Products of generalized n -projections
Bhagwati Prashad Duggal, I. H. Kim · Linear and Multilinear Algebra · 2021
A Hilbert space operator A∈B(H) is a generalized n-projection, A∈(G−n−P), if A∗n=A. The product AB of a commuting pair A, B of (G−n−P)-operators is a (G−n−P)-operator. The converse fails. We prove that if ‖AB‖=‖A‖‖B‖ and σa(AB)=σa(A)σa(B), then AB∈(G−n−P) implies A‖A‖,B‖B‖ are (G−n−P) if and only if A and B are normal operators. Translated to tensor products (and upon identifying the tensor product A⊗B with the left-right multiplication operator EA,B∗ acting on the Hilbert–Schmidt bimodule C2(H)) this says that A⊗B (resp., EA,B∗) is a (G−n−P)-operator implies A‖A‖,B‖B‖ are (G−n−P) if and only if A and B are normal operators.