Holomorphic functions (or maps) have been defined between Banach spaces by the use of a Taylor expansion involving Frechet derivatives. When the Banach spaces in question coincide with ℒ(ℋ), the space of linear operators over a Hilbert space ℋ. the set of holomorphic functions includes those arising from the Dunford functional calculus, but is certainly not limited to these. The holomorphic functions between Banach spaces share many of the properties of ordinary holomorphic functions from the complex plane ℄ into itself. However, in many aspects they behave differently. For example, the maximum modulus theorem implies that an ordinary holomorphic function with constant modulus must be a constant function. This is no longer true even for holomorphic functions of one complex variable taking values in a Banach space. In fact, the Thorp-Whitley Theorem states that if D is a domain in ℄, Y a Banach space, and F:D → Y holomorphic with ||F(ζ)|| = 1 on D, then F is a constant function if its range contains a complex extreme point of the unit ball of Y.
It is natural to ask which holomorphic functions between Banach spaces have constant norm. For the case where F:D ⊂ ℄ → Y, the problem was solved by Globevnik, who also specialized the result to the case F:D ⊂ ℄ → ℒ(ℋ). In addition, he determined under which conditions F might have constant norm in some norm equivalent to the given norm. This thesis solves the problem in the full case where F is now a holomorphic function between two Banach spaces. The following theorem analogous to Globevnik's is proved:
Theorem. Let χ, Y be Banach spaces, D a domain in χ and F:D → Y holomorphic. Then ||F(x)|| will be constant for all x ϵD if and only if
(i) The subspace ε(F(x)) is independent of x ϵD, i.e., ε(F(x)) = ε for all x ϵD,
(ii) F(x) - F(y) ϵε for all x, y ϵD, where for u ϵ Y the set ε(u) is defined to be ε(u) = {v ϵ Y|Ǝr > 0 such that ||u+ ζv|| ≤ ||u|| for all ζ ϵ ℄, |ζ| ≤ r}.
An immediate consequence is that the Thorp-Whitley Theorem also holds in this generality, that is, when F is a function between arbitrary Banach spaces.
When this theorem is applied to the case χ = Y = ℒ(ℋ) a simplified criterion is obtained. The norm constant functions are precisely those annihilated by certain projection operators on ℋ. As a corollary to this it is shown that the only holomorphic functions arising from the Dunford calculus with constant norm are the constant functions. In contrast to the above theorem, it is also shown that any holomorphic function F:D ⊂ ℒ(ℋ) → ℒ(ℋ) with Re(F) = 0 on D must be a constant function. A theorem analogous to Globevnik's for deciding when a function F:D ⊂ χ → Y can be norm constant under some equivalent norm is also obtained.
