Existence of inverses and square roots in locally Banach semigroups with identity

Abstract

Let S S be a multiplicative topological semigroup with identity e e . Suppose D D is an open subset containing e e and h h is a homeomorphism from D D onto a Banach space B B with h ( e ) = 0 h(e) = 0 . Define the function P P by P ( x , y ) = h [ h − 1 ( x ) ⋅ h − 1 ( y ) ] P(x,y) = h[{h^{ - 1}}(x) \cdot {h^{ - 1}}(y)] . A new implicit function theorem is applied to the function P P to show the existence of inverses and square roots of elements in a neighborhood of the identity. It is assumed that P P satisfies the following condition: There exist a one-one function A A from a subset of B B into B B and positive numbers r , M r,M , and c c such that (i) if | | x | | > r ||x|| > r then x ∈ dom ⁡ ( A − 1 ) x \in \operatorname {dom} ({A^{ - 1}}) and | | A − 1 ( x ) | | ⩽ M | | x | | ||{A^{ - 1}}(x)|| \leqslant M||x|| , (ii) c M > 1 cM > 1 , and (iii) if | | x i | | , | | y i | | > r ( i = 1 , 2 ) ||{x_i}||,||{y_i}|| > r(i = 1,2) then ( x i , y j ) ∈ dom ⁡ ( P ) ( i , j = 1 , 2 ) ({x_i},{y_j}) \in \operatorname {dom} (P)(i,j = 1,2) , \[ | | P ( x 1 , y 1 ) − P ( x 2 , y 2 ) − A ( y 1 − y 2 ) | | ⩽ c | | y 1 − y 2 | | , ||P({x_1},{y_1}) - P({x_2},{y_2}) - A({y_1} - {y_2})|| \leqslant c||{y_1} - {y_2}||, \] and \[ | | P ( x 1 , y 1 ) − P ( x 2 , y 1 ) − A ( x 1 − x 2 ) | | ⩽ c | | x 1 − x 2 | | . ||P({x_1},{y_1}) - P({x_2},{y_1}) - A({x_1} - {x_2})|| \leqslant c||{x_1} - {x_2}||. \]