Abstract
In this paper we provide some conditions under which the ascent and descent of the weighted conditional expectation operators of the form of \(M_wEM_u\) on \(L^p\)-spaces are finite. Moreover, we give some necessary and sufficient conditions for \(M_wEM_u\) to be power bounded. In the sequel we apply some results in operator theory on ascent and descent to \(M_wEM_u\). Finally we find that \(T=M_wEM_u\) is Cesaro bounded if and only if \({\widehat{T}}\) is Cesaro bounded.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Abramovich, Y.A., Aliprantis, C.D.: An Invitation to Operator Theory. American Mathematical Society, Providence (2002)
Douglas, R.G.: Contractive projections on an L1 space. Pac. J. Math. 15, 443–462 (1965)
Dodds, P.G., Huijsmans, C.B., De Pagter, B.: Characterizations of conditional expectation-type operators. Pac. J. Math. 141(1), 55–77 (1990)
Estaremi, Y.: On the algebra of WCE operators. Rocky Mt. J. Math. 48, 501–517 (2018)
Estaremi, Y.: Some classes of weighted conditional type operators and their spectra. Positivity 19, 83–93 (2015)
Estaremi, Y., Jabbarzadeh, M.R.: Weighted lambert type operators on L p-spaces. Oper. Matrices 1, 101–116 (2013)
Grabiner, S., Zemanek, J.: Ascent, descent and ergodic properties of linear operators. J. Oper. Theory 48, 69–81 (2002)
Grobler, J.J., De Pagter, B.: Operators representable as multiplication-conditional expectation operators. J. Oper. Theory 48, 15–40 (2002)
Lamber, A.: Lp multipliers and nested sigma-algebras. Oper. Theory Adv. Appl. 104, 147–153 (1998)
Moy, S.T.: Characterizations of conditional expectation as a transformation on function spaces. Pac. J. Math. 4, 47–63 (1954)
Rao, M.M.: Conditional Measure and Applications. Marcel Dekker, New York (1993)
Taylor, A.E., Lay, D.C.: Introduction to Functional Analysis, 2nd edn. Wiley, New York (1980)
Taylor, A.E.: Theorems on ascent, descent, nullity and defect of linear operators. Math. Ann. 163, 18–49 (1966)
Yood, B.: Ascent, descent and roots of Fredholm operators. Stud. Math. 158, 219–226 (2003)
Zaanen, A.C.: Integration, 2nd edn. North-Holland, Amsterdam (1967)
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Estaremi, Y. On a class of bounded operators and their ascent and descent. Rend. Circ. Mat. Palermo, II. Ser 69, 1393–1399 (2020). https://doi.org/10.1007/s12215-019-00478-1
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s12215-019-00478-1