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In a recent note, we proved a Fuglede-Putnam commutativity theorem for almost normal operators with nite modulus of C2-quasitriangularity modulo the HilbertSchmidt class. In this note we show how our proof can be adjusted to the case of normal operators to obtain an optimal norm estimate obtained by G. Weiss. The result is also reviewed in the case of almost normal operators with zero modulus of C2-quasitriangularity for the operator and its adjoint and an example is provided.

Remarks on Fuglede-Putnam theorem for normal operators modulo the Hilbert-Schmidt class