Reflexivity of Cowen-Douglas Operators

Subject Areas : Statistics

Ali Kashkooly ¹ , zahra fatahi ²

1 - Department of Mathematics, Yasuj University, Yasuj, Iran.
2 - Department of Mathematics, Yasuj University, Yasuj, Iran.

Received: 2019-05-11 Accepted : 2021-07-13 Published : 2021-07-23

Keywords: عملگر ضربی, الحاق, عملگر ون &ndash, نیومن, کلمات کلیدی: هسته‌ی برگمن, هم ارز یکانی,

Abstract :

For a connected open subset Ω of the plane and n a positive integer, let B_n (Ω) be the Cowen-Douglas class of operators. In this article, for a special case of Ω, we show that if T∈B_n (Ω) and its canonical model is a Von Neumann operator, then T is reflexive.In the main theorem of this paper we assume that the adjoint of the canonical model associated with g.B.K is a Von Neuman operator. We may replace this by the assumption that ‖M_P ‖≤c‖P‖_Ω or ‖M_P ‖=c‖P‖_Ω for every polynomial P. Actually K is the reproducing kernel for a coanalytic functional Hilbert space K on which we can define the operator M_z^* of multiplication byz.Note that if K is stricly positive kernel function on Λ, it gives rise to a functional Hilbert space on Λ with reproducing kernel K.A bounded linear operator T is said to be Von Neumann if the C^*-algebra generated by T is a Von Neumann algebra. We must point out that the operators of Cowen-Douglas class is not reflexive in general, since every operator of this class is unitarily equivalent to the adjoint of non-reflexive operator multiplication by Z ̅. We need to impose the additional conditions for reflexivity of T.