On ⟨A⟩-Normal Operators via Continuous Spatial Deformation

Recently, a new class of operators called A-normal operators was introduced. These operators satisfy the identity ATT♯A=AT♯AT, where A is a positive semi-definite operator on a complex Hilbert space K, T is a bounded linear operator that admits an A-adjoint, and T♯A denotes the distinguished A-adjoint of T. This notion extends the standard concept of A-normal operators in semi-Hilbertian spaces, which are defined by the strict algebraic commutation TT♯A=T♯AT. In this paper, we investigate this generalized class. Using continuous spatial deformation techniques extended to the entire Hilbert space, we provide an alternative global realization connecting A-normal operators to classically normal operators. This framework allows us to prove a generalization of the Fuglede–Putnam theorem. Furthermore, we completely characterize their behavior under tensor product and tensor sum operations, expanding upon recently established spectral properties.