Abstract
Given a Lebesgue measurable self-map \({\varphi}\) of the interval [0, 1], the Volterra- composition operator is defined as \({(V_\varphi)(x)=\int\limits_0^{\varphi(x)}f(t)\,dt, \quad f \in L^p[0,1],\,\, 1\leq p \leq \infty.}\) We develop the spectral theory of these operators. In particular, for a class of natural symbols \({\varphi}\) , finiteness of the spectrum is characterized and formulae for the trace and the convergence exponent of eigenvalues are provided. The positivity of the spectrum as well as the analyticity of the eigenfunctions are also treated. The theory of entire functions as well as solving some Cauchy Problems will play a fundamental role in this theory. We also supply some examples of symbols \({\varphi}\) to which the theory can be applied and, in particular, eigenvalues and eigenfunctions are computed explicitly.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Conway J.B.: A course in Functional Analysis, 2nd edn. Springer, New York (1990)
Gohberg, I., Kreĭ n, M.: Introduction to the theory of non-selfadjoint operators. Translations of Mathematical Monographs, vol. 18. American Mathematical Society, Providence (1969)
Halmos P.R.: A Hilbert Problem Book. Van Nostrand Inc., New York (1967)
Halmos P.R., Sunder V.S.: Bounded Integral Operators on L 2 spaces. Springer, New York (1978)
Herrero D.: Quasidiagonality, similarity and approximation by nilpotent operators. Indiana Univ. Math. J. 30, 199–233 (1981)
Karlin S.: Total Positivity. Stanford University Press, Stanford (1968)
Kreĭn, M., Rutman, M.: Linear operators leaving invariant a cone in a Banach space, vol. 26. Amer. Math. Soc., Providence (1950)
Lebedev N.N.: Special Fssunctions and their Applications. Prentice-Hall Inc., Englewood Cliffs (1965)
Levin B.Ja.: Distribution of Zeros of Entire Functions. AMS Providence, Rhode Island (1980)
Meyer-Nieberg P.: Banach Lattices. Springer, Berlin (1991)
Minc H.: Nonnegative Matrices. Wiley, New York (1987)
Newburgh J.D.: The variation of the spectra. Duke Math. J. 18, 165–176 (1951)
Pietsch A.: Eigenvalues and s-Numbers. Cambridge University Press, Cambridge (1987)
Rudin W.: Real and Complex Analysis. McGraw-Hill, New York (1971)
Rudin W.: Functional Analysis. McGraw-Hill, New York (1973)
Sadovnichiĭ V.A.: Theory of Operators. Consultants Bureau, New York (1991)
Schwarz L.: Analyse Mathematique. Hermann, Paris (1967)
Szegö G.: Orthogonal polynomials, vol. 23. American Mathematical Society Colloquium Publications, Buffalo (1939)
Tong Y.S.: Quasinilpotent integral operators. Acta Math. Sinica 32, 727–735 (1989)
Watson G.N.: A treatise on the Theory of Bessel functions. Cambridge University Press, New York (1944)
Whitley R.: The spectrum of a Volterra composition operator. Integral Equat. Oper. Theory 10, 146–149 (1987)
Author information
Authors and Affiliations
Corresponding author
Additional information
Partially supported by Plan Nacional I+D+I grant no. MTM2006-09060, Junta de Andalucía FQM-260 and P06-FQM-02225.
Rights and permissions
About this article
Cite this article
Montes-Rodríguez, A., Rodríguez-Martínez, A. & Shkarin, S. Spectral theory of Volterra-composition operators. Math. Z. 261, 431–472 (2009). https://doi.org/10.1007/s00209-008-0365-y
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00209-008-0365-y