In this study, some problems of operator theory on the reproducing kernel Hilbert space by using the Berezin symbols method are investigated. Namely, invariant subspaces of weighted composition operators on $H^2$ are studied. Moreover, some new inequalities for the Berezin number of operators are proved. In particular, new reverse inequalities for the Berezin numbers $\mathrm{b}\mathrm{e}\mathrm{r}\left({\left|A\right|}^2\right)$ and $\mathrm{b}\mathrm{e}\mathrm{r}\left(A\right)$ of operators ${\left|A\right|}^2=A^{\ast }A$ and A on the reproducing kernel Hilbert space are given. Also, reverse inequalities for the Berezin number of two operators are proved. Under some conditions we prove the power inequality $\mathrm{b}\mathrm{e}\mathrm{r}\left(A^n\right)\le {\left(\mathrm{b}\mathrm{e}\mathrm{r}\left(A\right)\right)}^n$ which is related to a well known analogue estimate of Halmos for the numerical radius.

In this paper, we investigate the fractional power $(-{\mathrm{\Delta }}_{\nu }{)}^{\gamma /2}$, $0<\gamma <2,$ of the Bessel operator in the form ${\mathrm{\Delta }}_{\nu }:=\frac{{\mathrm{d}}^2}{\mathrm{d}{x}^2}+\frac{2\nu +1}{x}\frac{\mathrm{d}}{\mathrm{d}x},\nu >-1/2.$ Our method uses a canonical representation for generalized Laplacian related to the structures of the Bessel–Kingman hypergroup. As a direct application, we solve the space-fractional diffusion equation.

Sentiment analysis plays an important role in understanding employee feedback and improving workplace culture. By leveraging NLP techniques to analyze this feedback accurately, organizations can pinpoint specific areas that need improvement, address employee concerns, and foster a positive work environment. These NLP-driven deep learning models offer valuable tools for E-Commerce HR and sales departments, enabling monitoring employee and users’ sentiment trends over time and assisting in implementing targeted interventions. Focusing on the e-commerce industry, this work utilizes NLP-driven deep learning methodologies to analyze employee and user feedback, aiming to identify sentiments. The proposed NLP-driven, deep learning-based framework is designed to classify user feedback into positive, negative, or neutral sentiments. The key steps in this framework include data collection, NLP-enhanced feature extraction using BERT-BiGRU, and final classification using a Graph Neural Network-based finite-state automata. The effectiveness of this NLP-centric approach was tested on diverse datasets of customer feedback from the e-commerce industry. The results demonstrate the framework’s efficacy, achieving an impressive 93.35% accuracy rate, surpassing existing benchmark methods. The research significantly benefits e-commerce by refining product portfolios and enhancing workplace culture.

We introduce the notion of Engliš algebras, defined in terms of reproducing kernels and Berezin symbols. Such algebras were apparently first investigated by Engliš (1995). Here we give some new results on Engliš $C^{\ast }$-algebras on abstract reproducin