In this paper, we introduce the concept of normality of a d-tuple of bounded linear operators acting on a complex Hilbert space $\mathcal{H}$ when an additional semi-inner product induced by a positive operator A is considered. We investigate this new class of operators which is called A-normal d-tuple of operators. Mainly, we study the tensor product and the tensor sum of two A-normal d-tuples of operators. Also, we define the A-joint spectral radius associated with a d-tuple $\mathbf{T}=(T_1,\dots ,T_d)$ denoted $r_A\left(\mathbf{T}\right)$ and we show that under some conditions we have $r_A\left(\mathbf{T}\right)=|||\mathbf{T}||{|}_A$, where $|||\mathbf{T}||{|}_A$ denotes the A-joint operator seminorm of $\mathbf{T}$.

The purpose of the present paper is to pursue further study of a class of linear bounded operators, known as $n$-quasi-$m$-isometric operators acting on an infinite complex separable Hilbert space ${\mathcal H}$. We give an equivalent condition for any $T$ to be $n$-quasi-$m$-isometric operator. Using this result we prove that any power of an $n$-quasi-$m$-isometric operator is also an $n$-quasi-$m$-isometric operator. In general the converse is not true. However, we prove that if $T^r$ and $T^{r+1}$ are $n$-quasi-$m$-isometries for a positive integer $r$, then T is an $n$-quasi-$m$-isometric operator. We study the sum of an $n$-quasi-$m$-isometric operator with a nilpotent operator. We also study the product and tensor product of two $n$-quasi-$m$-isometries. Further, we define $n$-quasi strict $m$-isometric operators and prove their basic properties.