چكيده فارسي :
P.R.halmos [5] has proved that similar operator on a Hilbert space H have the same spectrum, moreover they have the same point spectrum, approximate point spectrum and the same compression spectrum . In this paper we show that the situation is absolutely different for the numerical range by giving an example. Theorem.1/ gives a necessary condition on the similar operator S and T get that W(S)=W (T), where ) denotes closure, we also consider the case when H is finite dimensional. In theorm.2/ we prove that if T and S are similar operators on complex Hilbert space H such that W(T) is a closed polygon then W(S) contains the interior of W(T) , while when W(S) and W(T) are both closed polygon then W(S)=W(T). Moreover we prove theorm.3/which states that the numerical range of any operator Ton H unitarily invariant (i.e. W (U*TU) =W (T)), and we conclude that the numerical radius is also unitarily invariant. Finally, we extend thm.3/ to the case of n- tuples of operators on H. From theorm.4/ we conclude that the joint numerical radius is also unitarily invariant
