Cohyponormality of Unbounded Product of Composition Operators in L2 Spaces - Hang Zhou; Qing Su (Journal of Function Spaces)
Let X,A,μ be a σ-finite measure space. The composition operators Cϕ:DCϕ⟶L2μ are defined by Cϕf=f∘ϕ,f∈DCϕ, where ϕ are nonsingular transformations and DCϕ=f∈L2μ: f∘ϕ∈L2μ. Suppose that ϕ1,ϕ2,⋯,ϕn are nonsingular transformations of X, where n∈ℕ and n≥2. It is natural to define the product operator of composition operators Cϕ1,Cϕ2,⋯,Cϕn, denoted by Cϕn⋯Cϕ1 in L2μ. Equivalent conditions of cohyponormality of (not necessarily bounded) Cϕn⋯Cϕ1 in L2μ are given in this paper, where Cϕ1,Cϕ2,⋯,Cϕn are densely defined. In fact, the following statements are equivalent: (i) Cϕk⋯Cϕ1 is cohyponormal; (ii) ∫Xθk2f2dμ≤∫XEΦ~kf2dμ, where f∈L2μ; and (iii) EΦkθk2·f2≤EΦ~kf2 a.e. μΦk−1A∩Φ~k−1A, where f∈L2μ. Moreover, basic properties of product Cϕn⋯Cϕ1 in L2μ are conveyed, including the dense definiteness, boundedness, and adjointness.