CLARKSON-MCCARTHY INTERPOLATED
INEQUALITIES IN FINSLER NORMS
Clarkson-McCarthy Interpolated
Inequalities
CRISTIAN CONDE
Instituto de Ciencias
Universidad Nacional de General Sarmiento
J. M. Gutierrez 1150, (1613) Los Polvorines
Buenos Aires, Argentina.
Cristian Conde
vol. 10, iss. 1, art. 4, 2009
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EMail: cconde@ungs.edu.ar
Contents
Received:
29 August, 2008
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Accepted:
14 January, 2009
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Communicated by:
R. Bhatia
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2000 AMS Sub. Class.:
Primary 46B70, 47A30; Secondary 46B20, 47B10.
Key words:
p-Schatten class, Complex method, Clarkson-McCarthy inequalities.
Abstract:
We apply the complex interpolation method to prove that, given two spaces
(n)
(n)
Bp0 ,a;s0 , Bp1 ,b;s1 of n-tuples of operators in the p-Schatten class of a Hilbert
space H, endowed with weighted norms associated to positive and invertible op(n)
(n)
erators a and b of B(H) then, the curve of interpolation (Bp0 ,a;s0 , Bp1 ,b;s1 )[t]
of the pair is given by the space of n-tuples of operators in the pt -Schatten
class of H, with the weighted norm associated to the positive invertible element
γa,b (t) = a1/2 (a−1/2 ba−1/2 )t a1/2 .
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Contents
1
Introduction
3
2
Geometric Interpolation
6
3
Clarkson-Kissin Type Inequalities
12
Clarkson-McCarthy Interpolated
Inequalities
Cristian Conde
vol. 10, iss. 1, art. 4, 2009
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1.
Introduction
In [6], J. Clarkson introduced the concept of uniform convexity in Banach spaces
and obtained that spaces Lp (or lp ) are uniformly convex for p > 1 throughout the
following inequalities
2 kf kpp + kgkpp ≤ kf − gkpp + kf + gkpp ≤ 2p−1 kf kpp + kgkpp ,
Let (B(H), k · k) denote the algebra of bounded operators acting on a complex
and separable Hilbert space H, Gl(H) the group of invertible elements of B(H) and
Gl(H)+ the set of all positive elements of Gl(H).
If X ∈ B(H) is compact we denote by {sj (X)} the sequence of singular values
of X (decreasingly ordered). For 0 < p < ∞, let
kXkp =
X
p
sj (X)
p1
Cristian Conde
vol. 10, iss. 1, art. 4, 2009
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Contents
,
and the linear space
Bp (H) = {X ∈ B(H) : kXkp < ∞}.
For 1 ≤ p < ∞, this space is called the p−Schatten class of B(H) (to simplify
notation we use Bp ) and by convention kXk = kXk∞ = s1 (X). A reference for
this subject is [9].
C. McCarthy proved in [14], among several other results, the following inequalities for p-Schatten norms of Hilbert space operators:
(1.1)
Clarkson-McCarthy Interpolated
Inequalities
2(kAkpp + kBkpp ) ≤ kA − Bkpp + kA + Bkpp
≤ 2p−1 (kAkpp + kBkpp ),
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for 2 ≤ p < ∞, and
(1.2)
2p−1 (kAkpp + kBkpp ) ≤ kA − Bkpp + kA + Bkpp
≤ 2(kAkpp + kBkpp ),
for 1 ≤ p ≤ 2.
These are non-commutative versions of Clarkson’s inequalities. These estimates
have been found to be very powerful tools in operator theory (in particular they imply
the uniform convexity of Bp for 1 < p < ∞) and in mathematical physics (see [16]).
M. Klaus has remarked that there is a simple proof of the Clarkson-McCarthy inequalities which results from mimicking the proof that Boas [4] gave of the Clarkson
original inequalities via the complex interpolation method.
In a previous work [7], motivated by [1], we studied the effect of the complex
(n)
interpolation method on Bp (this set will be defined below) for p, s ≥ 1 and n ∈ N
with a Finsler norm associated with a ∈ Gl(H)+ :
kXkp,a;s := ka−1/2 Xa−1/2 ksp .
From now on, for the sake of simplicity, we denote with lower case letters the elements of Gl(H)+ .
As a by-product, we obtain Clarkson type inequalities using the Klaus idea with
(n)
(n)
the linear operator Tn : Bp −→ Bp given by
!
n
n
n
X
X
X
n−1
1
Tn (X̄) = (Tn (X1 , . . . , Xn ) =
Xj ,
θj Xj , . . . ,
θj Xj ,
j=1
j=1
j=1
where θ1 , . . . , θn are the n roots of unity.
Recently, Kissin in [12], motivated by [3], obtained analogues of the ClarksonMcCarthy inequalities for n-tuples of operators from Schatten ideals. In this work
Clarkson-McCarthy Interpolated
Inequalities
Cristian Conde
vol. 10, iss. 1, art. 4, 2009
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the author considers H n , the orthogonal sum of n copies of the Hilbert space H, and
each operator R ∈ B(H n ) can be represented as an n × n block-matrix operator
(n)
(n)
R = (Rjk ) with Rjk ∈ B(H), and the linear operator TR : Bp → Bp is defined
by TR (A) = RA. Finally we remark that the works [3] and [11] are generalizations
of [10].
In these notes we obtain inequalities for the linear operator TR in the Finsler norm
k·kp,a;s as by-products of the complex interpolation method and Kissin’s inequalities.
Clarkson-McCarthy Interpolated
Inequalities
Cristian Conde
vol. 10, iss. 1, art. 4, 2009
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2.
Geometric Interpolation
We follow the notation used in [2] and we refer the reader to [13] and [5] for details on the complex interpolation method. For completeness, we recall the classical
Calderón-Lions theorem.
Theorem 2.1. Let X and Y be two compatible couples. Assume that T is a linear
operator from Xj to Yj bounded by Mj , j = 0, 1. Then for t ∈ [0, 1]
Clarkson-McCarthy Interpolated
Inequalities
Cristian Conde
kT kX[t] →Y[t] ≤ M01−t M1t .
vol. 10, iss. 1, art. 4, 2009
Here and subsequently, let 1 ≤ p < ∞, n ∈ N, s ≥ 1, a ∈ Gl(H)+ and
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Bp(n) = {A = (A1 , . . . , An )t : Ai ∈ Bp },
Contents
(where with t we denote the transpose of the n-tuple) endowed with the norm
kAkp,a;s = (kA1 ksp,a + · · · + kAn ksp,a )1/s ,
and Cn endowed with the norm
s 1/s
|(a0 , . . . , an−1 )|s = (|a0 | + · · · + |an−1 | )
.
(n)
the space Bp
From now on,
endowed with the norm k(·, . . . , ·)kp,a;s .
From Calderón-Lions interpolation theory we get that for p0 , p1 , s0 , s1 ∈ [1, ∞)
(n)
(n)
(n)
= Bpt ,1;st ,
(2.1)
Bp0 ,1;s0 , Bp1 ,1;s1
[t]
where
1
1−t
t
=
+
pt
p0
p1
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s
(n)
we denote with Bp,a;s
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and
1
1−t
t
=
+ .
st
s0
s1
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Note that for p = 2, (1.1) and (1.2) both reduce to the parallelogram law
2(kAk22 + kBk22 ) = kA − Bk22 + kA + Bk22 ,
while for the cases p = 1, ∞ these inequalities follow from the triangle inequality
for B1 and B(H) respectively. Then the inequalities (1.1) and (1.2) can be proved
(for n = 2 via Theorem 2.1) by interpolation between the previous elementary cases
(2)
(2)
with the linear operator T2 : Bp,1;p −→ Bp,1;p , T2 (A) = (A1 + A2 , A1 − A2 )t as
observed by Klaus.
In this section, we generalize (2.1) for the Finsler norms k(·, . . . , ·)kp,a;s . In [7],
we have obtained this extension for the particular case when p0 = p1 = p and
s0 = s1 = s. For sake of completeness, we recall this result
Clarkson-McCarthy Interpolated
Inequalities
Cristian Conde
vol. 10, iss. 1, art. 4, 2009
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Theorem 2.2 ([7, Th. 3.1]). Let a, b ∈ Gl(H)+ , 1 ≤ p, s < ∞, n ∈ N and
t ∈ (0, 1). Then
(n)
(n)
(n)
Bp,a;s
, Bp,b;s
= Bp,γa,b (t);s ,
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where γa,b (t) = a1/2 (a−1/2 ba−1/2 )t a1/2 .
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Remark 1. Note that when a and b commute the curve is given by γa,b (t) = a1−t bt .
The previous corollary tells us that the interpolating space, Bp,γa,b (t);s can be regarded
as a weighted p-Schatten space with weight a1−t bt (see [2, Th. 5.5.3]).
We observe that the curve γa,b looks formally equal to the geodesic (or shortest
curve) between positive definitive matrices ([15]), positive invertible elements of
a C ∗ -algebra ([8]) and positive invertible operators that are perturbations of the pSchatten class by multiples of the identity ([7]).
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[t]
(n)
There is a natural action of Gl(H) on Bp , defined by
(2.2)
l : Gl(H) × Bp(n) −→ Bp(n) ,
lg (A) = (gA1 g ∗ , . . . , gAn g ∗ )t .
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(n)
Proposition 2.3 ([7, Prop. 3.1]). The norm in Bp,a;s is invariant for the action of the
(n)
group of invertible elements. By this we mean that for each A ∈ Bp , a ∈ Gl(H)+
and g ∈ Gl(H), we have
A
= lg (A)p,gag∗ ;s .
p,a;s
Now, we state the main result of this paper, the general case 1 ≤ p0 , p1 , s0 , s1 <
∞.
Theorem 2.4. Let a, b ∈ Gl(H)+ , 1 ≤ p0 , p1 , s0 , s1 < ∞, n ∈ N and t ∈ (0, 1).
Then
(n)
(n)
Bp(n)
,
B
= Bpt ,γa,b (t);st ,
p1 ,b;s1
0 ,a;s0
Clarkson-McCarthy Interpolated
Inequalities
Cristian Conde
vol. 10, iss. 1, art. 4, 2009
[t]
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where
1−t
t
1
1−t
t
1
=
+
and
=
+ .
pt
p0
p1
st
s0
s1
Proof. For the sake of simplicity, we will only consider the case n = 2 and omit the
transpose. The proof below works for n-tuples (n ≥ 3) with obvious modifications.
By the previous proposition, k(X1 , X2 )k[t] is equal to the norm of a−1/2 (X1 , X2 )a−1/2
interpolated between the norms k(·, ·)kp0 ,1;s0 and k(·, ·)kp1 ,c;s1 . Consequently it is
sufficient to prove our statement for these two norms.
(2)
Let t ∈ (0, 1) and (X1 , X2 ) ∈ Bpt such that k(X1 , X2 )kpt ,ct ;st = 1, and define
z t
−t z λ(z)
t z λ(z)
z2 − 2t
−
−
, U2 c 2 c 2 X2 c 2 c 2 g(z) = U1 c c X1 c 2 c 2 = (g1 (z), g2 (z)),
where
λ(z) = pt
1−z
z
+
p0
p1
st
1−z
z
+
s0
s1
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and Xi = Ui |Xi | is the polar decomposition of Xi for i = 1, 2.
(2)
(2)
Then for each z ∈ S, g(z) ∈ Bp0 + Bp1 and
!
2 s0
iy t
λ(iy) X
iy
t
Uk c 2 c− 2 Xk c− 2 c 2 kg(iy)ksp00 ,1;s0 =
≤
≤
p0
k=1
2 X
pt
iy2 − 2t
− 2t iy
2
c c Xk c c k=1
2
X
!
Clarkson-McCarthy Interpolated
Inequalities
pt
Cristian Conde
!
kXk kpptt ,ct
vol. 10, iss. 1, art. 4, 2009
=1
k=1
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and
kg(1 +
iy)ksp11 ,c;s1
≤
2
X
!
kXk kpptt ,ct
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= 1.
k=1
(2)
(2)
Since g(t) = (X1 , X2 ) and g = (g1 , g2 ) ∈ F Bp0 ,1;s0 , Bp1 ,c;s1 , we have
k(X1 , X2 )k[t] ≤ 1. Thus we have shown that
k(X1 , X2 )k[t] ≤ k(X1 , X2 )kpt ,ct ;st .
(2)
(2)
To prove the converse inequality, let f = (f1 , f2 ) ∈ F Bp0 ,1;s0 , Bp1 ,c;s1 ; f (t) =
(X1 , X2 ) and Y1 , Y2 ∈ B0,0 (H) (the set of finite-rank operators) with kYk kqt ≤ 1,
where qt is the conjugate exponent for 1 < pt < ∞ (or a compact operator and
q = ∞ if p = 1). For k = 1, 2, let
z
z
gk (z) = c− 2 Yk c− 2 .
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Consider the function h : S → C2 , |(·, ·)|st ,
h(z) = (tr(f1 (z)g1 (z)), tr(f2 (z)g2 (z))).
◦
◦
Since f (z) is analytic in S and bounded in S, then h is analytic in S and bounded in
S, and
t
t
t
t
h(t) = tr c− 2 X1 c− 2 Y1 , tr c− 2 X2 c− 2 Y2
= (h1 (t), h2 (t)).
Clarkson-McCarthy Interpolated
Inequalities
Cristian Conde
2
By Hadamard’s three line theorem applied to h and the Banach space (C , |(·, ·)|st ),
we have
|h(t)|st ≤ max sup |h(iy)|st , sup |h(1 + iy)|st .
y∈R
vol. 10, iss. 1, art. 4, 2009
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y∈R
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For j = 0, 1,
sup |h(j + iy)|st = sup
y∈R
y∈R
= sup
y∈R
≤ sup
y∈R
2
X
! s1
t
st
|tr(fk (j + iy)gk (j + iy))|
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k=1
2
X
! s1
|tr(c−j/2 fk (j + iy)c−j/2 gk (iy))|st
k=1
2
X
! s1
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t
t
kfk (j + iy)ksp,c
j
k=1
≤ kf kF B (2)
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t
(2)
p0 ,1;s0 ,Bp1 ,c;s1
,
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then
kX1 ksptt ,ct + kX2 ksptt ,ct =
=
sup
{|h1 (t)|st + |h2 (t)|st }
kY1 kqt ≤1,Y1 ∈B00 (H)
kY2 kqt ≤1,Y2 ∈B00 (H)
sup
kY1 kqt ≤1,Y1 ∈B00 (H)
kY2 kqt ≤1,Y2 ∈B00 (H)
|h(t)|sstt ≤ kf kst
(2)
(2)
,Bp1 ,c;s1
0 ,1;s0
F Bp
.
Clarkson-McCarthy Interpolated
Inequalities
(2)
(2)
Since the previous inequality is valid for each f ∈ F Bp0 ,1;s0 , Bp1 ,c;s1 with
f (t) = (X1 , X2 ), we have
vol. 10, iss. 1, art. 4, 2009
k(X1 , X2 )kpt ,ct ;st ≤ k(X1 , X2 )k[t] .
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Cristian Conde
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In the special case that p0 = p1 = p and s0 = s1 = s we obtain Theorem 2.2.
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3.
Clarkson-Kissin Type Inequalities
Bhatia and Kittaneh [3] proved that if 2 ≤ p < ∞, then
n
2
p
n
n
X
kAj k2p
≤
n
X
j=1
j=1
n
X
n
X
kAj kpp ≤
j=1
kBj k2p
≤n
2− p2
n
X
kAj k2p .
j=1
kBj kpp ≤ np−1
j=1
n
X
Clarkson-McCarthy Interpolated
Inequalities
kAj kpp .
Cristian Conde
j=1
vol. 10, iss. 1, art. 4, 2009
Pn
j
k=1 θk Ak
(for 0 < p ≤ 2, these two inequalities are reversed) where Bj =
θ1 , . . . , θn the n roots of unity.
If we interpolate these inequalities we obtain that
! s1
! s1
! s1
n
n
n
t
t
t
X
X
X
1
1
np
≤
kBj kspt
≤ n(1− p )
,
kAj kspt
kAj kspt
j=1
j=1
where
j=1
with
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1
1−t t
=
+ .
st
2
p
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st
Dividing by n , we obtain
n
n
1
p
1X
kAj kspt
n j=1
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! s1
n
t
≤
1X
kBj kspt
n j=1
n
1− p1
≤ n(
)
! s1
t
1X
kAj kspt
n j=1
Close
! s1
t
.
This inequality can be rephrased as follows, if µ ∈ [2, p] then
n
n
1
p
1X
kAj kµp
n j=1
! µ1
n
≤
1X
kBj kµp
n j=1
1− p1
≤ n(
)
! µ1
n
1X
kAj kµp
n j=1
! µ1
.
Clarkson-McCarthy Interpolated
Inequalities
Cristian Conde
n
In each of the following statements R ∈ Gl(H ) and we denote by TR the linear
operator
TR : Bp(n) −→ Bp(n)
TR (A) = RA = (B1 , . . . , Bn )t ,
P
with Bj = nk=1 Rjk Ak and α = kR−1 k, β = kRk (we use the same symbol to
denote the norm in B(H) and B(H n )).
We observe that if the norm of TR is at most M when
TR : Bp(n) , k(·, . . . , ·)kp,1,s → Bp(n) , k(·, . . . , ·)kp,1,r ,
then if we consider the operator TR between the spaces
TR : Bp(n) , k(·, . . . , ·)kp,a,s → Bp(n) , k(·, . . . , ·)kp,b,r ,
its norm is at most F (a, b)M with

 min{kb−1 kkak, ka1/2 b−1 a1/2 kka−1 kkak} if a 6= b,
F (a, b) =
 −1
ka kkak
if a = b.
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Remark 2. If a−1/2 ∈ Gl(H) commutes with R ∈ B(H n ), that is, if a−1/2 commutes
with Rjk for all 1 ≤ j, k ≤ n, then F is reduced to

 min{kb−1 kkak, ka1/2 b−1 a1/2 k} = ka1/2 b−1 a1/2 k if a 6= b,
F (a, b) =

1
if a = b.
In [12], Kissin proved the following Clarkson type inequalities for the n-tuples
(n)
A ∈ Bp . If 2 ≤ p < ∞ and λ, µ ∈ [2, p], or if 0 < p ≤ 2 and λ, µ ∈ [p, 2], then
! λ1
! µ1
n
n
X
X
1
1
(3.1)
≤
kBj kλp
n−f (p) α−1
kAj kµp
n j=1
n j=1
! µ1
n
X
1
,
≤ nf (p) β
kAj kµp
n j=1
where f (p) = p1 − 21 .
Remark 3. This result extends the results of Bhatia and Kittaneh proved for µ = λ =
2 or p and R = (Rjk ) where
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We use the inequalities (3.1) and the interpolation method to obtain the following
inequalities.
(n)
Theorem 3.1. Let a, b ∈ Gl(H)+ , A ∈ Bp , 1 ≤ p < ∞ and t ∈ [0, 1], then
! µ1
! λ1
! µ1
n
n
n
X
X
X
(3.2)
k̃
kAj kµp,a
≤
kBj kλp,γa,b (t)
≤ K̃
kAj kµp,a
j=1
Cristian Conde
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2π(j−1)(k−1)
) 1.
n
Rjk = e(i
j=1
Clarkson-McCarthy Interpolated
Inequalities
j=1
Close
where
k̃ = k̃(p, a, b, t) = F (a, a)t−1 F (b, a)−t n λ − µ −| p − 2 | α−1 ,
1
and
1
1
1
K̃ = K̃(p, a, b, t) = F (a, a)1−t F (a, b)t n λ − µ +| p − 2 | β,
if 2 ≤ p and λ, µ ∈ [2, p] or if 1 ≤ p ≤ 2 and λ, µ ∈ [p, 2].
1
1
1
1
Proof. We will denote by γ(t) = γa,b (t), when no confusion can arise.
(n)
Consider the space Bp with the norm:
kAkp,a;s = (kA1 ksp,a + · · · + kAn ksp,a )1/s ,
where a ∈ Gl(H)+ .
1
1
1
1
By (3.1), the norm of TR is at most F (a, a)n λ − µ +| p − 2 | β when
TR : Bp(n) , k(·, . . . , ·)kp,a;µ −→ Bp(n) , k(·, . . . , ·)kp,a;λ ,
1
1
1
1
and the norm of TR is at most F (a, b)n λ − µ +| p − 2 | β when
TR : Bp(n) , k(·, . . . , ·)kp,a;µ −→ Bp(n) , k(·, . . . , ·)kp,b;λ .
Therefore, using the complex interpolation, we obtain the following diagram of
interpolation for t ∈ [0, 1]
(n)
(Bp , k(·, . . . , ·)kp,a;λ )
ii4
TRiiiiii
(n)
i
iiii
iiii
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vol. 10, iss. 1, art. 4, 2009
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TR
/ (B (n) , k(·, . . . , ·)k
p
p,γ(t);λ )
UUUU
UUUUTR
UUUU
UUUU
*
(Bp , k(·, . . . , ·)kp,a;µ )
Clarkson-McCarthy Interpolated
Inequalities
(n)
(Bp , k(·, . . . , ·)kp,b;λ ).
By Theorem 2.1, TR satisfies
kTR (A)kp,γ(t);λ ≤ F (a, a)1−t F (a, b)t n λ − µ +| p − 2 | βkAkp,a;µ .
1
(3.3)
1
1
1
Now applying the Complex method to
(n)
(Bp , k(·, . . . , ·)kp,a;λ )
UUUU
UUUUTR−1
UUUU
UUUU
U*
T
R−1 /
(n)
(n)
(Bp , k(·, . . . , ·)kp,γ(t);λ )
(Bp , k(·, . . . , ·)kp,a;µ )
i4
TR−1 iiiiii
i
i
iiii
iiii
(n)
(Bp , k(·, . . . , ·)kp,b;λ )
Clarkson-McCarthy Interpolated
Inequalities
Cristian Conde
vol. 10, iss. 1, art. 4, 2009
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one obtains
(3.4)
kTR−1 (A)kp,a;µ ≤ F (a, a)1−t F (b, a)t n µ − λ +| p − 2 | αkAkp,γ(t);λ .
1
1
1
1
Replacing in (3.4) A by RA we obtain
kAkp,a;µ ≤ F (a, a)1−t F (b, a)t n µ − λ +| p − 2 | αkRAkp,γ(t);λ ,
1
(3.5)
1
1
1
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or equivalently
(3.6)
F (a, a)t−1 F (b, a)−t n λ − µ −| p − 2 | α−1 kAkp,a;µ ≤ kTR (A)kp,γ(t);λ .
1
1
1
1
Finally, the inequalities (3.3) and (3.6) complete the proof.
We remark that theprevious statement
is a generalization of Th. 4.1 in [7] where
2π(j−1)(k−1)
(i
)
n
1
and a−1/2 commutes with R for all
Tn = TR with R = e
1≤j,k≤n
a ∈ Gl(H)+ .
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On the other hand, it is well known that if x1 , . . . , xn are non-negative numbers,
1/s
P
s ∈ R and we denote Ms (x) = n1 ni=1 xsi
then for 0 < s < s0 , Ms (x) ≤
Ms0 (x).
If we denote kBk = (kB1 kp , . . . , kBn kp ) and we consider 1 < p ≤ 2, then it
+ qt that
holds for t ∈ [0, 1] and s1t = 1−t
p
Mst (kBk) ≤ Mq (kBk) ≤ r
2
−1
p
2
q
β n
n
X
−1
q
! p1
kAj kpp
,
j=1
Clarkson-McCarthy Interpolated
Inequalities
Cristian Conde
vol. 10, iss. 1, art. 4, 2009
or equivalently
(3.7)
n
X
! s1
t
kBj kspt
≤r
2
−1
p
2
q
β n
1
− 1q
st
n
X
! p1
kAj kpp
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.
Analogously, for 2 ≤ p < ∞ we get
(3.8)
n
X
j=1
Contents
j=1
j=1
! p1
kAj kpp
2
2
1
≤ ρ1− p α p n q
− s1
t
n
X
! s1
t
kBj kspt
j=1
where s1t = 1−t
+ pt .
q
Now we can use the interpolation method with the inequalities (3.7) and (3.1) (or
(3.8) and (3.1)).
If we consider the following diagram of interpolation with 1 < p ≤ 2 and t ∈
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[0, 1],
(n)
(Bp , k(·, . . . , ·)kp,1;p )
ii4
TR iiiii
i
iiii
iiii
TR
/ (B (n) , k(·, . . . , ·)k
p
p,1;st )
TTTT
TTTTTR
TTTT
TTTT
*
(n)
(Bp , k(·, . . . , ·)kp,1;p )
Clarkson-McCarthy Interpolated
Inequalities
(n)
(Bp , k(·, . . . , ·)kp,1;q ).
Cristian Conde
vol. 10, iss. 1, art. 4, 2009
By Theorem 2.1 and (3.1), TR satisfies
kTR (A)kp,1;st ≤ nf (p) β
(3.9)
1−t 2
2
r p −1 β q
t
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kAkp,1;p .
Contents
Finally, from the inequalities (3.7) and (3.9) we obtain
n
X
! s1
t
kBj kspt
n
n 2
o X
1
2
2
2
−1
≤ min r p −1 β q n st q , nf (p)(1−t) β 1+t( q −1) r( p −1)t
kAj kpp
j=1
! p1
.
j=1
(n)
j=1
kAj kpp
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Theorem 3.2. Let A ∈ Bp and B = RA, where R = (Rjk ) is invertible. Let
r = max kRjk k, ρ = max k(R−1 )jk k and q be the conjugate exponent of p. Then,
for t ∈ [0, 1] we get
! p1
II
Page 18 of 21
We can summarize the previous facts in the following statement.
n
X
JJ
n
n
o X
t+(1−t) p2 (1− p2 )(1−t)
1− p2 p2 1q − s1
f
(p)t
≤ min ρ α n t , n
α
ρ
kBj kspt
j=1
! 1s
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t
1
st
if 2 ≤ p and
n
X
1−t
q
=
! s1
t
kBj kspt
+ pt , or
n
n 2
o X
1
2
2
2
−1
≤ min r p −1 β q n st q , nf (p)(1−t) β 1+t( q −1) r( p −1)t
kAj kpp
j=1
! p1
,
j=1
if 1 < p ≤ 2 and
1
st
=
1−t
p
+ qt .
Finally using the Finsler norm k(·, . . . , ·)kp,a;s , Calderón’s method and the previous inequalities we obtain:
(n)
Corollary 3.3. Let a, b ∈ Gl(H)+ , A ∈ Bp and B = RA, where R = (Rjk ) is
invertible. Let r = max kRjk k, ρ = max k(R−1 )jk k and q be the conjugate exponent
of p. Then, for t, u ∈ [0, 1] we get
! p1
! s1
n
n
t
X
X
t
kAj kpp,a
≤ F (a, a)1−u F (b, a)u M1
kBj ksp,γ
,
a,b (u)
j=1
if 2 ≤ p,
1
st
j=1
1−t
q
=
+
t
p
and
o
n
1
2
2
2
2
−1
M1 = M1 (R, p, t) = min ρ1− p α p n q st , nf (p)t αt+(1−t) p ρ(1− p )(1−t)
or
n
X
! s1
t
t
kBj ksp,γ
a,b (u)
1
st
≤ F (a, a)
u
F (a, b) M2
kAj kpp
=
1−t
p
+
t
q
Cristian Conde
vol. 10, iss. 1, art. 4, 2009
Title Page
Contents
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! p1
,
j=1
j=1
if 1 < p ≤ 2,
1−u
n
X
Clarkson-McCarthy Interpolated
Inequalities
and
n 2
o
−1 2q s1 − 1q
f (p)(1−t) 1+t( 2q −1) ( p2 −1)t
p
t
M2 = M2 (R, p, t) = min r β n
,n
β
r
.
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References
[1] E. ANDRUCHOW, G. CORACH, M. MILMAN AND D. STOJANOFF,
Geodesics and interpolation, Revista de la Unión Matemática Argentina, 40(34) (1997), 83–91.
[2] J. BERGH AND J. LÖFSTRÖM, Interpolation Spaces. An Introduction,
Springer-Verlag, New York, 1976.
Clarkson-McCarthy Interpolated
Inequalities
Cristian Conde
[3] R. BHATIA AND F. KITTANEH, Clarkson inequalities with several operators,
Bull. London Math. Soc., 36(6) (2004), 820–832.
vol. 10, iss. 1, art. 4, 2009
[4] R. BOAS, Some uniformly convex spaces, Bull. Amer. Math. Soc., 46 (1940),
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Title Page
[5] A. CALDERÓN, Intermediate spaces and interpolation, the complex method,
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[6] J.A. CLARKSON, Uniformly convex spaces, Trans. Amer. Math. Soc., 40
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[9] I. GOHBERG AND M. KREIN, Introduction to the Theory of Linear Nonselfadjoint Operators, American Mathematical Society, Vol. 18, Providence,
R. I.,1969.
Contents
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[10] O. HIRZALLAH AND F. KITTANEH, Non-commutative Clarkson inequalities
for unitarily invariants norms, Paciic J. Math., 202 (2002), 363–369.
[11] O. HIRZALLAH AND F. KITTANEH, Non-commutative Clarkson inequalities
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[12] E. KISSIN, On Clarkson-McCarthy inequalities for n-tuples of operators, Proc.
Amer. Math. Soc., 135(8) (2007), 2483–2495.
[13] S. KREIN, J. PETUNIN AND E. SEMENOV, Interpolation of linear operators.
Translations of Mathematical Monographs, Vol. 54, American Mathematical
Society, Providence, R. I., 1982.
Clarkson-McCarthy Interpolated
Inequalities
Cristian Conde
vol. 10, iss. 1, art. 4, 2009
[14] C. MCCARTHY, cp , Israel J. Math., 5 (1967), 249–271.
Title Page
[15] G. MOSTOW, Some new decomposition theorems for semi-simple groups,
Mem. Amer. Math. Soc., 14 (1955), 31–54.
Contents
[16] B. SIMON, Trace Ideals and their Applications. Second edition, Mathematical
Surveys and Monographs, 120. American Mathematical Society, Providence,
RI, 2005.
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