Banach Journal of Mathematical Analysis

Energy functional of the Volterra operator

Yu-Xia Liang and Rongwei Yang

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

We define the energy functional Ef,A for a bounded linear operator A acting on a Hilbert space H through a newly defined non-Euclidean metric gf(z)|dz|2 on its resolvent set ρ(A), where the vector fH. We investigate the extremal values of Ef,A with respect to the change of f. We conduct an in-depth study of the case when A is the classical Volterra operator V on L2([0,1]). Our main theorem suggests a likely connection between the energy functional and the invariant subspace problem.

Article information

Source
Banach J. Math. Anal., Volume 13, Number 2 (2019), 255-274.

Dates
Received: 12 September 2018
Accepted: 17 September 2018
First available in Project Euclid: 26 January 2019

Permanent link to this document
https://projecteuclid.org/euclid.bjma/1548471622

Digital Object Identifier
doi:10.1215/17358787-2018-0029

Mathematical Reviews number (MathSciNet)
MR3927873

Zentralblatt MATH identifier
07045458

Subjects
Primary: 47A13: Several-variable operator theory (spectral, Fredholm, etc.)

Keywords
non-Euclidean metric Volterra operator energy functional quasinilpotent operator

Citation

Liang, Yu-Xia; Yang, Rongwei. Energy functional of the Volterra operator. Banach J. Math. Anal. 13 (2019), no. 2, 255--274. doi:10.1215/17358787-2018-0029. https://projecteuclid.org/euclid.bjma/1548471622


Export citation

References

  • [1] N. Aronszajn and K. T. Smith, Invariant subspaces of completely continuous operators, Ann. of Math. (2) 60 (1954), no. 2, 345–350.
  • [2] W. F. Donoghue, Jr., The lattice of invariant subspaces of a completely continuous quasi-nilpotent transformation, Pacific J. Math. 7 (1957), no. 2, 1031–1035.
  • [3] R. G. Douglas and R. Yang, “Hermitian geometry on resolvent set” in Operator Theory, Operator Algebras, and Matrix Theory, Oper. Theory Adv. Appl. 267, Birkhäuser/Springer, Cham, 2018, 167–184.
  • [4] B. Gustafsson and M. Putinar, “Linear analysis of quadrature domains, IV” in Quadrature Domains and Their Applications, Oper. Theory Adv. Appl. 156, Birkhäuser, Basel, 2005, 173–194.
  • [5] B. Gustafsson and M. Putinar, Hyponormal Quantization of Planar Domains: Exponential Transform in Dimension Two, Lecture Notes in Math. 2199, Springer, Cham, 2017.
  • [6] D. A. Herrero, The Volterra operator is a compact universal quasinilpotent, Integral Equations Operator Theory 1 (1978), no. 4, 580–588.
  • [7] H. J. Kim, Hyperinvariant subspace problem for quasinilpotent operators, Integral Equations Operator Theory 61 (2008), no. 1, 103–120.
  • [8] C. Lanczos, The Variational Principles of Mechanics, 4th ed., Math. Expo. 4, University of Toronto Press, Toronto, Ont., 1970.
  • [9] L. D. Landau and E. M. Lifshitz, Course of Theoretical Physics, I: Mechanics, 3rd ed., Pergamon, Oxford, 1976.
  • [10] Y. X. Liang and R. Yang, Quasinilpotent operators and non-Euclidean metrics, J. Math. Anal. Appl. 468 (2018), no. 2, 939–958.
  • [11] A. Montes-Rodríguez and S. A. Shkarin, “New results on a classical operator” in Recent Advances in Operator-Related Function Theory, Contemp. Math. 393, Amer. Math. Soc., Providence, 2006, 139–157.
  • [12] M. Putinar, Linear analysis of quadrature domains, Ark. Mat. 33 (1995), no. 2, 357–376.
  • [13] H. Radjavi and P. Rosenthal, Invariant Subspaces, Ergeb. Math. Grenzgeb. (3) 77, Springer, New York, 1973.
  • [14] C. J. Read, Quasinilpotent operators and the invariant subspace problem, J. Lond. Math. Soc. (2) 56 (1997), no. 3, 595–606.
  • [15] M. Tran and R. Yang, Hermitian geometry on resolvent set (III), in preparation.