On estimate for numerical radius of some contractions
For the numerical radius of an arbitrary nilpotent operator T on a Hilbert space H, Haagerup and de la Harpe proved the inequality \(w(T) \leqslant \left\| T \right\|cos\frac{\pi }{{n + 1}}\), where $n \geq 2$ is the nilpotency order of the operator T. In the present paper, we prove a Haagerup-de la...
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| author | Karaev, M. T. Караєв, М. Т. |
| author_facet | Karaev, M. T. Караєв, М. Т. |
| author_sort | Karaev, M. T. |
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| description | For the numerical radius of an arbitrary nilpotent operator T on a Hilbert space H, Haagerup and de la Harpe proved the inequality \(w(T) \leqslant \left\| T \right\|cos\frac{\pi }{{n + 1}}\), where $n \geq 2$ is the nilpotency order of the operator T. In the present paper, we prove a Haagerup-de la Harpe-type inequality for the numerical radius of contractions from more general classes. |
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UDC 517.97
M. T. Karaev (Inst. Math. and Mech. Azerbaijan Nat. Acad. Sci., Baku, Azerbaijan)
ON ESTIMATE OF NUMERICAL RADIUS
OF SOME CONTRACTIONS
PRO OCINKU ÇYSLOVOHO RADIUSA DEQKYX STYSNEN\
For a numerical radius of an arbitrary nilpotent operator T on a Hilbert space H, Haagerup and de la Harpe
proved the inequality w(T ) ≤ ‖T‖ cos
π
n + 1
, where n ≥ 2 is the nilpotency order of the operator T. In the
present paper, we prove a Haagerup – de la Harpe-type inequality for a numerical radius of contractions from
more general classes.
Xaaherun y Xarp dlq çyslovoho radiusa dovil\noho nil\potentnoho operatora T u hil\bertovomu pros-
tori H dovely nerivnist\ w(T ) ≤ ‖T‖ cos
π
n + 1
, de n ≥ 2 — porqdok nil\potentnosti operatora T.
U danij statti dovedeno nerivnist\ typu nerivnosti Xaaheruna – Xarpa dlq çyslovoho radiusa stysnen\
iz bil\ß zahal\nyx klasiv.
1. Introduction. Let B(H) be an algebra of bounded linear operators acting on a com-
plex Hilbert space H. The numerical range of the operator T ∈ B(H) is called the set
W (T ) =
{
(Tx, x) : x ∈ (H)1
}
,
where (H)1 =
{
x ∈ H : ‖x‖ = 1
}
is a unit sphere in the space H, and the numerical
radius of an operator T is defined by equality
w(T ) = sup
{
|λ| : λ ∈ W (T )
}
.
It is known that
‖T‖
2
≤ w(T ) ≤ ‖T‖ (1)
for any T ∈ B(H) (see, for instance, [1, 2]). For concrete operators, obtaining for their
numerical radius more subtle estimates than (1) is of special interest. So, in paper [3]
Haagerup and de la Harpe proved the inequality
w(T ) ≤ ‖T‖ cos
π
n + 1
(2)
for the numerical radius of an arbitrary nilpotent operator T ∈ B(H) with the power
of nilpotency n ≥ 2 (i.e., Tn = 0, but Tn−1 �= 0). In what follows other proofs of
inequality (2) of Haagerup and de la Harpe are given in the works [4 – 8] (in [7], an
evident description of the numerical range of any quadric operator on a Hilbert space is
also given).
In the present paper, we prove the inequalities of the type of inequality (2) of Haagerup
and de la Harpe for the numerical radius of contractions from the more general classes.
2. Notations and preliminaries. The notations in the paper are more or less standard.
C.0 denotes the class of all contractions T on H, for which limn T ∗n
x = 0 for all x ∈ H.
C00 is the class of all contractions T on H, for which limn T
n
x = limn T ∗n
x = 0 for
all x ∈ H. It is clear that C00 ⊂ C.0. It is well known (see, for instance, [9, 10] ) that if
T ∈ C.0, then it is unitary equivalent to the model operator MΘ,MΘf = PΘzf, f ∈ KΘ,
where PΘ is the orthogonal projection onto KΘ,
KΘ = H2(E) � ΘH2(E′),
c© M. T. KARAEV, 2006
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 10 1335
1336 M. T. KARAEV
where E = clos(I − TT ∗)H,E′ is a subspace of E, H2(E) is the Hardy space of E-
valued analytic functions on the unit circle D = {z ∈ C : |z| < 1} , and Θ is B(E′, E)-
valued bounded analytic function on D (the characteristic function of a contraction T ) for
which Θ(ξ)Θ(ξ)∗ = IE for almost all ξ ∈ T = ∂D, i.e., Θ is a ∗-inner function (see [9]).
In particular, if T ∈ C00, then its characteristic function Θ ∈ B(E) is a two-sided inner
function, that is Θ(ξ)∗Θ(ξ) = Θ(ξ)Θ(ξ)∗ = IE for almost all ξ ∈ T.
The proof of the following key lemma is contained, for instance, in [11].
Lemma 1. Let k ≥ 2 be an integer. Then
max
{
k−2∑
m=0
amam+1 :
k−1∑
m=0
a2
m = 1, am ≥ 0 (m = 0, 1, ..., k − 1)
}
= cos
π
k + 1
.
3. Estimates of a numerical radius. As is notied in [3], inequality (2) is sharp.
However, the following simple example shows that in some cases one can improve in-
equality (2):
Example. Let NS = S(I−SS∗), where S, Sf = zf, is a shift operator on the Hardy
space H2 = H2(D), and let N be a nilpotent operator on a H, with power of nilpotency
n, n ≥ 3, and ‖N‖ <
1
2
. We consider their orthogonal sum:
A = NS ⊕N.
It is clear that N2
S = 0, An = 0 and ‖A‖ = 1. Since W (NS) = D1/2 (see [12, 13]),
w(N) <
1
2
and W (A) is a convex hull of numerical ranges of addendums, we have
W (A) = D1/2, and therefore w(A) =
1
2
< cos
π
n + 1
. The latter shows that for the
numerical radius of the considered nilpotent operator A, a stronger estimate is valid.
Note that in the considered example, the numerical radius of the operator A is achieved
in the element
1 + z√
2
⊕ 0 belonging to ker(A2) (to make sure of that, it is sufficient to
note that |(NSf, f)| =
∣∣∧f(0)
∧
f(1)
∣∣ ≤ 1
2
for any f ∈ (H2)1). This suggests an idea
that conditions of achievability of the numerical radius of the operator on root subspaces
have to play decisive role in obtaining more subtle estimates for a numerical radius of
operators. The theorems stated below affirm such point of view.
The following our result can be also considered as an extension of already mentioned
Haagerup – de la Harpe’s result to some operators from the class C.0 that are not nilpotent
(recall that the unitary equivalance of operators preserves the numerical radius).
Theorem 1. Let Θ be a ∗-inner function (i.e., Θ(ξ)Θ(ξ)∗ = IE for a.a. ξ ∈ T), let
KΘ = H2(E) � ΘH2(E′) (E is some auxillary Hilbert space and E′ ⊂ E) be a model
subspace and let MΘ = PΘz | KΘ be a corresponding model operator. Suppose that for
some n ≥ 2, there exists x ∈ kerMn
Θ ∩ (KΘ)1 such that w(MΘ) =
∣∣(MΘx, x)
∣∣. Then
w(MΘ) ≤ cos
π
n + 1
.
Proof. In fact, the assertion x ∈ kerMn
Θ is equivalent to the assertion
Θ∗x ∈ H2
−(E) ∩ znH2(E′),
that is Θ∗x =
∑n
k=1 akz
k, and since ‖x‖ = 1 and Θ is an ∗-inner function,we have
‖Θ∗x‖ = 1, that is
∑n
k=1 ‖ak‖2 = 1; here, H2
−(E) df= L2(E) � H2(E). Now an
elementary argument together with Lemma 1 yields a required estimate:
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 10
ON ESTIMATE OF NUMERICAL RADIUS OF SOME CONTRACTIONS 1337
w(MΘ) = |(MΘx, x)| = |(zx, x)| =
= |(zΘ∗x,Θ∗x)| =
∣∣∣∣∣
n−1∑
k=1
(ak+1, ak)
∣∣∣∣∣ ≤
≤
n−1∑
k=1
‖ak‖ ‖ak+1‖ ≤ cos
π
n + 1
.
The theorem is proved.
In particular, it follows from this theorem that, for a numerical radius of the nilpotent
model operator MΘ with power of nilpotency n (n ≥ 3) satisfying the condition of
Theorem 1 for some k, 2 ≤ k < n, there is the estimate
w(MΘ) ≤ cos
π
k + 1
,
which is more subtle than the estimate w(MΘ) ≤ cos
π
n + 1
.
There are some more results on this direction.
Proposition 1. Let T ∈ B(H) be a contraction such that, for some integer n > 0,
there exists x ∈ kerTn ∩ (H)1 such that w(T ) =
∣∣(Tx, x)
∣∣. Then w(T ) ≤ cos
π
n + 1
.
Proof. Actually, the assertion of this proposition is a “nonmodel variant” of Theo-
rem 1. Really, for arbitrary r < 1, the operator rT has two-sided inner characteristic
function (because ‖rT‖ < 1, and therefore rT ∈ C00). Therefore, taking into account
that Theorem 1 is also true for the C00-class model operators, we obtain that
w(T ) = w(rT ) ≤ 1
r
cos
π
n + 1
.
It remains to pass to the limit as r → 1.
Proposition 2. Let T ∈ B(H) be a contraction such that, for any ε > 0, there
exists xε ∈ (H)1 such that ‖Tnxε‖ < ε for some n ≥ 2 and w(T ) ≤ |(Txε, xε)| + ε.
Then w(T ) ≤ cos
π
n + 1
.
Proof. In fact, using the Berberian construction [14], H → H0, T → T 0 (T ∈
∈ B(H), T 0 ∈ B(H0)), one can reduce the condition of a proposition to the condition
of Proposition 1. We recall that the space H0 is constructed by the following way:let us
take a quatient-space of l∞(H) of H-valued bounded sequences with respect to the linear
subspace
N(H) df=
{
x = (xm) : L.i.m.(xm, ym) = 0 for all y = (ym) ∈ l∞(H)
}
with the scalar product
(x + N(x), y + N(y)) = L.i.m.(xm, ym)
for x = (xm), y = (ym), where “L.i.m.” is generalized Banach limit on l∞. Then H0 is
the completion of the last prehilbertian space. Here, T 0 is defined as
T 0x = (Txn + N(H)).
It is well known (and easily verified) that
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 10
1338 M. T. KARAEV
‖T‖ =
∥∥T 0
∥∥ , σ(T ) = σ(T 0), w(T ) = w(T 0). (3)
Now pass to the reducing the condition of proposition to the condition of Proposition 1.
Choose the sequence of positive numbers εm tending to zero and a sequence of vectors
xεm satisfying the condition of proposition forε = εm such that
∣∣(Txεm , xεm)
∣∣ → w(T ).
Then for x = (xεm
) we have∣∣(T 0x, x)
∣∣ = |L.i.m.(Txεm
, xεm
)| =
= |lim(Txεm , xεm)| = lim |(Txεm , xεm)| ≥ w(T ) = w(T 0).
Since the inequality
∣∣(T 0x, x)
∣∣ ≤ w(T 0) is always true, it implies the equality w(T 0) =
=
∣∣(T 0x, x)
∣∣. Moreover,
((Tn)0x, (Tn)0x) = L.i.m.(Tnxεm , Tnxεm) =
= L.i.m. ‖Tnxεm‖2 ≤ L.i.m.ε2
m = 0,
that is, (Tn)0x = 0, whence subject to (Tn)0 = (T 0)n we get (T 0)nx = 0. Thus,
(T 0)nx = 0 and w(T 0) =
∣∣(T 0x, x)
∣∣ . Considering equality (3) and Proposition 1, we
get the desired inequality
w(T ) ≤ cos
π
n + 1
.
Proposition 3. Let Θ1, Θ2 be an inner functions. We consider the following ope-
rator:
KΘ1,Θ2
df=
[
TΘ̄1
, TΘ̄2
]
Θ1(MΘ2),
where
[
TΘ̄1
, TΘ2
]
= TΘ̄1
TΘ2 − TΘ2TΘ̄1
is the commutator of antianalytic Toeplitz ope-
rator TΘ̄1
and analytic Toeplitz operator TΘ2 , and Θ1(MΘ2) = PΘ2Θ1 | KΘ2 is the
function of model operator MΘ. Then w(KΘ1,Θ2) ≤
1
2
.
Proof. The simple calculations show that the operator KΘ1,Θ2 is the projection of the
operator TΘ̄1
NΘ2TΘ1 on a subspace KΘ2 = H2 � ΘH2, i.e.,
KΘ1,Θ2 = PΘ2(TΘ̄1
NΘ2TΘ1) | KΘ2 ,
where NΘ2
df= TΘ2PΘ2 = TΘ2(I − TΘ2TΘ̄2
). In fact, considering that
∣∣Θ2(ξ)
∣∣ = 1 a.a.
ξ ∈ T (i.e., TΘ2 is an isometry ), for each f ∈ KΘ2 we have
PΘ2(TΘ̄1
NΘ2TΘ1)f = PΘ2TΘ̄1
TΘ2PΘ2Θ1f =
= (I − TΘ2TΘ̄2
)TΘ̄1
TΘ2Θ1(MΘ2)f =
= (TΘ̄1
TΘ2 − TΘ2TΘ̄2
TΘ̄1
TΘ2)Θ1(MΘ2)f =
= (TΘ̄1
TΘ2 − TΘ2TΘ2Θ1 Θ2
)Θ1(MΘ2)f =
= (TΘ̄1
TΘ2 − TΘ2TΘ̄1
)Θ1(MΘ2)f =
= [TΘ̄1
, TΘ2 ]Θ1(MΘ2)f = KΘ1,Θ2f.
Then for any x ∈ (KΘ2)1 we have
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 10
ON ESTIMATE OF NUMERICAL RADIUS OF SOME CONTRACTIONS 1339
(KΘ1,Θ2x, x) = (PΘ2(TΘ̄1
NΘ2TΘ1)x, x) = (NΘ2Θ1x,Θ1x).
Taking into account that Θ1x ∈ (H2)1, N2
Θ2
= 0, and
∥∥NΘ2
∥∥ = 1, we get the following
relation from the last equality and inequality (2):
w(KΘ1,Θ2) ≤ cos
π
3
=
1
2
.
The proof is completed.
I am grateful to Yu. V. Turowski for useful discussions.
1. Halmos P. R. A Hilbert space problem book. – New York: Springer, 1982.
2. Gustafson K. E., Rao D. K. M. Numerical range. The field of values of linear operators and matrices. –
New York: Springer, 1997.
3. Haagerup U., de la Harpe P. The numerical radius of a nilpotent operator on a Hilbert space // Proc. Amer.
Math. Soc. – 1992. – 115. – P. 371 – 379.
4. Pop C. On a result of Haagerup and de la Harpe // Rev. roum. math. pures et appl. – 1998. – 93. – P. 9 – 10.
5. Badea C, Cassier G. Constrained von Neumann inequalities // Adv. Math. – 2002. – 166. – P. 260 – 297.
6. Suen C. Y. WA contactions // Positivity. – 1998. – 2. – P. 301 – 310.
7. Karaev M. T. The numerical range of a nilpotent operator on a Hilbert space // Proc. Amer. Math. Soc. –
2004. – 132, # 8. – P. 2321 – 2326.
8. Karaev M. T. New proofs of de la Haagerup – Harpe’s inequality // Mat. Zametki. – 2004. – 75, # 5. –
P. 787 – 788.
9. Szökefalvi-Nagy B., Foias C. Harmonic analysis of operators on Hilbert space. – Amsterdam: North-
Holland, 1970.
10. Nikolski N. K. Treatise on the shift operator. – New York: Springer, 1986.
11. Polya G., Szego G. Problems and theorems in analysis. – Berlin: Springer, 1976. – Vol 2.
12. Karaev M. T. On numerical characteristics of some operators associated with isometries // Spectral Theory
Operators and Appl. – 1997. – 11. – P. 90 – 98.
13. Tso S-H., Wu P. Y. Matrical ranges of quadratic operators // Rocky Mountain J. Math. – 1999. – 29. –
P. 1139 – 1152.
14. Berberian S. K. Approximate proper vectors // Proc. Amer. Math. Soc. – 1962. – 13. – P. 111 – 114.
Received 05.11.2003,
after revision — 03.05.2006
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 10
|
| id | umjimathkievua-article-3536 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:44:21Z |
| publishDate | 2006 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/1b/c49a658de896e1dd96170d1f19c9eb1b.pdf |
| spelling | umjimathkievua-article-35362020-03-18T19:57:10Z On estimate for numerical radius of some contractions Про оцінку числового радіуса деяких стиснень Karaev, M. T. Караєв, М. Т. For the numerical radius of an arbitrary nilpotent operator T on a Hilbert space H, Haagerup and de la Harpe proved the inequality \(w(T) \leqslant \left\| T \right\|cos\frac{\pi }{{n + 1}}\), where $n \geq 2$ is the nilpotency order of the operator T. In the present paper, we prove a Haagerup-de la Harpe-type inequality for the numerical radius of contractions from more general classes. Хаагерун и Харн для числового радіуса довільного нільпотентного оператора $T$ у гільбертовому простторі $H$ довели нерівність $w(T) \leq ||T||\, \cos\cfrac{\pi}{n+1}$, де $n \geq 2$ — порядок нільпотентності оператора $T$. У даній статті доведено нерівність типу нерівності Хаагеруна-Харпа для числового радіуса стиснень із більш загальних класів. Institute of Mathematics, NAS of Ukraine 2006-10-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/3536 Ukrains’kyi Matematychnyi Zhurnal; Vol. 58 No. 10 (2006); 1335–1339 Український математичний журнал; Том 58 № 10 (2006); 1335–1339 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/3536/3808 https://umj.imath.kiev.ua/index.php/umj/article/view/3536/3809 Copyright (c) 2006 Karaev M. T. |
| spellingShingle | Karaev, M. T. Караєв, М. Т. On estimate for numerical radius of some contractions |
| title | On estimate for numerical radius of some contractions |
| title_alt | Про оцінку числового радіуса деяких стиснень |
| title_full | On estimate for numerical radius of some contractions |
| title_fullStr | On estimate for numerical radius of some contractions |
| title_full_unstemmed | On estimate for numerical radius of some contractions |
| title_short | On estimate for numerical radius of some contractions |
| title_sort | on estimate for numerical radius of some contractions |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/3536 |
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