A bounded linear operator T: H1 → H2, where H1, H2 are Hilbert spaces, is said to be norm attaining if there exists a unit vector x ∈ H1 such that kT xk = kT k and absolutely norm attaining (or AN -operator) if T |M: M → H2 is norm attaining for every closed subspace M of H1. We prove a structure theorem for positive operators in β(H):= {T ∈ B(H): T |M: M → M is norm attaining for all M ∈ RT }, where RT is the set of all reducing subspaces of T . We also compare our results with those of absolutely norm attaining operators. Later, we characterize all operators in this new class. © 2021 University of Szeged. All rights reserved.