## Banach Journal of Mathematical Analysis

### Energy functional of the Volterra operator

#### Abstract

We define the energy functional $E_{f,A}$ for a bounded linear operator $A$ acting on a Hilbert space ${\mathcal{H}}$ through a newly defined non-Euclidean metric $g_{f}(z)|dz|^{2}$ on its resolvent set $\rho (A)$, where the vector $f\in \mathcal{H}$. We investigate the extremal values of $E_{f,A}$ 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 $L^{2}([0,1])$. 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
Accepted: 17 September 2018
First available in Project Euclid: 26 January 2019

https://projecteuclid.org/euclid.bjma/1548471622

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

Mathematical Reviews number (MathSciNet)
MR3927873

Zentralblatt MATH identifier
07045458

#### 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

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