Let ß(H) be the Banach algebra of bounded linear operators acting on the complex separable Hilbert space H. For an operator T є. ß (H), )w;(T) is the weakly closed subaJgebra of ß(H) which is generated by T. A vector x in H is a separating vector for an algebra w(T) T є. ß(H), ' if the map A -> Ax , A є w(T), is injective. And x is a strictly separating vector for an algebra w(T), if the map A -» Ax , A є w (T), is bounded below. In this thesis, we study separating vectors and strictly separating vectors, for the algebra w(T). We discuss with some details,, the existence of such kind of vectors for algebras that are generated by some operators that belong to some classes of operators such as algebraic operators, normal operators, triangular operators and some operators matrices. We study the. relation between the set of separating vectors and cyclic vectors for algebra w (T). We illustrate this study by some examples. We give the details of proofs for many of the known results and we supply proofs for some known results that appeared without proofs. Also we add some results that seem to be new to the best of our knowledge many of these results concern algebras generated by operator matrices. One of our main results shows that for any similar operators T, , T2, the algebra w (Tı) has a separating vector if and only if w (T2) has a separating vector. This result gives us our main tool for giving a sufficient condition for the weak closed algebra w (S), S = [A E] in ß (H Ө ¢ⁿ) to have a separating vector. [ 0 B ]