Extended Eigenvalues and Eigenoperators of Some Weighted Shift Operators.

A complex number µ is called an extended eigenvalue for an operator T ∈ B(H) on a Hilbert space H if there exists a nonzero operator T ∈B(H)such that: TX = µXT, such X is called an extended eigenoperator corresponding to. The goal of this paper is to calculate extended eigenvalues and extended eigenoperators for the weighted unilateral (Forward and Backward) shift operators. We also find an extended eigenvalues for weighted bilateral shift operator. Moreover, the closed ness of extended eigenvalues for the weighted unilateral (Forward and Backward) shift operators under multiplication is proven. [ABSTRACT FROM AUTHOR]