On the structure and spectral properties of an extension of normal operators

Let $ \mathbb{H} $ be a complex Hilbert space, and let $ \mathcal{P} $ be a non-zero positive semi-definite operator on $ \mathbb{H} $. Denote by $ \mathscr{L}_{\mathcal{P}}(\mathbb{H}) $ the class of bounded linear operators on $ \mathbb{H} $ admitting a $ \mathcal{P} $-adjoint, and by $ \mathrm{Q}^{\sharp_{\mathcal{P}}} $ the distinguished $ \mathcal{P} $-adjoint of $ \mathrm{Q} \in \mathscr{L}_{\mathcal{P}}(\mathbb{H}) $. In this paper, we study the algebraic and spectral properties of $ \langle\mathcal{P}\rangle $-normal operators. By definition, these operators satisfy the relation $ \mathcal{P}\mathrm{Q}\mathrm{Q}^{\sharp_{\mathcal{P}}} = \mathcal{P}\mathrm{Q}^{\sharp_{\mathcal{P}}}\mathrm{Q} $. We establish equivalent metric and algebraic characterizations for this class. We prove that every $ \langle\mathcal{P}\rangle $-normal operator is $ \mathcal{P} $-normaloid. To overcome the restrictive closed range assumption of $ \mathcal{P} $ in classical spectral theory, we introduce a topological formulation of $ \mathcal{P} $-invertibility and the $ \mathcal{P} $-spectrum. Using this framework, without assuming that the range of $ \mathcal{P} $ is closed, we generalize several $ \mathcal{P} $-numerical radius inequalities and spectral inclusion theorems. Finally, we establish the null space symmetry and prove that the $ \mathcal{P} $-eigenspaces of these operators are mutually $ \mathcal{P} $-orthogonal.