation pour les opérateurs linéaires à valeurs dans les espaces $ {L^p}$, Astérisque 11 (1974).
  • [S] V.S. Sunder, Absolutely bounded matrices, Indiana Univ. Math. J. 27 (1978), 919-927. MR 511247 (80d:47052)
  • [Z] A.C. Zaanen, Riesz spaces II, North-Holland, 1983. MR 704021 (86b:46001)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 47B38, 47A30

Retrieve articles in all journals with MSC: 47B38, 47A30

Additional Information

Keywords: Operator norms, positive linear operator, Volterra operator
Article copyright: © Copyright 1990 American Mathematical Society

American Mathematical Society