Weighted shift operators and extended eigenvalues

Abstract

Let H be a Hilbert space. A complex number is named the extended eigenvalue for an operator T ∈ B(H), if there is operator not equal zero X ∈ B(H) so that: TX = mXT and X are named as extended eigen operator for an operator T opposite to m. The goal of this work is to find extended eigenvalues and extended eigen operators for shift operators J, Ja, K, Ka such that J : I2 (ℕ) → I2 (ℕ) and K : I2 (ℕ) → I2 (ℕ) defined by:           Jen = e2n , and Ken = { (en/2 if n even) (0 if n odd), for all x ∈ l2(ℕ). Furthermore, the closeness of extended eigenvalues for all of these shift operators under multiplication has been proven.