Avector $x$in a Banachspace $\cal B$ iscalled hypercyclic for a bounded linear operator $T:\cal B \rightarrow\calB $ if the orbit $\{T^n x : n \geq 1 \}$ is dense in $\cal B$. We prove that if $T$is hereditarily hypercyclic and its essential spectrum meets the closed unit disk, then there is an infinite-dimensionalclosed subspace consisting, except for zero, entirelyof hypercyclic vectors for $T$. The converse is also true even if $T$ is not hereditarily hypercyclic. In this way, we improve and extend to Banach spaces a recent result for Hilbert spaces.We also investigate the corresponding problem for supercyclic operators. In this case we obtain a description of the norm closure of the class of all supercyclic operators that have an infinite-dimensional closed subspace of hypercyclic vectors.Moreover, for certain kinds of supercyclic operators we can characterize when they have an infinite-dimensional closed subspace of supercyclic vectors. 1991 Mathematics Subject Classification: 47B38, 47B99.
