Inequalities for Power Series in Banach Algebras

for any a; b 2 B: The normed algebra (B; k k) is a Banach algebra if k k is a complete norm. We assume that the Banach algebra is unital, this means that B has an identity 1 and that k1k = 1: Let B be a unital algebra. An element a 2 B is invertible if there exists an element b 2 B with ab = ba = 1: The element b is unique; it is called the inverse of a and written a 1 or 1 a : The set of invertible elements of B is denoted by InvB. If a; b 2InvB then ab 2InvB and (ab) 1 = b a : For a unital Banach algebra we also have: (i) If a 2 B and limn!1 kank < 1; then 1 a 2InvB; (ii) fa 2 B: k1 bk < 1g InvB; (iii) InvB is an open subset of B; (iv) The map InvB 3 a 7 ! a 1 2InvB is continuous. For simplicity, we denote 1; where 2 C and 1 is the identity of B, by : The resolvent set of a 2 B is de…ned by