Methods of Functional Analysis
and Topology

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.

In this paper, we introduce generalized multivalued $F-$proximal contraction mappings within the partial metric spaces framework and establish best proximity point results for such mappings. The best proximity point theorem for multivalued $F-$proximal contraction mappings involving $\alpha-$admissibility is also obtained. Several related results in the literature are unified and generalized by our new best proximity point results. We also provide nontrivial examples to support our findings. Finally, we derive an existence of a solution to an integral equation that validates our finding.

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.

$(\tau_1,\tau_2)$-star-Lindelöfness ensures that for every pair $(\mathcal U_1 \subseteq \tau_1,\mathcal U_2 \subseteq \tau_2)$ of open covers, a countable subcover of $\mathcal U_1$, can spread through $\mathcal U_2 $ via the star operation to cover the entire bitopological space $(X,\tau_1,\tau_2)$. Giving a positive answers to the questions of Choudhury et. al. [12], DCCC and meta-Lindelöf like characterization of star-Lindelöf bitopological spaces are presented in this paper. It has been established that a JDCCC bitopological space is both $(\tau_1,\tau_2)$-2-star-Lindelöf and $(\tau_2,\tau_1)$-2-star-Lindelöf. And if a bitopological space which is both $(\tau_1,\tau_2)$-$n$-star-Lindelöf ($n \in \mathbb N$) and $(\tau_1,\tau_2)$-meta-Lindelöf, then $(X, \tau_2)$ is $(n-1)$-star Lindelöf ($0$-star-Lindelöfness represents Lindelöfness).