On (α, β)-Class (Qn) Operators

In this paper, we introduce a new class of operators, called the class of (α, β)-Class (Qn) operators acting on a complex Hilbert space H. For an integer n ≥ 1, an operator T ∈ B(H) is said to belong to the (α, β)-Class (Qn) if αnT∗nTn ≤ (TT)n ≤ βnTnT∗n, for scalars 0 ≤ α ≤ 1 ≤ β. This definition extends the earlier notion of (α, β)-Class (Q) operators, which corresponds to the case n = 2. We investigate several fundamental properties of this generalized class through rigorous operator-theoretic analysis, employing techniques involving polar decompositions, unitary conjugations, and commutativity conditions. A key finding establishes that every (α, β)-normal operator automatically belongs to the (α, β)-Class (Qn) for all integers n ≥ 1, with the same coefficients α2 and β2 appearing uniformly across all values of n. Additional results demonstrate closure properties under scalar multiplication, unitary equivalence, and specific composition operations, while also characterizing the behavior of adjoints under polar decomposition conditions. These findings provide a comprehensive foundation for understanding the structural properties and operator-theoretic consequences that arise from this natural extension of the classical framework.