Abstract. Some interesting algebras of operators are not complete in any algebra norm. Two examples: The algebra of all compact operators on an incomplete normed linear space; and the algebra of all bounded Carleman integral operators on $L^2[0,1]$. Although not Banach algebras, both of these algebras are functional algebras, i.e. the usual holomorphic functional calculus operates in them. This paper is concerned with functional algebras and related topics. Examples and applications involving algebras of bounded linear operators are given.

AMS Subject Classification (1991): 46H30, 46H35, 47A10

Keyword(s): holomorphic functional calculus, left and right ideals, range inclusion

Received February 15, 1999. (Registered under 2731/2009.)