Vol. 40, No. 1, 1972

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Maximum modulus theorems for algebras of operator valued functions

Kenneth O. Leland

Vol. 40 (1972), No. 1, 121–138

Let F be a family of functions on subsets of a real Euclidean space E into a commutative subalgebra with identity T0 of the algebra T of linear transformations of E into itself. If a suitable integration condition, motivated by Morera’s theorem in complex function theory is placed on the elements of F,F becomes an algebra of “integrable” functions which can be realized as the derivatives of transformations of E into itself. It is asked what properties of the algebra of complex analytic functions from the complex plane K into K are satisfied by such algebras F. Simple examples show that analyticity and even differentiability are lost. However various forms of the maximum modulus theorem are still satisfied. Three such theorems are presented here:

(A) If commutivity of T0 is replaced by the requirement that the elements of T0 are “orientation preserving” then the elements of F are maximized on the boundary of a sphere.

(B) There exists N > 0, such that for all f F, U = {t E;t1}⊆ domain f,x U , implies

∥f(x)∥ ≦ N sup{∥f(t)∥;∥t∥ = 1}.

(C) For all f F,U domain f,x U, implies

∥f(x)∥s ≦ sup{∥f(t)∥s;∥t∥ = 1},

where for A T0,As is the spectral norm of A.

Mathematical Subject Classification 2000
Primary: 46J99
Secondary: 30A90
Received: 2 December 1970
Published: 1 January 1972
Kenneth O. Leland