Our goal in this article is to use ideas from symmetric q-calculus operator theory in the study of meromorphic functions on the punctured unit disc and to propose a novel symmetric q-difference operator for these functions. A few additional classes of meromorphic functions are then defined in light of this new symmetric q-difference operator. We prove many useful conclusions regarding these newly constructed classes of meromorphic functions, such as convolution, subordination features, integral representations, and necessary conditions. The technique presented in this article may be used to produce a wide variety of new types of generalized symmetric q-difference operators, which can subsequently be used to investigate a wide variety of new classes of analytic and meromorphic functions related to symmetric quantum calculus.

In this study, we introduce novel concepts within the framework of fuzzy bipolar b-metric spaces, focusing on various mappings such as ${\psi }_{\alpha }$-contractive and ${\mathcal{ϝ}}_{\eta }$-contractive mappings, which are essential for quantifying distances between dissimilar elements. We establish fixed-point theorems for these mappings, demonstrating the existence of invariant points under certain conditions. To enhance the credibility and applicability of our findings, we provide illustrative examples that support these theorems and expand the existing knowledge in this field. Furthermore, we explore practical applications of our research, particularly in solving integral equations and fractional differential equations, showcasing the robustness and utility of our theoretical advancements. Symmetry, both in its traditional sense and within the fuzzy context, is fundamental to our study of fuzzy bipolar b-metric spaces. The introduced contractive mappings and fixed-point theorems expand the theoretical framework and offer robust tools for addressing practical problems where symmetry is significant.

This article explores the analysis of various spectral properties pertaining to totally paranormal closed operators, extending beyond the confines of boundedness and encompassing operators defined in a Hilbert space. Within this class, closed symmetric operators are included. Initially, we establish that the spectrum of such an operator is non-empty and provide a char-acterization of closed-range operators in terms of the spectrum. Building on these findings, we proceed to prove Weyl’s theorem, demonstrating that for a densely defined closed totally paranormal operator T, the difference between the spectrum σ(T) and the Weyl spectrum σw(T) equals the set of all isolated eigenvalues with finite multiplicities, denoted by π00(T). In the final section,we establish the self-adjointness of the Riesz projection Eμ corresponding to any non-zero isolated spectral value μ of T. Furthermore, we show that this Riesz projection satisfies the relationships ran(Eμ) = ker(T − μI) = ker(T − μI)∗. Additionally, we demonstrate that if T is a closed totally paranormal operator with a Weyl spectrum σw(T) = 0, then T qualifies as a compact normal operator.

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