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We study the operators A(X) = Y" M, XN, and A*(X) = M,* XN,* which map the algebra of all bounded linear operators on a separable Hilbert space to itself, where ( M,) j" and ( N,) , are separately commuting sequences of normal operators. We prove that (1) when m < 2, the Hilbert-Schmidt norms of A(X) and A*(X) are equal (finite or infinite); (2) for m 2 3, if A(X) and A*(X) are Hilbert-Schmidt operators, then their Hilbert-Schmidt norms are equal; (3) if A, A* have the property that for each X, A( X) = 0 implies A*( X) = 0, then for each X, if A( X) is a Hilbert-Schmidt operator then A*2( X) is also and the latter has the same Hilbert-Schmidt norm as A2( X). Note that Fuglede's Theorem is immediate from (1) in the case mi = 2, M1 = N2 and N1 = I -M2. The proofs employ the duality between the trace class and the class of all bounded linear operators and, unlike the early proofs of Fuglede's Theorem, they are free of complex function theory. Let H denote a separable, finite-dimensional, complex Hilbert space. Let L(H) D K(H) D C D F(H), C2, C1,11 * Ilp, 11 11 denote, respectively, the class of all bounded linear operators acting on H, the compact operators, the Schatten p-class, the finite rank operators, the Hilbert-Schmidt class, the trace class, the C -norm, and the operator norm. DEFINITION. Let K Mn ) n 1 and K Nn ) m 1 be two separately commuting sequences of normal operators. Define A, A*: L(H) -L(H) by

An extension of the Fuglede commutativity theorem modulo the Hilbert-Schmidt class to operators of the form ∑𝑀_{𝑛}𝑋𝑁_{𝑛}