Banach Journal of Mathematical Analysis

Energy functional of the Volterra operator

Yu-Xia Liang and Rongwei Yang

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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

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

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

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Zentralblatt MATH identifier

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

non-Euclidean metric Volterra operator energy functional quasinilpotent operator


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.

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