We introduce the notions of lacunary -convergence and lacunary -Cauchy in the topology induced by random -normed spaces and prove some important results.

In this paper, we introduce the notion of $\mathcal{I}$ -statistical convergence and $\mathcal{I}$ -lacunary statistical convergence with respect to the intuitionistic fuzzy norm $\left( \mu,v\right) $ , investigate their relationship, and make some

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.