Logarithms and exponentials in the matrix algebra \({\mathcal M}_2(A)\) (Q1745951)

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 and describes their logarithms. Finally, provided that the set \(\mathcal{A}^{-1}\) of invertible elements is connected, Theorem 7.1 shows that every invertible matrix in \(\mathcal{M}_2(\mathcal{A})\), though it may not be an exponential itself, can be expressed as a product of at most four exponentials. Under various additional conditions, this number might be reduced to three or even two (see Theorem 7.2, for instance). Though the proofs can be somewhat technical, the results are quite satisfying and make this paper a worthwhile read.