This paper addresses the spectral analysis of operators acting on reproducing kernel Hilbert spaces equipped with a semi-inner product induced by a positive operator $\mathbb{A}$. A fundamental challenge in this setting is the geometric discrepancy between the normalized reproducing kernels and the unit $\mathbb{A}$-sphere, which renders classical numerical radius techniques inapplicable. By overcoming this structural obstacle, we establish sharp inequalities for the $\mathbb{A}$-Berezin number and $\mathbb{A}$-Berezin norm. Our main contribution involves the derivation of multiparametric estimates for triple operator products of the form ${\mathrm{P}}^{\alpha }X{\mathrm{R}}^{\alpha }$ involving Schatten-type exponents. These results generalize and refine existing bounds in the literature. Furthermore, we provide a qualitative analysis of the obtained bounds through weighted Toeplitz operators on Hardy spaces and verify the theoretical findings with concrete matrix examples involving the geometric behavior of weight functions.

We developed an Attouch-Wets convergence theory for nonempty closed subsets of $\star$-metric spaces, associated with an admissible scalarization and a bounded testing family. Since the nonlinear triangle structure of a $\star$-metric space does not permit a direct transfer of the classical bounded-Hausdorff framework, we first introduced admissible scalarizations that convert bounded-region distance-profile comparisons into a workable additive setting. On this basis, we defined profile and truncated excess functionals associated with a scalarization, established equivalent formulations of the resulting convergence, and showed that it is pseudometrizable whenever the testing family has a countable cofinal subfamily. We then compared this convergence with the corresponding bornological convergences and with Wijsman convergence, and showed that the latter is strictly weaker in general, while equivalence holds under properness of the linearized metric. We also proved restriction and product results, together with relative compactness and sequential completeness theorems for the induced hyperspace structure. The examples showed that the theory is nontrivial in genuinely nonlinear $\star$-metric settings and that the main assumptions used in the comparison and compactness results are essential.

In this work, we investigate a new type of convergence known as relative statistical convergence through the use of the deferred Nörlund and deferred Riesz means. We demonstrate that the idea of deferred Nörlund and deferred Riesz statistically relative uniform convergence is significantly stronger than deferred Nörlund and deferred Riesz statistically uniform convergence. We provide some interesting examples which explain the validity of the theoretical results and effectiveness of constructed sequence spaces. Furthermore, as an application point of view we prove the Korovkin-type approximation theorem in the context of relative equi-statistical convergence for real valued functions and demonstrate that our theorem effectively extends and most of the earlier existing results. Finally, we present an example involving the Meyer-König-Zeller operator of real sequences proving that our theorem is a stronger approach than its classical and statistical version.

This paper introduces $\varsigma$-neutrosophic fuzzy metric spaces ( $\varsigma$-NFMSs), a significant generalization of neutrosophic fuzzy metric spaces (NFMSs). By extending the parameter space from a single dimension $(0,\mathrm{\infty })$ to a multi-dimensional vector space $(0,\mathrm{\infty }{)}^{\varsigma }$, this framework offers enhanced flexibility for modeling complex systems where uncertainty depends on multiple factors simultaneously. The study investigates the topological properties of $\varsigma$-NFMSs, rigorously proving that their topology is first-countable and that the associated space is Hausdorff. Furthermore, a generalized fixed-point theorem is established within this new framework, extending previous results in NFMSs.

We investigated geometric limit sets of double sequences in finite dimensional $2$-normed spaces when negligibility of index sets was measured by a density induced by a nonnegative Robison–Hamilton (RH)-regular family $\mathfrak{B}$ of four-dimensional matrices. First, using the $\mathfrak{B}$-density, we defined $\mathfrak{B}$-statistical limit superior and limit inferior for the scalar reductions generated by the seminorms $x\mapsto \Vert x,u\Vert$, and we studied the associated real cluster behavior. Next, we introduced $\mathfrak{B}$-statistical cluster points and the $\mathfrak{B}$-statistical core of a double sequence as the intersection of all closed convex sets that contained the sequence outside a $\mathfrak{B}$-density zero set. We obtained a disk-type representation of the core via intersections of sets of the form $\{x\in X:\Vert x-z,u\Vert \le r\}$, where the radii were governed by $\mathfrak{B}$-statistical $lim sup$. We also proved a Knopp-type inclusion theorem: for a family of transforms satisfying a natural $\mathfrak{B}$-regularity condition, the Knopp core of each transform was contained in the $\mathfrak{B}$-statistical core of the original sequence. Finally, replacing $\mathfrak{B}$-density zero sets by a strongly admissible ideal ${\mathcal{I}}_2$ on $\mathbb{N}\times \mathbb{N}$, we defined ideal cores, established disk representations, compared the density-based and ideal cores, and identified an explicit condition under which the ${\mathcal{I}}_2$-core and the ${\mathcal{I}}_2^{\ast }$-core coincided.

This paper introduced a novel class of bivariate $\lambda$-Bernstein operators defined on triangular domain, denoted as $B_m^{{\lambda }_1,{\lambda }_2}(f;x,y)$. These operators leverage a new class of bivariate Bézier basis functions defined on triangular domain with shape parameters ${\lambda }_1$ and ${\lambda }_2$. A Korovkin-type approximation theorem for $B_m^{{\lambda }_1,{\lambda }_2}(f;x,y)$ was established, with the convergence rate being characterized by both the complete and partial moduli of continuity. Additionally, a local approximation theorem and a Voronovskaja-type asymptotic formula were derived for $B_m^{{\lambda }_1,{\lambda }_2}(f;x,y)$. Finally, the convergence of $B_m^{{\lambda }_1,{\lambda }_2}(f;x,y)$ to $f(x,y)$ was illustrated through graphical representations and numerical examples, highlighting instances where they surpass the performance of standard bivariate Bernstein operators defined on triangular domain, $B_m(f;x,y)$.