Generalized normal operators and the $\mathcal{P}$-absolute value of operators

In this paper, a generalization of normal operators in Hilbert spaces is introduced. Specifically, a bounded linear operator T on a complex Hilbert space \(\mathsf{H}\) as \([\mathcal{P}]\) -normal if \(T^{*}\mathcal{P}T = T\mathcal{P}T^{*}\) is defined, where \(\mathcal{P}\) is a bounded positive linear operator on \(\mathsf{H}\) , and \(T^{*}\) denotes the adjoint operator of T . Several properties and characterizations involving this new class of operators are examined. Additionally, the \(\mathcal{P}\) -absolute value in semi-Hilbert spaces is introduced, and several applications are presented, including a generalization of the polar decomposition and Bohr’s inequality for Hilbert space operators. Other related results are also discussed.