Page 151 - Textos de Matemática Vol. 44

ON ALMOST NORMAL MATRICES 141
Proof. The Aluthge transform of A is unitarily similar to ∆(An) ⊕ ∆(Aa). Since ∆(An) = An is normal, ∆(A) is normal if and only if Aa is either absent (that is, A itself is normal) or ∆(Aa) is normal. According to Lemma 6.1, this cannot happen if the size of Aa is bigger than 2. On the other hand, for a 2-by-2 matrix Aa the normality of its Aluthge transform is given by [11, Corollary 1] and amounts exactly to the condition μ1μ2 ≤ 0 on its eigenvalues μ1, μ2.
6.2. Almost normal matrices in the range of ∆. The Aluthge transform is a non-linear mapping of Cn×n into itself. We now address the question of which pure almost normal matrices lie in its range. To formulate the result, an additional notion has to be introduced. We partition the set {λ1, . . . , λn−1} from the representation (1.1) into equivalence classes according to the equiva- lence relation
λi ≡λj ifandonlyif argλi =argλj modπ,
and call them clusters. In other words, each cluster is formed by all λi lying on the same line passing through the origin. Let J1,...,Jk be the respective partition of the index set {1,...,n−1}.
Theorem 6.4. A pure almost normal matrix A lies in the range of the Aluthge transform if and only if it is invertible,
βj2 (6.8) ci:= λ ̸=0
j∈Ji j
for each cluster consisting of more than one element, and all the eigenvalues
of A−1A∗ have absolute value one.
Proof. Suppose A = ∆(B) for some B. If A is not invertible then neither is B, since the spectrum is invariant under Aluthge transform. This would imply unitary reducibility of A, a contradiction. Thus, invertibility of A is necessary, and will be imposed in the rest of the proof.
According to [11, Corollary 3], an invertible matrix A lies in the range of ∆ if and only if A−1A∗ is similar to a unitary matrix, that is, it is diagonalizable with all the eigenvalues having absolute value one. So, it remains to show that conditions (6.8) are necessary and sufficient for A−1A∗ to be diagonalizable.
For technical reasons, it is more convenient to consider A∗A−1 in place of A−1A∗; the two matrices are similar, so this switch is allowed. Since A