On a class of bounded operators and their ascent and descent
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On a class of bounded operators and their ascent and descent Y. Estaremi1 Received: 8 November 2019 / Accepted: 11 December 2019 © Springer-Verlag Italia S.r.l., part of Springer Nature 2019
Abstract In this paper we provide some conditions under which the ascent and descent of the weighted conditional expectation operators of the form of Mw E Mu on L p -spaces are finite. Moreover, we give some necessary and sufficient conditions for Mw E Mu to be power bounded. In the sequel we apply some results in operator theory on ascent and descent to Mw E Mu . Finally is Cesaro bounded. we find that T = Mw E Mu is Cesaro bounded if and only if T Keywords Weighted conditional expectation · Ascent · Descent Mathematics Subject Classification 46E30 · 47A05
1 Preliminaries and introduction Let X be a linear space and T : X −→ X be a linear operator with domain D(T ) and range R(T ) in X . The null space of the iterates of T , T n , is denoted by N (T n ), and we know that the null spaces of T n ’s form an increasing chain of subspaces {0} = N (T 0 ) ⊂ N (T ) ⊂ N (T 2 ) ⊂ . . .. Also the ranges of iterates of T form a nested chain of subspaces X = R(T 0 ) ⊃ R(T ) ⊃ R(T 2 ) ⊃ . . .. Note that if N (T k ) coincides with N (T k+1 ) for some k, it coincides with all N (T n ) for n > k. The smallest non-negative integer k such that N (T k ) = N (T k+1 ) is called the ascent of T and denotes by α(T ). If there is no such k, then we set α(T ) = ∞. Also if R(T k ) = R(T k+1 ), for some non-negative integer k, then R(T n ) = R(T k ) for all n > k. The smallest non-negative integer k such that R(T k ) = R(T k+1 ) is called descent of T and denotes by δ(T ). We set δ(T ) = ∞ when there is no such k. When ascent and descent of an operator are finite, then they are equal and the linear space X can be decomposed into the direct sum of the null and range spaces of a suitable iterates of T . The ascent and descent of an operator can be used to characterize when an operator can be broken into a nilpotent piece and an invertible one; see, for example, [1,12]. For some results on ascent and descent of bounded operators in general setting see, for example, [13,14].
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Y. Estaremi [email protected] Department of Mathematics, Payame Noor University (PNU), P. O. Box: 19395-3697, Tehran, Iran
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Y. Estaremi
The operator T is called power bounded if the norms of T k , k ≥ 0, are uniformly bounded n−1 (supk T k < ∞), and Cesaro bounded if the Cesaro means An (T ) = n −1 i=0 T i are uniformly bounded. Let (X , F , μ) be a complete σ -finite measure space. All sets and functions statements are to be interpreted as holding up to sets of measure zero. We denote the collection of (equivalence classes modulo sets of zero measure of) F -measurable complex-valued functions on X by L 0 (F ). For a σ -sub-algebra A of F , the conditional expectation operator associated with A is the mapping f → E A f , defined for all non-negative function f as well as for all f ∈ L p (F ) = L p (X , F , μ), 1 ≤ p ≤ ∞, where E A f is the unique A-measurable functi
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