Some properties of MCE operators between different Orlicz spaces
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Some properties of MCE operators between different Orlicz spaces Yousef Estaremi1 Received: 9 August 2018 / Revised: 29 May 2019 / Accepted: 11 August 2019 © Springer Nature Switzerland AG 2019
Abstract We study some basic properties, like boundedness and closedness of range, of multiplication conditional expectation operators between different Orlicz spaces. Keywords Multiplication operator · Conditional expectation · Continuous operators · Closed-range operators · Finite-rank operators · Orlicz space Mathematics Subject Classification 47B38
1 Introduction Our concern in this paper is to provide some necessary conditions, sufficient conditions, and some simultaneously necessary and sufficient conditions for the multiplication conditional expectation or briefly MCE operators between distinct Orlicz spaces to be bounded or to have closed range or finite rank. MCE operators are a generalization of multiplication and in some cases, a generalization of weighted composition operators. One of the most important properties of MCE operators is that a large class of bounded operators on measurable functions spaces as well as on L p -spaces are of the form of MCE operators [3]. In [15] the authors studied multipliation and composition operators between different L p -spaces. In particular, they have given a complete description of bounded and closed range multiplication and composition operators. Also, in [2] we studied some basic properties of multiplication and composition operators between distinct Orlics spaces. In addition, we investigated bounded and compact MCE operators on Orlicz spaces in [5]. One can find many papers about MCE operators on L p -spaces, for instance [4,6,7,10–12]. In this paper we continue our project to characterize closed range MCE operators between different Orlicz spaces. In Sect. 3 we provide various necessary conditions and sufficient conditions under which the
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Yousef Estaremi [email protected] Department of Mathematics, Payame Noor University (PNU), P.O. Box 19395-3697, Tehran, Iran
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Y. Estaremi
MCE operator E Mu between distinct Orlicz spaces is bounded. In Sect. 4 we investigate closed-range MCE operators between distinct Orlicz spaces. Our results will generalize some results in the literature like some results in [2].
2 Preliminaries and basic lemmas In this section, for the convenience of the reader, we gather some essential facts on Orlicz spaces and prove two basic lemmas for later use. For more details on Orlicz spaces, see [9,14]. A function : R → [0, ∞] is called a Young function if is convex, even, and (0) = 0; we will also assume that is neither identically zero nor identically infinite on (0, ∞). The fact that (0) = 0, along with the convexity of , implies that lim x→0+ (x) = 0; while = 0, again along with the convexity of , implies that lim x→∞ (x) = ∞. We set a ..= sup {x 0 : (x) = 0} and b ..= sup {x > 0 : (x) < ∞}. Then it can be checked that is continuous and nondecreasing on [0, b ) and strictly increasing on [a , b ). We also ass
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