Polynomially Quasi-mth-Helton class of Hilbert spaces operators
- 1Department of Mathematics, College of Science, Qassim University, Buraydah, Saudi Arabia
Abstract
In this paper, we introduce and investigate a new class of bounded linear operators on a Hilbert space, called the polynomially quasi-mth Helton class of operators of order m∈ℕ. This class is defined through a vanishing operator relation involving a polynomial perturbation of the Helton operator map, namely. 𝒫(𝒜)*Φ(ℬ;𝒜)m(ℐ)𝒫(𝒜)=0, for some nonconstant polynomial 𝒫, where Φ(ℬ;𝒜) denotes the Helton operator map given by Φ(ℬ;𝒜)(X)=ℬX−X𝒜, so that Φ(ℬ;𝒜)m(ℐ)=∑j=0m(−1)j(mj)ℬj𝒜m−j. This framework extends the classical Helton class and the quasi-Helton class of order m by incorporating polynomial operator relations. We establish several fundamental properties of this class and investigate its structural behavior. In particular, we obtain characterizations, study stability properties, and discuss connections with well-known operator classes such as m-isometric and m-symmetric operators. Our results provide a unified framework that highlights the interplay between polynomial operator expressions and quasi-Helton-type relations in operator theory.
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BibTeX
@article{rwaily2026polynomially,
title = {{Polynomially Quasi-mth-Helton class of Hilbert spaces operators}},
author = {Asma Al Rwaily},
journal = {PLOS One},
year = {2026},
month = sep,
doi = {10.1371/journal.pone.0355301},
url = {https://doi.org/10.1371/journal.pone.0355301},
note = {CC BY 4.0},
}
RIS
TY - JOUR
TI - Polynomially Quasi-mth-Helton class of Hilbert spaces operators
AU - Asma Al Rwaily
PY - 2026
DA - 2026/09/02
JO - PLOS One
DO - 10.1371/journal.pone.0355301
UR - https://doi.org/10.1371/journal.pone.0355301
AB - In this paper, we introduce and investigate a new class of bounded linear operators on a Hilbert space, called the polynomially quasi-mth Helton class of operators of order m∈ℕ. This class is defined through a vanishing operator relation involving a polynomial perturbation of the Helton operator map, namely. 𝒫(𝒜)*Φ(ℬ;𝒜)m(ℐ)𝒫(𝒜)=0, for some nonconstant polynomial 𝒫, where Φ(ℬ;𝒜) denotes the Helton operator map given by Φ(ℬ;𝒜)(X)=ℬX−X𝒜, so that Φ(ℬ;𝒜)m(ℐ)=∑j=0m(−1)j(mj)ℬj𝒜m−j. This framework extends the classical Helton class and the quasi-Helton class of order m by incorporating polynomial operator relations. We establish several fundamental properties of this class and investigate its structural behavior. In particular, we obtain characterizations, study stability properties, and discuss connections with well-known operator classes such as m-isometric and m-symmetric operators. Our results provide a unified framework that highlights the interplay between polynomial operator expressions and quasi-Helton-type relations in operator theory.
SN - 1932-6203
ER -