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, a new class of $H$-monotone in Banach spaces is considered and studied. The resolvent operator and Cayley approximation operator associated with the $H$-monotone are defined, and the Lipschitz continuity of Cayley approximation operator is also established. An application involves the solvability of a class of generalized Cayley inclusions with $H$-monotone in Banach spaces. By utilizing the technique of resolvent, an iterative algorithm is developed for solving such a class of generalized Cayley inclusions in Banach spaces. The convergence of the iterative sequence generated by the algorithm is proven under certain suitable conditions. The results are justified by means of a numerical example analytically and graphically using Python(matplotlib).

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.