The necessary and sufficient conditions that guarantee the boundedness and compactness of integral operators with positive kernels from $L^p(a,b)$ to $L^q_{\nu}(a,b)$, where $p,q\in(1,\infty)$ or $0lt q\leq1lt plt\infty$, for a non-negative Borel measure $\nu$ on $(a,b)$ are found.

AMS 2000 Mathematics subject classification: Primary 46B50; 47B34; 47B38

We work with interpolation methods for $N$-tuples of Banach spaces associated with polygons. We compare necessary conditions for interpolating closed operator ideals with conditions required to interpolate compactness. We also establish a formula for the measure of non-compactness of interpolated operators.

A vector $x$ in a Banach space $\cal B$ is called hypercyclic for a bounded linear operator $T: \cal B \rightarrow \cal B $ 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-dimensional closed subspace consisting, except for zero, entirely of 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.