On Stability of Polynomially Bounded Operators
We prove that if T is a polynomially bounded operator and the peripheral spectrum of T has zero measure, then Tⁿx → 0 for all x in X if and only if T* has no nontrivial invariant subspace on which it is invertible and doubly power bounded.
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Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
2007
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irk-123456789-1064472016-09-29T03:02:17Z On Stability of Polynomially Bounded Operators Muraz, G. Quoc Phong Vu We prove that if T is a polynomially bounded operator and the peripheral spectrum of T has zero measure, then Tⁿx → 0 for all x in X if and only if T* has no nontrivial invariant subspace on which it is invertible and doubly power bounded. 2007 Article On Stability of Polynomially Bounded Operators азвание / G. Muraz, Quoc Phong Vu // Журнал математической физики, анализа, геометрии. — 2007. — Т. 3, № 2. — С. 234-240. — Бібліогр.: 13 назв. — англ. 1812-9471 http://dspace.nbuv.gov.ua/handle/123456789/106447 en Журнал математической физики, анализа, геометрии Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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We prove that if T is a polynomially bounded operator and the peripheral spectrum of T has zero measure, then Tⁿx → 0 for all x in X if and only if T* has no nontrivial invariant subspace on which it is invertible and doubly power bounded. |
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Article |
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Muraz, G. Quoc Phong Vu |
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Muraz, G. Quoc Phong Vu On Stability of Polynomially Bounded Operators Журнал математической физики, анализа, геометрии |
| author_facet |
Muraz, G. Quoc Phong Vu |
| author_sort |
Muraz, G. |
| title |
On Stability of Polynomially Bounded Operators |
| title_short |
On Stability of Polynomially Bounded Operators |
| title_full |
On Stability of Polynomially Bounded Operators |
| title_fullStr |
On Stability of Polynomially Bounded Operators |
| title_full_unstemmed |
On Stability of Polynomially Bounded Operators |
| title_sort |
on stability of polynomially bounded operators |
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Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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2007 |
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http://dspace.nbuv.gov.ua/handle/123456789/106447 |
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On Stability of Polynomially Bounded Operators азвание / G. Muraz, Quoc Phong Vu // Журнал математической физики, анализа, геометрии. — 2007. — Т. 3, № 2. — С. 234-240. — Бібліогр.: 13 назв. — англ. |
| series |
Журнал математической физики, анализа, геометрии |
| work_keys_str_mv |
AT murazg onstabilityofpolynomiallyboundedoperators AT quocphongvu onstabilityofpolynomiallyboundedoperators |
| first_indexed |
2025-07-07T18:30:19Z |
| last_indexed |
2025-07-07T18:30:19Z |
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1837013946401292288 |
| fulltext |
Journal of Mathematical Physics, Analysis, Geometry
2007, vol. 3, No. 2, pp. 234�240
On Stability of Polynomially Bounded Operators
G. Muraz
Institut Fourier, B.P. 74, 38402 St-Martin-d'H�eres Cedex, France
E-mail:Gilbert.Muraz@ujf-grenoble.fr
Quoc Phong Vu
Department of Mathematics, Ohio University
Athens, OH 45701, USA
E-mail:qvu@math.ohiou.edu
Received December 26, 2005
We prove that if T is a polynomially bounded operator and the peripheral
spectrum of T has zero measure, then Tnx! 0 for all x in X if and only if
T
� has no nontrivial invariant subspace on which it is invertible and doubly
power bounded.
Key words: polynomially bounded operator, Banach space, invariant sub-
space.
Mathematics Subject Classi�cation 2000: 47A15.
1. Introduction
Let X be a Banach space. A linear bounded operator T on X is called poly-
nomially bounded if there exists a constant M such that
kp(T )k �M sup
jzj�1
kp(z)k; (1)
for every polynomial p.
It is a well known theorem of Sz. Nagy and C. Foias [8] that if X is a Hilbert
space and T is a completely nonunitary contraction on X with spectrum �(T )
such that m(�(T )\�) = 0, where � denotes the unit circle and m is the Lebesgue
measure on �, then kT nxk ! 0 as n ! 1, for all x in X. According to von
Neumann's inequality (see e.g. [8]), every contraction operator T satis�es (1)
with M = 1, hence every contraction is a power bounded operator. However,
G. Pisier [9] has shown that not every polynomially bounded operator on a Hilbert
space is similar to a contraction. The proof of the above result of Sz. Nagy and
C. Foias uses the theory of unitary dilations of contractions and, therefore, cannot
be extended to polynomially bounded operators on a Hilbert space.
c
G. Muraz and Q.Ph. Vu, 2007
On Stability of Polynomially Bounded Operators
In this note, we extend the Nagy�Foias theorem to polynomially bounded
operators on Banach spaces.
Throughout the paper, D is the open unit dist, � is the unit circle and A(D)
is the disk algebra of functions analytic in D and continuous in D.
2. The Limit Isometry
Let T be a power bounded operator on a Banach space X, i.e., T satis�es
the condition supn�0 kT
nk < 1. By introducing the equivalent norm kjxkj =
sup
n�0 kT
nxk, we can always assume, without loss of generality, that T is a con-
traction. This implies that limn!1 kT nxk exists for all x in X.
The following construction associates with T another Banach space E, a natu-
ral homomorphism Q from X to E and an isometry V on E such that QT = V Q
and �(V ) � �(T ). This construction has proved useful in many investigations on
the asymptotic behavior of semigroups of operators (see [2, 7, 10�13]).
Lemma 1. Let T be a power bounded on a Banach space X. There exists
a Banach space E, a bounded linear map Q of X into E with dense range, and
an isometric operator V on E, with the following properties:
1) Qx = 0 if and only if infn�0 kT
nxk = 0;
2) QT = V Q (s 2 S);
3) �(V ) � �(T ); P�(V �) � P�(T �).
The operator V in Lem. 1 is called the limit isometry of T . Recall the con-
struction of E, Q and V . First, a seminorm on X is de�ned by
l(x) = lim
n!1
kT nxk; x 2 X:
Let L = ker(l) = fx 2 X : l(x) = 0g. Consider the quotient space bX = X=L,
the canonical homomorphism Q : X ! bX; Qx = x̂, and de�ne a norm in bX by
l̂(x̂) = l(x); x 2 X:
The operators T generate the corresponding operator bT on bX in the natural way,
namely bT x̂ := cTx; x 2 X:
Clearly, bT is an isometric operator on the normed space bX , since
l̂( bT x̂) = lim
n!1
kT n(Tx)k = l̂(x̂); x 2 X:
We denote by E the completion of bX in the norm l̂, and by V the continuous
extension of bT from bX to E. All properties 1)�3) can be veri�ed directly.
Journal of Mathematical Physics, Analysis, Geometry, 2007, vol. 3, No. 2 235
G. Muraz and Q.Ph. Vu
An operator T is called stable, if the discrete semigroup fT ngn�0 is stable,
i.e., limn!1 kT nxk = 0 for all x 2 X. Note that in the above construction
the subspace E is nonzero if and only if T is nonstable. On the other hand, if
infn�0 kT
nxk > 0 for all x 2 X; x 6= 0, then T is said to be of class C1. From
�(V ) � �(T ) it follows that if �(T ) does not contain the unit circle, then �(V )
also does not contain the unit circle, so that V is an invertible isometry.
3. Stability of fT ng
An important property of polynomially bounded invertible isometries is that
they possess a functional calculus for continuous functions on their spectra.
Lemma 2. Let V be a polynomially bounded invertible isometry on a Banach
space E. Then the algebra A(V ) is isomorphic to C(�(V )).
P r o o f. It was shown in [6] that there is a homomorphism ' : C(�)! L(E)
such that k'k � M , i.e., there is a functional calculus on C(�) which satis�es:
kf(T )k � Mkfk1. Moreover, f(T ) is completely determined by its values on
�(V ), and the spectral mapping theorem holds: �(f(V )) = f(�(V )). Therefore,
the functional calculus can be de�ned for C(�(V )), and we have
sup
�2�(V )
jf(�)j � kf(V )k �M sup
�2�(V )
jf(�)j;
i.e., the homomorphism is in fact an isomorphism.
Now let T be a polynomially bounded operator on a Banach space X. Assume
that T is not stable, i.e., there exists x 2 X such that kT nxk does not converge
to 0. Then the Banach space E, de�ned in Lemma 1, is nonzero, and we can
speak about the limit isometry V . Assume that V is invertible (which holds, e.g.,
if T has a dense range or �(T ) does not contain the whole unit circle).
Lemma 3. Let T be polynomially bounded, nonstable, and let E and V be as
in Lemma 1 such that V is an invertible isometry. Then there exists a family of
measures �z;z�, where z 2 E; z� 2 E�, such that
hf(V )z; z�i =
Z
�(V )
f(�)d�z;z�(�) (2)
for every function f in C(�(V )).
P r o o f. Since T also is polynomially bounded, it follows easily that V also
is polynomially bounded. In fact, we have
l̂(p( bT )x̂) = limn!1 kT np(T )xk � kp(T )k limn!1 kT nxk
= kp(T )kl̂(x̂) �M supjzj�1 jp(z)jl̂(x̂);
236 Journal of Mathematical Physics, Analysis, Geometry, 2007, vol. 3, No. 2
On Stability of Polynomially Bounded Operators
which implies kp( bT )k � M supjzj�1 jp(z)j, hence kp(V )k � M supjzj�1 jp(z)j,
i.e., V is polynomially bounded. Lemma 2 implies that A(V ) is isomorphic to
C(�(V )). Therefore, for each z 2 E; z� 2 E�, the mapping f 7! hf(V )z; z�i is
a continuous linear functional on C(�(V )). Hence, by Riesz's theorem, for every
z 2 E; z 2 E�, there exists a measure �z;z� on �(V ) such that (2) holds.
Note that, in general, V does not have a spectral measure, i.e., it is not a spec-
tral operator in the sense of N. Dunford [4]. But formula (2), which resembles
the functional calculus for spectral operators of scalar type and holds in our case
only for continuous functions f on the spectrum of V , will be one of the main
ingredients in the proof of Lemma 5 below.
Lemma 4. Suppose that T is a polynomially bounded operator on a Banach
space X. Then for every function f 2 A(D) one can de�ne a bounded linear
operator f(T ) on X such that:
1) If f = 1, then f(T ) = I;
2) If f(�) = �, then f(T ) = T ;
3) The mapping f 7! f(T ) is an algebra homomorphism from A(D) into L(X)
satisfying kf(T )k �Mkfk1.
The proof of Lemma 4 is straightforward. In fact, we �rst de�ne f(T ) for
polynomials f in the standard way. Then, using von Neumann's inequality, we
can extend this de�nition to the functions of the class A(D) using approximations.
In the sequel, an invertible operator S on X is called doubly power bounded
provided that both S and S�1 are power bounded, i.e., if supn2Z kS
nk <1. It is
easy to see that if S is doubly power bounded, then S is an (invertible) isometry
in the equivalent norm kjxkj = supn2Z kS
nxk, x 2 X.
Lemma 5. Assume that:
1) T is polynomially bounded operator on a Banach space X.
2) There does not exist an invariant subspace K with respect to T � such that
T �jK is invertible and doubly power bounded.
Then the measures �z;z� are absolutely continuous with respect to the Lebesgue
measure.
P r o o f. Assuming the contrary, i.e., there exist z 2 E, z� 2 E� such that
�z;z� is not absolutely continuous with respect to the Lebesgue measure m. This
implies that there exists a compact set K with m(K) = 0 and �z;z�(K) 6= 0.
By Fatou's theorem (see e.g. [6, p. 80]), there exists a function h 2 A(D)
such that
h(�) = 1; if � 2 K and jh(�)j < 1 if � 2 D nK < 1: (3)
Journal of Mathematical Physics, Analysis, Geometry, 2007, vol. 3, No. 2 237
G. Muraz and Q.Ph. Vu
Let ~h(�) := h(��). Then ~h 2 A(D); k~hk = 1. Since V � also is a polynomially
bounded invertible isometry, ~hn(V �) is de�ned and satis�es
sup
n�0
k~hn(V �)k �M <1: (4)
Fix a nonzero functional z� in E�. By (4) and the weak� compactness of the unit
ball in E�, there exists a subsequence nk such that ~hnk(V �)z� ! z�0 in the (E�; E)-
topology. De�ne two functionals x� and x�0 in E� by
x�(x) = z�(x̂); x�0(x) = z�0(x̂); x 2 X: (5)
Then, for every vector x in X, (5) implies that
(~hnk(T �)x�)(x) = x�(hnk(T )x) = z�( \(hnk(T )x))
= z�(hnk(V )x̂) = (~hnk(V �)z�)(x̂):
Therefore,
limk!1(hnk(T �)x�)(x) = limk!1(~hnk(V �)z�)(x̂)
= z�0(x̂) = x�0(x);
i.e., ~hnk(T �)x� converges to x�0 in the (X;X�)-topology. Now we have, by adopting
(4)�(6) and the Dominated Convergence Theorem,
x�0(y) = lim
k!1
(~hnk(T �)x�)(y)
= lim
k!1
x�0(h
nk(T )y) = lim
k!1
z�(hnk(V )ŷ)
= lim
k!1
�Z
��
hnk(ei�)d�ŷ;z�(�) = �ŷ;z�(K):
Since �z;z�(K) 6= 0, and bX is dense in E, there exists ŷ such that �ŷ;z�(K) 6= 0,
so that x�0 6= 0.
By Rudin�Carleson's theorem (see e.g. [6, p. 80]), there exists a function
� 2 A(D) such that
�(ei�) = e�i� for � 2 K and k�k1 = 1: (6)
We show that
T ��(T �)x�0 = x�0: (7)
238 Journal of Mathematical Physics, Analysis, Geometry, 2007, vol. 3, No. 2
On Stability of Polynomially Bounded Operators
Indeed, we have, in view of (4)�(6),
([I � T ��(T �)]x�0)(y) = x�0([I � T�(T )]y)
= z�0([I � V �(V )]ŷ) = limk!1
~[hnk(V �)z�]([I � V �(V )]ŷ)
= limk!1 z�(hnk(V )[I � V �(V )]ŷ)
= limk!1
�R
��
hnk(ei�)(1� ei��(ei�))d�ŷ;z�(�)
=
R
K
(1� ei��(ei�))d�ŷ;z�(�) = 0; for all y 2 X;
which implies that (7) holds.
Now let W := �(T �). Then (6) and (7) imply that sup
n�0 kW
nk � M , and
(WT �)n(T �)kx�0 = (T �)kx�0, k = 0; 1; 2; : : : . Let K := spanf(T �)kx�0 : k � 0g.
ThenK is invariant subspace for T �, T �jK is invertible (with the inverse equalW )
and supn2Z k(T
�jK)nk �M , which is a contradiction.
R e m a r k. Lemma 5 has been proved in [10, Prop. 2.1], for contractions on
Hilbert space, and here we generalized this proof.
From Lemma 5 we obtain the following result which is a generalization of
the Nagy�Foias theorem.
Theorem 1. Let T be a polynomially bounded operator on a Banach space X
such that �(T ) \ � has measure 0. Then the following are equivalent:
(i) T nx! 0 for every x 2 X;
(ii) T � does not have an invariant subspace K 6= f0g on which T �jK is
invertible and doubly power bounded.
P r o o f. Since �(T ) \ � has measure zero, it follows that �(V ) \ � also has
measure zero, hence V is an invertible isometry.
Suppose that (ii) holds, we show that (i) holds. Assuming the contrary, we
have E 6= f0g. By Lemma 5, the measures mz;z�; z 2 E; z� 2 E�; are absolutely
continuous with respect to the Lebesgue measure. From m(�(V )) = 0 it follows
that mz;z�(�(V )) = 0, i.e., all the measures mz;z� are zero, which is an absurd.
Now suppose that (i) holds but (ii) does not hold. Thus, there is a nonzero
subspace K of X� which is invariant under T � and such that T �jK is invertible
and supn2Z k(T
�jK)nk < 1. Let S = T �jK. Fix an element x� in K; x� 6= 0.
Then fS�nx� : n � 0g are uniformly bounded, hence x�(x) = (Sn(S�n)x�)(x) =
[(T �)nS�nx�](x) = [S�nx�](T nx) ! 0; for all x 2 X, which is a contradiction.
Note that Theorem 1 can be regarded as an analogue of the stability results in
[1, 7, 10] (see also [11�13])where the condition that m(�(T ) \ �) = 0 is replaced
Journal of Mathematical Physics, Analysis, Geometry, 2007, vol. 3, No. 2 239
G. Muraz and Q.Ph. Vu
by countability of �(T ) \ �, and condition
T � does not have an invariant subspace K 6= f0g
such that T �jK is invertible and doubly power bounded
is replaced by
T �does not have eigenvalues on the unit circle.
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