Polynomially Quasi-mth-Helton class of Hilbert spaces operators

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.