الفهرس 
The invariant subspace problem and hypercyclic operatorsOnly 14 pages are availabe for public view
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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 .