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Publié par | profil-urra-2012 |
Nombre de lectures | 17 |
Langue | English |
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Quasi-hyperbolic semigroups
Yuri Tomilov (joint work with C. Batty (Oxford))
IM PAN, Warsaw and Nicholas Copernicus University, Torun´
Lille, 31 May, 2010
Yuri Tomilov (IM PAN, Warsaw and Nicholas CopernicusQuasi-hUniveryperbolic sity,semigrTorun)´ oups Lille, 31 May, 2010 1 / 20The class of contraction (power bounded) operators T (or operator
semigroups) on X :
nkTk 1 (supkT k = c <1:)
n0
is comparatively well-understood.
Our aim is to try to understand an opposite class of expansion
operators (operator semigroups) satisfying
nkT xk ckxk
at least in a certain sense to be made precise.
Let T be a bounded operator on a Banach space X:
Yuri Tomilov (IM PAN, Warsaw and Nicholas CopernicusQuasi-hUniveryperbolic sity,semigrTorun)´ oups Lille, 31 May, 2010 2 / 20Our aim is to try to understand an opposite class of expansion
operators (operator semigroups) satisfying
nkT xk ckxk
at least in a certain sense to be made precise.
Let T be a bounded operator on a Banach space X:
The class of contraction (power bounded) operators T (or operator
semigroups) on X :
nkTk 1 (supkT k = c <1:)
n0
is comparatively well-understood.
Yuri Tomilov (IM PAN, Warsaw and Nicholas CopernicusQuasi-hUniveryperbolic sity,semigrTorun)´ oups Lille, 31 May, 2010 2 / 20Let T be a bounded operator on a Banach space X:
The class of contraction (power bounded) operators T (or operator
semigroups) on X :
nkTk 1 (supkT k = c <1:)
n0
is comparatively well-understood.
Our aim is to try to understand an opposite class of expansion
operators (operator semigroups) satisfying
nkT xk ckxk
at least in a certain sense to be made precise.
Yuri Tomilov (IM PAN, Warsaw and Nicholas CopernicusQuasi-hUniveryperbolic sity,semigrTorun)´ oups Lille, 31 May, 2010 2 / 20In other words,
n n n nkT xk Ckxk (x2 X ; n2 ); kT xk c kxk (x2 X ; n2 )s u
where< 1 and > 1:
n n n nFor non-zero x2 X eitherkT xk c orkT xk C (n2 ):x x
T is hyperbolic if and only if(T )\ = ; ( is the unit circle).
Hyperbolic operators
Definition A bounded linear operator T on a Banach space X is said
to be hyperbolic if
X = X X ;s u
where X and X are closed T -inv. subspaces of X, T is invertible,s u Xu
and
1 1n nk(T )k ; k(T ) k for some n2 :X Xs u2 2
Yuri Tomilov (IM PAN, Warsaw and Nicholas CopernicusQuasi-hUniveryperbolic sity,semigrTorun)´ oups Lille, 31 May, 2010 3 / 20NNNNT is hyperbolic if and only if(T )\ = ; ( is the unit circle).
Hyperbolic operators
Definition A bounded linear operator T on a Banach space X is said
to be hyperbolic if
X = X X ;s u
where X and X are closed T -inv. subspaces of X, T is invertible,s u Xu
and
1 1n nk(T )k ; k(T ) k for some n2 :X Xs u2 2
In other words,
n n n nkT xk Ckxk (x2 X ; n2 ); kT xk c kxk (x2 X ; n2 )s u
where< 1 and