Semigroups of EP linear transformations
type: Item
instance of: scholarly article
title: Semigroups of EP linear transformations
zbMATH Open document ID: 0946.15009
DOI: 10.1016/S0024-3795(99)00192-5
author: Gorazd Lešnjak
published in: Linear Algebra and its Applications
publication date: 2000-10-12T00:00:00Z
review text: 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).
reviewed by: John D. Dixon
Mathematics Subject Classification ID: 15A30
Mathematics Subject Classification ID: 15A04
zbMATH DE Number: 1420034
zbMATH Keywords: algebra of linear transformations
zbMATH Keywords: finite dimensional inner product space
zbMATH Keywords: EP linear transformation
zbMATH Keywords: semigroups
zbMATH Keywords: Hilbert spaces
MaRDI profile type: Publication
cites work: Q4061081
cites work: Theorems on products of \(EP_{r}\) matrices
cites work: Q3207128
cites work: On products of EP matrices
cites work: Wiegmann type theorems for EPr matrices
cites work: A simple proof of the product theorem for EP matrices
cites work: On normal and EPr-matrices
cites work: Q5800366
Wikidata QID: Q127675440
Recommended article: EP properties of (b, c)-invertible matrices
Recommended article: On products of EP matrices
Recommended article: Q4020479
Recommended article: Characterizations of EP, normal, and Hermitian matrices
Recommended article: Q3353266
Recommended article: On \(k\)-\(EP\) matrices
Recommended article: Separation theorems
Recommended article: Q4344250
Recommended article: REPRESENTATION OF BOUNDED LINEAR OPERATORS WITH EQUAL SPECTRAL PROJECTIONS AT ZERO
Recommended article: Q3731774
description: scientific article; zbMATH DE number 1420034