The investigation of new operators belonging to some specific classes has been quite fashionable since the beginning of the century, and sometimes it is indeed relevant. In this study, we introduce and study a new class of operators called k‐quasi‐(m, n)‐isosymmetric operators on Hilbert spaces. This new class of operators emerges as a generalization of the (m, n)‐isosymmetric operators. We give a characterization for any operator to be k‐quasi‐(m, n)‐isosymmetric operator. Using this characterization, we prove that any power of an k‐quasi‐(m, n)‐isosymmetric operator is also an k‐quasi‐(m, n)‐isosymmetric operator. Furthermore, we study the perturbation of an k‐quasi‐(m, n)‐isosymmetric operator with a nilpotent operator. The product and tensor products of two k‐quasi‐(m, n)‐isosymmetries are investigated.

In this article, we introduce and study a new class of operators, larger than (α, β) − A‐normal operators and different than (α, β) − A‐normal operators, named m‐quasi‐(α, β) − A‐normal operators. Considering the semi‐inner product induced by a positive operator A, the m‐quasi‐(α, β) − A‐normal operators turn into a generalization (for this new structure) of classical m‐quasi‐(α, β)‐normal operators. Several results concerning properties of this kind of operators are presented in the paper. Several inequalities for the A‐numerical radius and A‐operator norm for members of this class are established.

We study the existence and large time behavior of positive solutions for the semilinear parabolic equation $\frac{\partial u}{\partial t}(x,t) =\Delta u(x,t)+ V(x)u(x,t) +f(x,u(x,t))$ with initial Dirichlet boundary conditions on $(D\setminus E)\times (0,\infty)$, where $D$ is a bounded Lipschitz domain in $\mathbb{R}^n$, $ n\geq 3$, $E$ is a prescribed compact set of $D$, and $V$ and $f$ are real-valued functions satisfying some general conditions. Our results cover various types of nonlinearities and extend known results proved for the power nonlinearity $f(x,u)=W(x)u^p$ and a singular one-point set $E$.