Logarithms and exponentials in the matrix algebra \({\mathcal M}_2(A)\)
type: Item
instance of: scholarly article
title: Logarithms and exponentials in the matrix algebra \({\mathcal M}_2(A)\)
zbMATH Open document ID: 1412.30145
author: Raymond Mortini
author: Rudolf Rupp
published in: Computational Methods and Function Theory
publication date: 2018-04-18T00:00:00Z
review text: It is a classical result in functional analysis that, if 0 is in the unbounded connected component of the complement of the spectrum of an invertible element of a complex Banach algebra, then that element is an exponential, and, hence, has a logarithm. In this paper, the authors explore this problem for \(2\times 2\) matrices whose entries belong to the disc algebra \(A(\mathbb{D})\) of holomorphic functions in the open unit disc with continuous extensions to the closed disc. In other words, given \(M\in\mathcal{M}_2(A(\mathbb{D}))\), when is there a matrix \(L\in\mathcal{M}_2(A(\mathbb{D}))\) such that \(M=e^L\)? When possible, the disc algebra is replaced by a general commutative Banach algebra. A trace criterion for the existence of a logarithm is given in Proposition 3.2. This shows that, if \(M\in\mathcal{M}_2 (\mathcal{A})\), where \(\mathcal{A}\) is a unital commutative Banach algebra, satisfies \(\det{M}=\mathbf{1}\) and the spectrum of the trace \(\mathrm{tr}(M)\) does not intersect the interval \((-\infty,-2]\), then \(M\) is an exponential. For example (see Example 3.3), for \(0<\varepsilon\leq 1\), the matrix \[ M=\left(\begin{matrix} 1 & \varepsilon\\ z &\varepsilon z+1\end{matrix}\right) \] satisfies the criteria of the proposition. In Proposition 3.4, the authors present an explicit formula for the logarithm of an invertible matrix with trace 0. In Section 5, the authors explore logarithms of the identity matrix. Among the requirements for a \(2\times 2\) matrix \(L\) to satisfy \(e^L=I_2\) is that the sum of the diagonal entries of \(L\) should have the form \(2k\pi i\mathbf{1}\), for some integer \(k\) (see Proposition 5.3). Corollary 5.5 then shows that, for any pair of elements of \(\mathcal{A}\), there is a logarithm of \(I_2\) in \(\mathcal{M}_2(\mathcal{A})\) that has those elements as its off-diagonal entries. Proposition 6.1 gives a characterization of those upper triangular matrices in \(\mathcal{M}_2(A(\mathbb{D}))\) that are exponentials a…
Mathematics Subject Classification ID: 30H50
Mathematics Subject Classification ID: 46J05
Mathematics Subject Classification ID: 46J10
Mathematics Subject Classification ID: 15A16
Mathematics Subject Classification ID: 15A54
zbMATH DE Number: 6861515
zbMATH Keywords: disc algebra
zbMATH Keywords: logarithms of matrices
zbMATH Keywords: matrix exponentials
zbMATH Keywords: commutative Banach algebra
zbMATH Keywords: matrix algebra
reviewed by: Timothy G. Feeman
describes a project that uses: MathOverflow
MaRDI profile type: Publication
full work available at URL: s40315-017-0209-z
OpenAlex ID: W2726067750
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DOI: 10.1007/S40315-017-0209-Z
description: scientific article; zbMATH DE number 6861515