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Reducibility of M u C φ L 2 ( Σ )
Abstract
In this note we study reducing subspaces for weighted composition operators defined on L²(Σ). Some necessary and sufficient conditions are given for such operators to have two types of reducing subspaces of the forms L²(ΣA) and L²(A). This is basically discussed by using conditional expectation properties.
In this paper we characterize the closed subspaces of $L^2(\mathcal{F})$ that reduce the operators of the form $E^{\mathcal{A}}M_u$, in which $\mathcal{A}$ is a $\sigma$- subalgebra of $\mathcal{F}$. We show that $L^2(A)$ reduces $E^{\mathcal{A}}M_u$ if and only if $E^{\mathcal{A}}(\chi_A)=\chi_A$ on the support of $E^{\mathcal{A}}(|u|^2)$, where $A\in \mathcal{F}$. Also, some necessary and sufficient conditions are provided for $L^2(\mathcal{B})$ to reduces $E^{\mathcal{A}}M_u$, for the $\sigma$- subalgebra $\mathcal{B}$ of $\mathcal{F}$.
Let ( X , Σμ) denote a complete a-finite measure space and T : X → X a measurable ( T ⁻¹ A ε Σ each A ε Σ) point transformation from X into itself with the property that the measure μ° T ⁻¹ is absolutely continuous with respect to μ. Given any measurable, complex-valued function w ( x ) on X , and a function f in L ² (μ), define W T f ( x ) via the equation
As is well-known, conditional expectation operators on various function spaces exhibit a number of remarkable properties related either to the underlying order structure of the given function space, or to the metric structure when the function space is equipped with a norm. Such operators are necessarily positive projections which are averaging in a precise sense to be described below and in certain normed function spaces are contractive for the given norm.
This paper presents the fundamental operator-theoretic properties of products of conditional expectation and multiplication operators. It is shown that boundedness of such a product need not depend on the boundedness of the multiplication operator. The spectrum is described, as is the unique polar decomposition. It is also shown that compactness implies the existence of an atom in the underlying σ -subalgebra. An algebra containing such operators is shown to be weakly closed and, when the underlying space is of finite measure, its commutant is an algebra of multiplication operators with suitably measurable symbol.
Given a sigma finite measure space (X, , m) and a measurable transformation T: X → X, the reducibility of the composition operator C: ƒ → ƒ ∘ T on L2 is examined. It is shown that if C is reducible then either there is a subset A of X such that L2(A, , m) reduces C, or there is a sigma subalgebra of such that L2(X, , m) reduces C. Several examples are then presented illustrating various levels of reducibility for composition operators.
Given a measure-preserving transformation T acting on a σ-finite measure space (X, A, m) and a σ-finite sigma algebra $\mathscr{B} \subset \mathscr{A}$, the conditional expectations E(·∣B) acting on L∞(A) and E(·∣ T-1B) acting on L∞(T-1A) are known to be related by the formula [ E(f∣B)] ⚬ T = E(f⚬ T∣ T-1B). In this note the conditional expectation E(·∣ T-1B) is investigated in the non-measure-preserving case, and those transformations for which the above equation holds are characterized in terms of measurability conditions for d(m ⚬ T-1)/dm. It is precisely in the non-measure-preserving case that the measurability of d(m ⚬ T-1)/dm plays an important role. Relatedly, it is shown that if composition by T intertwines E(·∣B) and any mapping Λ, then Λ is a conditional expectation induced by a measure equivalent to m. These results were motivated by a result concerning induced conditional expectation operators on C*-algebras, and the paper concludes with a brief description of this C*-algebra setting.
Let (X, ∑, μ) denote a σ-finite measure space. We show that the kernel condition on a weighted composition operator acting on L2(X, ∑, μ), which is necessary for hyponormality of the adjoint, implies that a certain subset of X has the localising property defined by Lambert. For operators satisfying this condition, we find a reducing subspace whose orthocomplement in L2 is annihilated by both the operator and its adjoint, allowing us to obtain characterisations of seminormality for the operator by looking only at the restriction to the reducing subspace. This simplifies the analysis significantly, giving transparent characterisations for the hyponormality and quasinormality of the adjoint, as well as a characterisation of normality for the operator which does not require the computation of any conditional expectations. Several examples are given. We then characterise the semi-hyponormal class for both the operator and its adjoint.