A study of operator classes on Hilbert spaces

This paper introduces and studies a new class of operators on Hilbert spaces, called polynomially quasi- $m$-complex Helton operators, which extend classical Helton-type operator classes through polynomial perturbations and higher-order structural relations. Motivated by recent developments in operator theory, an operator $\mathcal{A}\in \mathcal{L}(\mathcal{K})$ is said to belong to this class if there exists a nonconstant polynomial $p$ such that $p(\mathcal{A}{)}^{\ast }\mathrm{\Theta }(\mathcal{B},\mathcal{A},C{)}^m(I)p(\mathcal{A})=0,$ where $\mathrm{\Theta }(\mathcal{B},\mathcal{A},C)$ represents the corresponding $m$-complex Helton-type operator expression. We establish fundamental properties of this class and investigate its structural behavior, including stability under direct sums, invariance under closed invariant subspaces, and stability under powers of operators. In particular, we show that, under suitable conditions, the order of the class may increase from $m$ to higher levels of the hierarchy. Several equivalent characterizations are also obtained in terms of kernels of operator polynomials, providing deeper insight into the underlying algebraic structure.