Methods of Functional Analysis
and Topology

Let \( \mathcal{H} \) be a complex Hilbert space of dimension at least 3, and let \( \mathcal{B}(\mathcal{H}) \) denote the algebra of all bounded linear operators on \( \mathcal{H} \). Based on results by Molnar, this paper revisits the problem addressed in [18], which characterizes surjective maps \( \phi: \mathcal{B}(\mathcal{H}) \to \mathcal{B}(\mathcal{H}) \) that preserve the set of partial isometric operators in both directions. We focus exclusively on the linear case, rather than the more general additive case. Furthermore, we provide an alternative proof of the main result in [9] from a different point of view. Finally, we propose new directions for exploring maps that preserve higher-order partial isometric operators in both directions.

This paper introduces and investigates the class of $k$-quasi $n$-power posinormal operators in Hilbert spaces, generalizing both posinormal and $n$-power posinormal operators. We establish fundamental properties including matrix representations in $2 \times 2$ block form, tensor product preservation ($T\otimes S$ remains in the class when $T,S$ are), and complete characterizations for weighted conditional type operators $T_{w,u} := wE(uf)$ on $L^2(\Sigma)$. Key theoretical contributions include a structural decomposition theorem for operators with non-dense range, spectral properties, invariant subspace behavior, and interactions with isometric operators. For weighted operators, we derive explicit conditions for $k$-quasi $n$-power posinormality in terms of weight functions $w,u$ and their conditional expectations. The work bridges abstract operator theory with concrete applications, particularly in conditional expectation analysis, while significantly extending posinormal operator theory. The results provide new tools for operator analysis with potential applications in spectral theory, functional calculus and mathematical physics. Concrete examples throughout the paper illustrate the theory and the framework opens new research directions in operator theory and its applications, offering both theoretical insights and practical computational tools for analyzing this important class of operators in Hilbert spaces.

In this paper, we extends the classical theory of variational inequalities to the hyperbolic scalar setting using the structure of $\mathbb{D}$-Hilbert spaces. We introduce and analyze a new class of variational inequalities, termed general mildly $\mathbb{D}$-nonlinear variational inequalities, which generalize classical formulations by incorporating $\mathbb{D}$-nonlinear and product-type mappings. We characterize these problems in terms of their idempotent components and demonstrate that several known variational inequality problems, including Stampacchia-type and complementarity problems, emerge as special cases.

In this paper, we study uniform ergodicity for $C_0$-semigroups of universally bounded operators acting on locally convex spaces. Characterizations of uniform ergodic $C_0$-semigroups are given. Importantly, we give a $C_0$-semigroups version of F. Pater, T. Binzar [14] theorem.