The purpose of this paper is to find new inclusion relations of the harmonic class $\mathcal{HF}(\varrho ,\gamma )$ with the subclasses ${\mathcal{S}}_{\mathcal{HF}}^{*},$${\mathcal{K}}_{\mathcal{HF}}$ and $\mathcal{T}{\mathfrak{N}}_{\mathcal{HF}}\left(\tau \right)$ of harmonic functions by applying the convolution operator $\Theta (\Im )$ associated with the Mittag-Leffler function. Further for $\varrho =0$, several special cases of the main results are also obtained.

Our aim in this study is to consider a generalization of the concept of m-complex symmetric transformations to n-quasi-m-complex symmetric transformations. A map $S\in \mathcal{B}\left(\mathcal{Y}\right)$ is said to be an n-quasi-m-complex symmetric transformation if there exists a conjugation C on $\mathcal{Y}$ such that S satisfies the condition $S^{*n}\left({\displaystyle \sum _{0\le k\le m}}{(-1)}^{m-k}\left(\genfrac{}{}{0pt}{}{m}{k}\right)S^{*k}C{S}^{m-k}C\right)S^n=0$, for some positive integers n and m. This class of transformation contains the class of m-complex symmetric transformations as a proper subset. Some basic structural properties of n-quasi-m-complex symmetric linear transformations are established with the help of transformation matrix representation. In particular, we obtain that a power of an n-quasi-m-complex symmetric is again an n-quasi-m-complex symmetric operator. Moreover, if T and S are such that T is an $n_1$-quasi-$m_1$-complex symmetric and S is an $n_2$-quasi-$m_2$-complex symmetric, their product $TS$ is an $max\{n_1,n_2\}$-quasi-$(m_1+m_2-1)$-complex symmetric under suitable conditions. We examine the stability of n-quasi-m-complex symmetric operators under perturbation by nilpotent operators.

This paper is concerned with studying a new class of multivariable operators know as joint \(n\)-normal \(q\)-tuple of operators. Some structural properties of some members of this class are given.