Volume 10 (2009), Issue 1, Article 4, 10 pp.
CLARKSON-MCCARTHY INTERPOLATED INEQUALITIES IN FINSLER NORMS
CRISTIAN CONDE
I NSTITUTO DE C IENCIAS
U NIVERSIDAD NACIONAL DE G ENERAL S ARMIENTO
J. M. G UTIERREZ 1150
(1613) L OS P OLVORINES
B UENOS A IRES , A RGENTINA .
cconde@ungs.edu.ar
Received 29 August, 2008; accepted 14 January, 2009
Communicated by R. Bhatia
(n)
A BSTRACT. We apply the complex interpolation method to prove that, given two spaces Bp0 ,a;s0 ,
(n)
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 operators a and b of B(H) then, the curve
(n)
(n)
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 .
Key words and phrases: p-Schatten class, Complex method, Clarkson-McCarthy inequalities.
2000 Mathematics Subject Classification. Primary 46B70, 47A30; Secondary 46B20, 47B10.
1. I NTRODUCTION
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
X
1
p p
kXkp =
sj (X)
,
and the linear space
Bp (H) = {X ∈ B(H) : kXkp < ∞}.
243-08
2
C RISTIAN C ONDE
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 pSchatten norms of Hilbert space operators:
(1.1)
2(kAkpp + kBkpp ) ≤ kA − Bkpp + kA + Bkpp
≤ 2p−1 (kAkpp + kBkpp ),
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 interpolation
(n)
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 the linear
(n)
(n)
operator Tn : Bp −→ Bp given by
!
n
n
n
X
X
X
Tn (X̄) = (Tn (X1 , . . . , Xn ) =
Xj ,
θj1 Xj , . . . ,
θjn−1 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 Clarkson-McCarthy
inequalities for n-tuples of operators from Schatten ideals. In this work 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 R = (Rjk ) with Rjk ∈ B(H), and the linear
(n)
(n)
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.
2. G EOMETRIC I NTERPOLATION
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]
kT kX[t] →Y[t] ≤ M01−t M1t .
J. Inequal. Pure and Appl. Math., 10(1) (2009), Art. 4, 10 pp.
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C LARKSON -M C C ARTHY I NTERPOLATED I NEQUALITIES
3
Here and subsequently, let 1 ≤ p < ∞, n ∈ N, s ≥ 1, a ∈ Gl(H)+ and
Bp(n) = {A = (A1 , . . . , An )t : Ai ∈ Bp },
(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
|(a0 , . . . , an−1 )|s = (|a0 |s + · · · + |an−1 |s )1/s .
(n)
(n)
From now on, we denote with Bp,a;s the space Bp 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
1
1−t
t
=
+
and
=
+ .
pt
p0
p1
st
s0
s1
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 with the linear operator
(2)
(2)
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
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 ,
[t]
where γa,b (t) = a1/2 (a−1/2 ba−1/2 )t a1/2 .
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 p-Schatten class by multiples of the
identity ([7]).
(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 .
(n)
Proposition 2.3 ([7, Prop. 3.1]). The norm in Bp,a;s is invariant for the action of the group of
(n)
invertible elements. By this we mean that for each A ∈ Bp , a ∈ Gl(H)+ and g ∈ Gl(H), we
have
A
lg (A)
=
.
p,a;s
p,gag ∗ ;s
Now, we state the main result of this paper, the general case 1 ≤ p0 , p1 , s0 , s1 < ∞.
J. Inequal. Pure and Appl. Math., 10(1) (2009), Art. 4, 10 pp.
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4
C RISTIAN C ONDE
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
[t]
where
1−t
t
1
=
+
pt
p0
p1
1
1−t
t
=
+ .
st
s0
s1
and
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
λ(z) −t z λ(z)
z2 − 2t
2 −2
− 2t z2 2
2
g(z) = U1 c c X1 c c , U2 c c X2 c c = (g1 (z), g2 (z)),
where
1−z
z
1−z
z
λ(z) = pt
+
st
+
p0
p1
s0
s1
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
s0 !
2 iy t
X
λ(iy)
iy
Uk c 2 c− 2 Xk c− 2t c 2 kg(iy)ksp00 ,1;s0 =
p0
k=1
≤
2 X
pt
iy2 − 2t
− 2t iy
2
c c Xk c c pt
k=1
≤
2
X
!
!
kXk kpptt ,ct
=1
k=1
and
kg(1 + iy)ksp11 ,c;s1 ≤
2
X
!
kXk kpptt ,ct
= 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 .
Consider the function h : S → C2 , |(·, ·)|st ,
h(z) = (tr(f1 (z)g1 (z)), tr(f2 (z)g2 (z))).
J. Inequal. Pure and Appl. Math., 10(1) (2009), Art. 4, 10 pp.
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C LARKSON -M C C ARTHY I NTERPOLATED I NEQUALITIES
◦
5
◦
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)).
By Hadamard’s three line theorem applied to h and the Banach space (C2 , |(·, ·)|st ), we have
|h(t)|st ≤ max sup |h(iy)|st , sup |h(1 + iy)|st .
y∈R
y∈R
For j = 0, 1,
2
X
sup |h(j + iy)|st = sup
y∈R
y∈R
! s1
t
st
|tr(fk (j + iy)gk (j + iy))|
k=1
2
X
= sup
y∈R
! s1
t
|tr(c
−j/2
fk (j + iy)c
−j/2
st
gk (iy))|
k=1
2
X
≤ sup
y∈R
! s1
t
t
kfk (j + iy)ksp,c
j
k=1
≤ kf kF B (2)
(2)
p0 ,1;s0 ,Bp1 ,c;s1
,
then
kX1 ksptt ,ct + kX2 ksptt ,ct =
{|h1 (t)|st + |h2 (t)|st }
sup
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
(2)
(2)
Bp0 ,1;s0 , Bp1 ,c;s1
Since the previous inequality is valid for each f ∈ F
we have
k(X1 , X2 )kpt ,ct ;st ≤ k(X1 , X2 )k[t] .
.
with f (t) = (X1 , X2 ),
In the special case that p0 = p1 = p and s0 = s1 = s we obtain Theorem 2.2.
3. C LARKSON -K ISSIN T YPE I NEQUALITIES
Bhatia and Kittaneh [3] proved that if 2 ≤ p < ∞, then
2
np
n
X
kAj k2p ≤
j=1
n
n
X
j=1
n
X
2
kBj k2p ≤ n2− p
j=1
kAj kpp
≤
n
X
j=1
n
X
j=1
kBj kpp
≤n
p−1
n
X
kAj kpp .
j=1
(for 0 < p ≤ 2, these two inequalities are reversed) where Bj =
n roots of unity.
J. Inequal. Pure and Appl. Math., 10(1) (2009), Art. 4, 10 pp.
kAj k2p .
Pn
j
k=1 θk Ak
with θ1 , . . . , θn the
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6
C RISTIAN C ONDE
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
j=1
where
1−t t
1
=
+ .
st
2
p
Dividing by nst , we obtain
n
n
1
p
1X
kAj kspt
n j=1
! s1
n
t
≤
1X
kBj kspt
n j=1
1− p1
≤ n(
)
! s1
t
n
1X
kAj kspt
n j=1
! s1
t
.
This inequality can be rephrased as follows, if µ ∈ [2, p] then
! µ1
! µ1
n
n
X
X
1
1
1
np
kAj kµp
≤
kBj kµp
n j=1
n j=1
n
1− p1
≤ n(
)
1X
kAj kµp
n j=1
! µ1
.
In each of the following statements R ∈ Gl(H n ) 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.
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.
J. Inequal. Pure and Appl. Math., 10(1) (2009), Art. 4, 10 pp.
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C LARKSON -M C C ARTHY I NTERPOLATED I NEQUALITIES
7
(n)
In [12], Kissin proved the following Clarkson type inequalities for the n-tuples A ∈ Bp . If
2 ≤ p < ∞ and λ, µ ∈ [2, p], or if 0 < p ≤ 2 and λ, µ ∈ [p, 2], then
! µ1
! λ1
n
n
X
X
1
1
(3.1)
kAj kµp
≤
kBj kλp
n−f (p) α−1
n j=1
n j=1
! µ1
n
X
1
,
≤ nf (p) β
kAj kµp
n j=1
where f (p) = p1 − 12 .
Remark 3. This result extends the results of Bhatia and Kittaneh proved for µ = λ = 2 or p
and R = (Rjk ) where
2π(j−1)(k−1)
) 1.
n
R = e(i
jk
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
where
j=1
j=1
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 ,
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 .
1
1
1
1
Therefore, using the complex interpolation, we obtain the following diagram of interpolation
for t ∈ [0, 1]
(n)
(Bp , k(·, . . . , ·)kp,a;λ )
ii4
iiii
i
i
i
iii
iiii
TR
(n)
/ (B (n) , k(·, . . . , ·)k
(Bp , k(·, . . . , ·)kp,a;µ )
p
p,γ(t);λ )
UUUU
UUUUTR
UUUU
UUUU
*
TR
(n)
(Bp , k(·, . . . , ·)kp,b;λ ).
J. Inequal. Pure and Appl. Math., 10(1) (2009), Art. 4, 10 pp.
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8
C RISTIAN C ONDE
By Theorem 2.1, TR satisfies
(3.3)
kTR (A)kp,γ(t);λ ≤ F (a, a)1−t F (a, b)t n λ − µ +| p − 2 | βkAkp,a;µ .
1
1
1
1
Now applying the Complex method to
(n)
(Bp , k(·, . . . , ·)kp,a;λ )
UUUU
UUUTUR−1
UUUU
UUUU
*
TR−1
(n)
(n)
/
(Bp , k(·, . . . , ·)kp,γ(t);λ )
(Bp , k(·, . . . , ·)kp,a;µ )
i4
TR−1 iiiii
i
i
i
i
ii
iiii
(n)
(Bp , k(·, . . . , ·)kp,b;λ )
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
(3.5)
kAkp,a;µ ≤ F (a, a)1−t F (b, a)t n µ − λ +| p − 2 | αkRAkp,γ(t);λ ,
1
1
1
1
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
the previous
statement is a generalization of Th. 4.1 in [7] where Tn = TR
that
2π(j−1)(k−1)
(i
)
n
with R = e
1
and a−1/2 commutes with R for all a ∈ Gl(H)+ .
1≤j,k≤n
On the other hand, it is well known that if x1 , . . . , xn are non-negative numbers, s ∈ R and
1/s
P
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 holds for
t ∈ [0, 1] and s1t = 1−t
+ qt that
p
Mst (kBk) ≤ Mq (kBk) ≤ r
2
−1
p
2
q
β n
n
X
−1
q
! p1
kAj kpp
,
j=1
or equivalently
(3.7)
n
X
! s1
t
kBj kspt
≤r
2
−1
p
2
q
β n
1
− 1q
st
j=1
Analogously, for 2 ≤ p < ∞ we get
! p1
n
X
1
2
2
−1
(3.8)
kAj kpp
≤ ρ1− p α p n q st
j=1
n
X
! p1
kAj kpp
.
j=1
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)).
J. Inequal. Pure and Appl. Math., 10(1) (2009), Art. 4, 10 pp.
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C LARKSON -M C C ARTHY I NTERPOLATED I NEQUALITIES
9
If we consider the following diagram of interpolation with 1 < p ≤ 2 and t ∈ [0, 1],
(n)
(Bp , k(·, . . . , ·)kp,1;p )
j4
jjjj
j
j
j
jjjj
jjjj
TR
(n)
/ (B (n) , k(·, . . . , ·)k
(Bp , k(·, . . . , ·)kp,1;p )
p
p,1;st )
TTTT
TTTTTR
TTTT
TTTT
*
TR
(n)
(Bp , k(·, . . . , ·)kp,1;q ).
By Theorem 2.1 and (3.1), TR satisfies
kTR (A)kp,1;st ≤ n
(3.9)
f (p)
β
1−t r
2
−1
p
β
2
q
t
kAkp,1;p .
Finally, from the inequalities (3.7) and (3.9) we obtain
! s1
! p1
n
n
n 2
o X
t
X
1
1
2
2
2
−
kBj kspt
≤ 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
j=1
We can summarize the previous facts in the following statement.
(n)
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
! 1s
n
n
t
n
o X
X
1
1
2
2
2
2
−
1−
t+(1−t)
(1−
)(1−t)
p
f
(p)t
s
t
q
s
p
p
p
p
kAj kp
α
≤ min ρ α n t , n
ρ
kBj kp
j=1
j=1
if 2 ≤ p and
n
X
1
st
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
1
st
if 1 < p ≤ 2 and
=
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
! s1
! p1
n
n
t
X
X
t
≤ F (a, a)1−u F (b, a)u M1
kBj ksp,γ
,
kAj kpp,a
a,b (u)
j=1
j=1
if 2 ≤ p,
1
st
=
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)
≤ F (a, a)1−u F (a, b)u M2
j=1
J. Inequal. Pure and Appl. Math., 10(1) (2009), Art. 4, 10 pp.
n
X
! p1
kAj kpp
,
j=1
http://jipam.vu.edu.au/
10
C RISTIAN C ONDE
if 1 < p ≤ 2,
1
st
=
1−t
p
+
t
q
and
n 2
o
1
2
2
2
−1
M2 = M2 (R, p, t) = min r p −1 β q n st q , nf (p)(1−t) β 1+t( q −1) r( p −1)t .
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J. Inequal. Pure and Appl. Math., 10(1) (2009), Art. 4, 10 pp.
http://jipam.vu.edu.au/