Characterizations of conditional expectation-type operators

Iterates of Conditional Expectation Operators

On Convergence of Conditional Expectation Operators

Given an operator T : UX(Σ) → Y or T : C(H,X) → Y , one may consider the net of conditional expectation operators (Tπ) directed by refinement of the partitions π. It has been shown previously that (Tπ) does not always converge to T . This paper gives several conditions under which this convergence does occur, including complete characterizations when X = R or when X∗ has the Radon-Nikodým prope...

Iterates of a Product of Conditional Expectation Operators

Let (Ω,F , μ) be a probability space and let T = P1P2 · · ·Pd be a finite product of conditional expectations with respect to the sub σ-algebras F1,F2, . . . ,Fd. Since conditional expectations are contractions of all Lp(μ) spaces, p ∈ [1,∞], so is T . When d = 2, Burkholder and Chow [2] proved that for every f ∈ L2(μ) the iterates T f converge a.s. (and thus also in L2-norm) to the conditional...

Beals-cordes-type Characterizations of Pseudodifferential Operators

We show that, if U is the representation of SOe(n+1, 1) on L2(Sn) given by (2.11), and P is a bounded operator on L2(Sn), then P belongs to OPS0 1,0(S n) if and only if P (g) = U(g)PU(g)−1 is a C∞ function on SOe(n+1, 1) with values in the Banach space L(L2(Sn)). Introduction Let M be a compact C∞ manifold. Denote by OPS 1,0(M) the space of pseudodifferential operators on M , whose symbols in l...

CONDITIONAL EXPECTATION IN THE KOPKA'S D-POSETS

The notion of a $D$-poset was introduced in a connection withquantum mechanical models. In this paper, we introduce theconditional expectation of  random variables on theK^{o}pka's $D$-Poset and prove the basic properties ofconditional expectation on this  structure.