An integral representation for strictly continuous linear operators

Martin Bartelt

Vol. 56 (1975), No. 1, 21–34

DOI: 10.2140/pjm.1975.56.21

Abstract

Let B denote the algebra of bounded analytic functions on the open unit disc D in the complex plane. Let (B,τ) denote B endowed with the topology τ, where τ is chosen from κ,β or σ, respectively, the topology of uniform convergence on compact subsets of D, the strict topology and the topology of uniform convergence on D. This note obtains an integral representation of the form Tf(z) = ∫ _Γf(w)K(z,w)dw where Γ = {z : |z| = 1} for the linear operators which are continuous from (B,κ) into (B,σ). This representation is then used to study the convergence of operators in the full algebra of all continuous linear operators from (B,β) into (B,β).

Mathematical Subject Classification 2000

Primary: 47B37

Secondary: 46E10

Milestones

Received: 5 December 1973

Published: 1 January 1975