CYCLIC MULTIPLICATION OPERATORS ON ^-SPACES

Let (X, Σ, μ) be a measure space. Suppose / is in L^X, Σ, μ). The operator Mf on LP(X, Σ, μ) defined by Mf(g) = f g, for g in LP{X, Σ, μ) is called a multiplication operator. The purpose of this paper is to characterize cyclic multiplication operators and to relate their structure to the properties of the measure space on which the underlying Lp-space is defined. In L2(Xf Σ, μ) any maximal abelian self-adjoint algebra of bounded operators may be transformed isometrically to the algebra of all multiplication operators on some L2-space (see, e.g. [9]). This is due to the fact that among the Z^-spaces, only L2-possesses a sufficiently rich collection of orthogonal projections. In fact, if p Φ 2, the only "orthogonal " projections on Lp are multiplications by characteristic functions (shown by Sullivan [10] for real Lp). As a consequence, isometries between Lp-spaces are related to σ-isomorphisms between the underlying measure spaces when p Φ 2. These relationships may be exploited to characterize certain multiplication operators on L^-spaces where 1 ^ p < oo. In § 1, we present Sullivan's theorem along with applications to direct sum decompositions of I^-spaces and to surjective isometries between L^-spaces. Section 2 deals with the characterization of the operator Mz on Lp(v), where v is a finite Borel measure with compact support in the plane, defined by MJ(z) = zf(z) for / in Lp(v). In § 3 the concept of a normal measure space (introduced by Halmos and von Neumann [3]) is used to relate the structure of certain measure spaces (X, Σ, μ) to the structure of cyclic multipli-cation operators on LP(X, Σ, μ). We mention that throughout, all measure spaces are assumed to be tf-finite. For notational ease, we denote LP(X, Σ, μ) by Lp(μ) when no confusion arises. The algebra of all multiplication operators on Lp(μ) is denoted by ^ffμ. Also if / is a measurable function on (X, Σ \ then supp (/) = {x e X \ \ f(x) |> 0} is called the support of /. I * Structural and isometric properties of L^spaces* DEFINITION 1.1. A closed subspace R of Lp(μ) is a p-dίrect 54