Abstract
A necessary and sufficient condition is given for a positive bounded linear operator with an integral kernel to be trace class on L 2(μ) for a σ-finite measure μ. The condition refines earlier criteria for positive Hilbert–Schmidt operators and positive integral operators with continuous kernels on a locally compact space.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Arthur, J.: An introduction to the trace formula, Harmonic analysis, the trace formula, and Shimura varieties, Clay Math. Proc. 4. Am. Math. Soc., Providence, RI, 1–263 (2005)
Bogachev V.: Measure Theory. Springer, Berlin (2007)
Brislawn C.: Kernels of trace class operators. Proc. Am. Math. Soc. 104, 1181–1190 (1988)
Brislawn C.: Traceable integral kernels on countably generated measure spaces. Pac. J. Math. 150, 229–240 (1991)
Carey A., Sukochev F.: Dixmier traces and some applications in non-commutative geometry. Russ. Math. Surv. 61, 1039–1099 (2006)
Castro M., Menegatto V., Peron A.: Integral operators generated by Mercer-like kernels on topological spaces. Colloq. Math. 126, 125–138 (2012)
Castro M., Menegatto V., Peron A.: Traceability of positive integral operators in the absence of a metric. Banach J. Math. Anal. 6, 98–112 (2012)
Delgado J.: Trace formulas for nuclear operators in spaces of Bochner integrable functions. Monatsh. Math. 172, 259–275 (2013)
Delgado J.: The trace of nuclear operators on L p(μ) for σ-finite Borel measures on second countable spaces. Integral Equ. Oper. Theory 68, 61–74 (2010)
Delgado, J.: A trace formula for nuclear operators on L p, Pseudo-differential operators: complex analysis and partial differential equations. Oper. Theory Adv. Appl. 205205:181–193
Ferreira J., Menegatto V., Oliveira C.: On the nuclearity of integral operators. Positivity 13, 519–541 (2009)
Gohberg, I., Goldberg, S., Krupnik, N.: Traces and determinants of linear operators. Operator Theory: Advances and Applications vol. 116, Birkhäuser Verlag, Basel (2000)
Gohberg, I., Kreĭn, M.: Introduction to the theory of linear nonselfadjoint operators, Translations of Mathematical Monographs vol. 18, American Mathematical Society, Providence, R.I. (1969)
Grafakos, L.: Classical Fourier analysis, Graduate Texts in Mathematics vol. 249, 2nd Ed., Springer, New York (2008)
Halmos, P., Sunder, V. Bounded integral operators on L 2 spaces, Ergebnisse der Mathematik vol. 96, Springer, Berlin-New York (1978)
Meyer-Neiberg P.: Banach lattices. Springer, Berlin (1991)
Pietsch A.: Nuclear Locally Convex Spaces, Ergebnisse der Mathematik und ihrer Grenzgebiete 66. Springer, New York-Heidelberg (1972)
Schwartz L.: Radon Measures on Arbitrary Topological Spaces and Cylindrical Measures, Tata Institute Publications. Oxford University Press, Bombay (1973)
Simon, B.: Trace ideals and their applications. 2nd Ed., Mathematical Surveys and Monographs, vol. 120 American Mathematical Society, Providence, RI (2005)
Weidman J.: Integraloperatoren der spurklasse. Math. Ann. 163, 340–345 (1966)
Zaanen A.: Integration. North Holland, Amsterdam (1967)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Jefferies, B. Traceability of Positive Integral Operators. Integr. Equ. Oper. Theory 84, 419–428 (2016). https://doi.org/10.1007/s00020-015-2267-9
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00020-015-2267-9