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Conditional expectation type operators and modular inequalities

In this paper we discuss the connection between conditional expectation type operators and integral operators. A variant of Schur’s lemma is established and we obtain modular inequalities for a class of conditional expectation type operators. | Turk J Math (2018) 42: 3117 – 3122 © TÜBİTAK doi:10.3906/mat-1610-85 Turkish Journal of Mathematics http://journals.tubitak.gov.tr/math/ Research Article Conditional expectation type operators and modular inequalities Dah-Chin LUOR∗, Department of Financial and Computational Mathematics, I-Shou University, Dashu District, Kaohsiung City, Taiwan Received: 21.10.2016 • Accepted/Published Online: 14.10.2018 • Final Version: 27.11.2018 Abstract: In this paper we discuss the connection between conditional expectation type operators and integral operators. A variant of Schur’s lemma is established and we obtain modular inequalities for a class of conditional expectation type operators. Key words: Conditional expectation, modular inequalities, norm inequalities 1. Introduction Let (Ω, S, P) be a probability space and let X be a real-valued random variable on Ω . The expectation ∫ EX of X is defined as Ω XdP if the integral exists. Let A be a sub- σ -algebra of S . The conditional expectation of X given A is defined as a random variable E(X|A), measurable for A , such that for all A ∈ A , ∫ ∫ E(X|A)dP = A XdP , if such a E(X|A) exists. For any X ∈ L1 (Ω, S, P) and any sub- σ -algebra A of S , A a conditional expectation E(X|A) exists, and if Y and Z are conditional expectations of X given A, then Y = Z almost everywhere (see [5, 10.1.1 Theorem]). The operator E(·|A) : L1 (Ω, S, P) → L1 (Ω, A, P) is called the conditional expectation operator induced by A. If X is also A-measurable, then E(X|A) = X and hence E(·|A) is a projection from L1 (Ω, S, P) onto L1 (Ω, A, P) . It is known that E(·|A) is a bounded linear operator and for each 1 ≤ p ≤ ∞ , if X ∈ Lp (Ω, S, P), then E(X|A) ∈ Lp (Ω, A, P) and ∥E(X|A)∥p ≤ ∥X∥p . For more important properties and detailed discussion, we refer the readers to [1,3–5,16]. Recently, Estaremi and Jabbarzadeh established the boundedness and compactness properties for weighted conditional expectation type operators. Let (Ω, S, µ) be a .