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Abstract Chapter one: In this chapter, we studied the invariant subspace problem and the classes in which it was solved then we mentioned Eno and Reads works. We also discussed the relation between the dynamical system to be irreducible (The space cannot be divided into two disjoint in- variant subspaces) and to be topologically transitive. Moreover, we discussed topologically transitive, mixing and weakly mixing proper- ties and the relation between them. We presented the de nition of cyclic, supercyclic and hypercyclic operators and we noted that the hypercyclicity has more structure than others. Moreover, if the op- erator is hypercyclic this implies that it is supercyclic, which in turn implies that the operator is cyclic. We showed that the hypercyclic- ity is su¢ cient condition for topological transitivity. Moreover, the equivalence is valid if the space is separable and complete. Chapter Two: In this chapter, we presented the de nition of chaos in sense of Devaney (three conditions) and by using theorem that was introduced by Banks, Brooks, Cairns, Davis and Stacey. We also discussed the conditions for weighted shift operator to be hypercyclic, hereditarily hypercyclic and nally chaotic. Chapter Three: In this chapter, we represented the unilateral weighted shift op- erators Rz; Rw; RzRwand Sz; Sw (forward and backward) as formal power series: we gave the exact estimations of s-numbers of the op- erators Rz and Rw (also Sz; Sw ) then we evaluated the upper and lower estimations of s-numbers of the operator RzRw. Moreover, we evaluated an upper estimation to the s-numbers of the unilateral for- ward shift operator of the form of an in nite series 1P m=0 cmRm z and 1P n;m=0 cnmRn zRm w on the space Hp ; . We considered the multiplying formal Taylor power series inm-variables X I aI zI (where I = (i1; i2; ; im) is an index set of m natural numbers) by zj make a right shift operator in dimension j ). We gave upper and lower estimations of s-numbers for multiplication of m- right weighted shift operators RJ . This allowed us to estimate upper bounds for s- numbers of in nite series of m-right weighted shift operators P J cJRJ . |