Products of generalized n-projections

Abstract

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 σ a ( AB ) = σ a ( A ) σ a ( 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 E A , B ∗ acting on the Hilbert–Schmidt bimodule C 2 ( H ) ) this says that A ⊗ B (resp., E A , 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.