Application of symmetric analytic functions to spectra of linear operators

I. Burtnyak; I. Chernega; V. Hladkyi; O. Labachuk; Z. Novosad

doi:10.15330/cmp.13.3.701-710

I. Burtnyak Vasyl Stefanyk Precarpathian National University, 57 Shevchenka str., 76018, Ivano-Frankivsk, Ukraine https://orcid.org/0000-0002-9440-1467
I. Chernega Institute for Applied Problems of Mechanics and Mathematics, 3b Naukova str., 79060, Lviv, Ukraine
V. Hladkyi Vasyl Stefanyk Precarpathian National University, 57 Shevchenka str., 76018, Ivano-Frankivsk, Ukraine
O. Labachuk Vasyl Stefanyk Precarpathian National University, 57 Shevchenka str., 76018, Ivano-Frankivsk, Ukraine
Z. Novosad Lviv University of Trade and Economics, 10 Tuhan-Baranovskyi str., 79005, Lviv, Ukraine

https://doi.org/10.15330/cmp.13.3.701-710

Keywords: symmetric analytic function on a Banach space, $p$-nuclear operator, Fredholm determinant

Published online: 2021-12-11

Abstract

The paper is devoted to extension of the theory of symmetric analytic functions on Banach sequence spaces to the spaces of nuclear and $p$-nuclear operators on the Hilbert space. We introduced algebras of symmetric polynomials and analytic functions on spaces of $p$-nuclear operators, described algebraic bases of such algebras and found some connection with the Fredholm determinant of a nuclear operator. In addition, we considered cases of compact and bounded normal operators on the Hilbert space and discussed structures of symmetric polynomials on corresponding spaces.