F. Ayatollah Zadeh Shirazi

arXiv2025

Topological Stability and Shadowing in Uniform Transformation Semigroups Modulo an Ideal

In this paper, we introduce and analyze several key dynamical properties-namely shadowing modulo an ideal, expansivity modulo an ideal, and topological stability modulo an ideal-within the framework of uniform transformation semigroups. Given an ideal $\mathcal I$ on semigroup $T$, we investigate the interplay between these properties in compact Hausdorff transformation semigroup $(T,X,\mathfrak{X})$. Our main result establishes that if a compact Hausdorff transformation semigroup exhibits the shadowing property modulo $\mathcal I$ and is expansive modulo $\mathcal I$ then it is also topologically stable modulo $\mathcal I$. This extends known stability theorems in classical topological dynamics to the setting of ideal-constrained dynamics. Additionally, we explore the relationship between shadowing modulo $\mathcal I$ and the conventional shadowing property.

math.DS↗

arXiv2024

On adjoint invariant classes of shift operators

In the following text we compute the adjoint of weighted generalized shift operators over Hilbert spaces. We show for a conjugate invariant subset $A$ of $\mathbb C$, the additive semigroup generated by $A\cup\{0\}-$weighted generalized shifts over Hilbert space $\mathcal H$ is adjoint invariant if and only if $\mathcal H$ is a finite dimensional Hilbert space or $0$ is not a limit point of $A$.

math.FA↗