Logarithms and exponentials in Banach algebras
نویسندگان
چکیده
Let A be a complex Banach algebra. If the spectrum of an invertible element a ∈ A does not separate the plane, then a admits a logarithm. We present two elementary proofs of this classical result which are independent of the holomorphic functional calculus. We also discuss the case of real Banach algebras. As applications, we obtain simple proofs that every invertible matrix over C has a logarithm and that every real matrixM inMn(R) with detM > 0 is a product of two real exponential matrices.
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تاریخ انتشار 2017