M.Sc Thesis | |

M.Sc Student | Ivtsan Alon |
---|---|

Subject | Topics in Interpolation Spaces |

Department | Department of Mathematics |

Supervisor | PROFESSOR EMERITUS Michael Cwikel |

Full Thesis text |

In this work we carry out two projects in the field of interpolation spaces.

For the first
project, let (B_{0},B_{1}) be a Banach pair. Stafney showed
that, in the definition of the norm in the Calderón complex
interpolation method on the strip, one can replace the space F(B_{0},B_{1})
with its subspace G(B_{0},B_{1}) if the element belongs to the
intersection of the spaces B_{i}. We shall extend this result to a more
general setting, which contains several well-known interpolation methods,
including the Calderón complex interpolation method on the annulus, an
appropriate version of the Lions-Peetre real method, and the Peetre
“plus-minus” method.

For the second
project, let (A_{0},A_{1}) and (B_{0},B_{1}) be
Banach couples such that A_{0} is contained in A_{1} and let T
be a possibly nonlinear Lipschitz operator mapping A_{1 }into B_{1},
which also maps A_{0 }into B_{0} boundedly and compactly. It
is known that T maps (A_{0},A_{1})_{s,q} boundedly into
(B_{0},B_{1})_{s,q} for each s satisfying 0<s<1
and each q satisfying 1≤q≤∞, and that this map is also
compact if T is linear. We present examples which show that in general T is not
compact as a map from (A_{0},A_{1})_{s,q} into (B_{0},B_{1})_{s,q}.
However, this map will be compact if we also assume that (B_{0},B_{1})
satisfies A. Persson's approximation condition (H) and if we assume that T
satisfies an appropriate quantitative compactness condition.