Logarithms and Exponentials in the Matrix Algebra $${\mathcal M}_2(A)$$ M 2 ( A )

Abstract

It is well known that in the disk algebra $$A({ \mathbb D})$$ every zero-free function has a logarithm in $$A({ \mathbb D})$$ . This is no longer true if we look at invertible matrices over $$A({ \mathbb D})$$ . In this paper, we give a sufficient condition on the trace of a $$2\times 2$$ -matrix M so that $$M=e^L$$ for some matrix $$L\in A({ \mathbb D})$$ . We compute all the logarithms of the identity matrix in $${\mathcal M}_2(A({ \mathbb D}))$$ and observe that the anti-diagonal elements can be arbitrarily prescribed. We also characterize those upper (or lower) triangular matrices which are exponentials in $${\mathcal M}_2(A({ \mathbb D}))$$ and determine all their logarithms. This will enable us to prove that $$\exp {\mathcal M}_2(A({ \mathbb D}))$$ is neither closed nor open within the principal component of $${\mathcal M}_2(A({ \mathbb D}))^{-1}$$ . Finally, we show that every invertible matrix in $${\mathcal M}_2(A({ \mathbb D}))$$ is a product of four exponential matrices and give conditions for reducing this number. These results will be put into the more general setting of commutative Banach algebras whenever possible.