Semigroups of EP linear transformations
Let \(\mathcal{L}(X)\) denote the algebra of linear transformations of a finite dimensional inner product space \(X\) over \(\mathbb{R}\) or \(\mathbb{C}.\) Then \(A\in\mathcal{L}(X)\) is defined to be an EP linear transformation if the ranges \(R(A)\) and \(R(A^{\ast})\) of \(A\) and its adjoint are equal; equivalently, \(X\) is the orthogonal sum of \(R(A)\) and the null space \(N(A)\). (EP matrices were originally defined through rather different conditions by \textit{H. Schwerdtfeger} in ``Introduction to linear algebra and the theory of matrices'', P. Noordhoff, Groningen, 1950; Zbl 0040.29503). It was shown by \textit{R. E. Hartwig} and \textit{I. J. Katz} [Linear Algebra Appl. 252, 339-345 (1997; Zbl 0868.15015)] that the product \(AB\) of two EP linear transformations \(A\) and \(B\) is EP if and only if \(R(A)\) is a \(B^{\ast}\)-invariant subspace and \(R(B)\) is \(A\)-invariant. In the present paper the author characterizes semigroups of EP linear transformations as follows (Theorem 2.2). Let \(\mathcal{A}\) be a semigroup in \(\mathcal{L}(X).\) Then \(\mathcal{A}\) consists of EP linear transformations if and only if \(X=\bigoplus X_{i}\) is a direct sum of nonzero orthogonal subspaces \(X_{i}\) which are invariant under all elements of \(\mathcal{A}\) such that for each \(A\in\mathcal{A}\) the restriction \(A_{|X_{i}}\) is either \(0\) or invertible. The author also proves analogous results for infinite-dimensional Hilbert spaces (it is necessary to replace the ranges \(R(A)\) and \(R(A^{\ast})\) by their closures).
scientific article; zbMATH DE number 1420034
scientific article