Generalization of Gracia's Results

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Jun Liao
Heguo Liu
Yulei Wang
Zuohui Wu
Xingzhong Xu


Let α be a linear transformation of the m à n-dimensional vector space M_{mÃn}(C) over the complex field C such that α(X) = AX âXB, where A and B are mÃm and nÃn complex matrices, respectively. In this paper, the dimension formulas for the kernels of the linear transformations α^2 and α^3 are given, which generalizes the work of Gracia in [J.M. Gracia. Dimension of the solution spaces of the matrix equations [A, [A, X]] = 0 and [A[A, [A, X]]] = 0. Linear and Multilinear Algebra, 9:195â200, 1980.].

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