Quasi hyperbolic semigroups

Nicholas Copernicus

Description

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 Copernicus University, Torun)Quasi-hyperbolic semigroups Lille, 31 May, 2010 1 / 20

• ?t? ≤


• ?t nx?


• expansion operators


• quasi-hyperbolic semigroups


• bounded linear operator


• made precise

Sujets

• Copernicus
• Bounded operator

Informations

• Publié par: profil-urra-2012
• Nombre de lectures: 17
• Langue: English

Extrait

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