Karim HEDAYATIAN, Masoumeh FAGHIH-AHMADI

Commutants and hyper-reflexivity of multiplication operators

We characterize the commutants of some multiplication operators on a Banach space of analytic functions defined on a bounded domain in the plane. Under certain conditions on the symbol of a multiplication operator, we show that its commutant is a set of multiplication operators. This partially answers a question of Axler, Cuckovic and Rao. Next, the hyper-reflexivity of these multiplication operators are proved. The paper is concluded by proving the hyper-reflexivity of the multiplication operators with symbols j (z) = zk, k=1, 2,... .

Anahtar Kelimeler:

Bergman space, multiplication operators, hyper-reflexive operator, commutant, hyper-invariant

Commutants and hyper-reflexivity of multiplication operators

Keywords:

Bergman space, multiplication operators, hyper-reflexive operator, commutant, hyper-invariant,

PDF

___

Axler, S., Cuckovic, Z. and Rao, N. V.: Commutant of analytic Toeplitz operators on the Bergman space, Proc. Amer. Math. Soc. 128, 1951–1953 (2000).
Cao, G.: On a problem of Axler, Cuckovic and Rao, Proc. Amer. Math. Soc. 136, 931–935 (2008).
Chevereau, B., Pearcy, C. and Shields, A.: Finitely connected domains G , representations of H ∞ (G) , and invariant subspaces, J. Operator Theory, 6, 375–405 (1981).
Conway, J. B.: A Course in Functional Analysis, Springer-Verlag, 1990.
Conway, J. B.: The Theory of Subnormal Operators, Amer. Math. Soc. Surveys and Monographs, 36, 1991.
Cuckovic, Z.: Commutant of Toeplitz operators on the Bergman spaces, Paciﬁc J. Math. 162, 277–285 (1994). Eschmeier, J.: Multiplication operators on Bergman spaces are reﬂexive, Operator Theory: Advances and Applications, 43, 165–184 (1990).
Gamelin, T.: Uniform Algebras, New York, 1984.
Khani Robati, B., Vaezpour, S. M.: On the commutant of operators of multiplication by univalent functions, Proc. Amer. Math. Soc. 129, 2379–2383 (2001).
Khani Robati, B., Vaezpour, S. M.: On the commutant of multiplication operators with analytic symbols, Rocky Mountain J. Math. 33, 1049–1056 (2003).
Khani Robati, B.: On the structure of certain operators on spaces of analytic function, Ital. J. Pure Appl. Math. 18, 175–180 (2005).
Sarason, D.: Weak star generator of H ∞ , Paciﬁc J. Math. 17, 519–528 (1966).
Shields, A. L., Wallen, L.: The commutant of certain Hilbert space operators, Indiana Univ. Math. J. 20, 777–788 (1971).