Weighted shift operators and extended eigenvalues

Let H be a Hilbert space. A complex number is named the extended eigenvalue for an operator T is an element of B(H) if there is operator not equal zero X is an element of B(H) so that TX =mu XT and X are named as extended eigen operator for an operator T opposite to mu. The goal of this work is to find extended eigenvalues and extended eigen operators for shift operators J,J(alpha) , K, K-alpha such that J: l(2) (N) -> l(2) (N) and K: l(2) (N) -> l(2) (N) defined by: Je(n) = e(2n), and Ke(n) ={ En/2 if n even 0 if n odd for all x is an element of l(2) (N). Furthermore, the closeness of extended eigenvalues for all of these shift operators under multiplication has been proven.