On reducibility of weighted composition operators

Document Type : Research Paper


1 Department of Mathematics, University of Maragheh, Maragheh, Iran.

2 faculty of mathematical sciences, university of tabriz, p. o. box: 5166615648, tabriz,


In this paper, we study two types of the reducing subspaces for the weighted composition operator $W: f\rightarrow u\cdot f\circ \varphi$ on $L^2(\Sigma)$. A necessary and sufficient condition is given for $W$ to possess the reducing subspaces of the form $L^2(\Sigma_B)$ where $B\in \Sigma_{sigma(u)}$. Moreover, we pose some necessary and some sufficient conditions under which the subspaces of the form $L^2(\mathcal{A})$ reduce $W$. All of these are basically discussed by using the conditional expectation properties. To explain the results some examples are then presented.


Main Subjects

C. Burnap and A. Lambert, Reducibility of composition operators on L2, J. Math. Anal. Appl. 178 (1993), no. 1, 87--101.
J.T. Campbell, M. Embry-Wardrop, R.J. Fleming and S.K. Narayan, Normal and quasinormal weighted composition operators, Glasg. Math. J. 33 (1991), no. 3, 275--279.
J.T. Campbell and W.E. Hornor, Localising and seminormal composition operators on L2, Proc. Roy. Soc. Edinburgh Sect. A 124 (1994), no. 2, 301--316.
J.T. Campbell and J. Jamison, On some classes of weighted composition operators, Glasg. Math. J. 32 (1990), no. 2, 87--94.
J.B. Conway, A Course in Functional Analysis, Springer-Verlag, New York, 1990.
P. Dodds, C. Huijsmans and B. de Pagter, Characterizations of conditional expectation type operators, Pacific J. Math. 141 (1990), no. 1, 55--77.
J. Herron, Weighted conditional expectation operators, Oper. Matrices 5 (2011), no. 1, 107--118.
T. Hoover, A. Lambert and J. Quinn, The Markov process determined by a weighted composition operator, Studia Math. 72 (1982), no. 3, 225--235.
A. Lambert and B.M. Weinstock, Descriptions of conditional expectations induced by non-measure preserving transformations, Proc. Amer. Math. Soc. 123 (1995), no. 3, 897--903.
H. Radjavi and P. Rosenthal, Invariant Subspaces, , Dover Publications, 2nd edition, Mineola, NY, 2003.
M.M. Rao, Conditional measure and applications, Marcel Dekker, New York, 1993.
R.K. Singh and J.S. Manhas, Composition Operators on Function Spaces, North-Holland Mathematics Studies, North-Holland Publishing Co. Amsterdam, 1993.