CLARKSON-MCCARTHY INTERPOLATED INEQUALITIES IN FINSLER NORMS

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Clarkson-McCarthy Interpolated Inequalities

Cristian Conde vol. 10, iss. 1, art. 4, 2009

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CLARKSON-MCCARTHY INTERPOLATED INEQUALITIES IN FINSLER NORMS

CRISTIAN CONDE

Instituto de Ciencias

Universidad Nacional de General Sarmiento J. M. Gutierrez 1150, (1613) Los Polvorines Buenos Aires, Argentina.

EMail:cconde@ungs.edu.ar

Received: 29 August, 2008

Accepted: 14 January, 2009

Communicated by: R. Bhatia

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 Bp⁽ⁿ⁾₀,a;s₀, B_p⁽ⁿ⁾

1,b;s₁ ofn-tuples of operators in thep-Schatten class of a Hilbert spaceH, endowed with weighted norms associated to positive and invertible op- eratorsaandbofB(H)then, the curve of interpolation(B⁽ⁿ⁾p₀,a;s₀, B_p⁽ⁿ⁾

1,b;s₁)[t]

of the pair is given by the space ofn-tuples of operators in thept-Schatten class ofH, with the weighted norm associated to the positive invertible element γa,b(t) =a^1/2(a^−1/2ba^−1/2)^ta^1/2.

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1. Introduction

In [6], J. Clarkson introduced the concept of uniform convexity in Banach spaces and obtained that spacesL_p (or l_p) are uniformly convex forp > 1 throughout the following inequalities

2

kfk^p_p+kgk^p_p

≤ kf −gk^p_p+kf+gk^p_p ≤2^p−1

kfk^p_p+kgk^p_p ,

Let (B(H),k · k) denote the algebra of bounded operators acting on a complex and separable Hilbert spaceH,Gl(H)the group of invertible elements ofB(H)and Gl(H)⁺the set of all positive elements ofGl(H).

IfX ∈ B(H)is compact we denote by{s_j(X)}the sequence of singular values ofX (decreasingly ordered). For0< p < ∞, let

kXk_p =X

s_j(X)^p_p¹ ,

and the linear space

B_p(H) ={X ∈B(H) :kXk_p <∞}.

For 1 ≤ p < ∞, this space is called the p−Schatten class of B(H) (to simplify notation we use B_p) and by conventionkXk = kXk_∞ = s₁(X). A reference for this subject is [9].

C. McCarthy proved in [14], among several other results, the following inequalities forp-Schatten norms of Hilbert space operators:

2(kAk^p_p+kBk^p_p)≤ kA−Bk^p_p+kA+Bk^p_p (1.1)

≤2^p−1(kAk^p_p+kBk^p_p),

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for2≤p <∞, and

2^p−1(kAk^p_p+kBk^p_p)≤ kA−Bk^p_p+kA+Bk^p_p (1.2)

≤2(kAk^p_p+kBk^p_p), for1≤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 ofB_pfor1< 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 method onB⁽ⁿ⁾p (this set will be defined below) forp, s≥1andn ∈N with a Finsler norm associated witha∈Gl(H)⁺:

kXk_p,a;s:=ka^−1/2Xa^−1/2k^s_p.

From now on, for the sake of simplicity, we denote with lower case letters the elements ofGl(H)⁺.

As a by-product, we obtain Clarkson type inequalities using the Klaus idea with the linear operatorT_n:Bp⁽ⁿ⁾ −→Bp⁽ⁿ⁾given by

T_n( ¯X) = (T_n(X₁, . . . , X_n) =

n

X

j=1

X_j,

n

X

j=1

θ¹_jX_j, . . . ,

n

X

j=1

θⁿ⁻¹_j X_j

! ,

whereθ₁, . . . , θ_nare thenroots of unity.

Recently, Kissin in [12], motivated by [3], obtained analogues of the Clarkson- McCarthy inequalities forn-tuples of operators from Schatten ideals. In this work

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the author considersHⁿ,the orthogonal sum ofncopies of the Hilbert spaceH,and each operator R ∈ B(Hⁿ) can be represented as an n ×n block-matrix operator R = (R_jk)withR_jk ∈ B(H),and the linear operator T_R : Bp⁽ⁿ⁾ →Bp⁽ⁿ⁾ is defined byT_R(A) =RA. Finally we remark that the works [3] and [11] are generalizations of [10].

In these notes we obtain inequalities for the linear operatorT_Rin the Finsler norm k·k_p,a;sas by-products of the complex interpolation method and Kissin’s inequalities.

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2. Geometric Interpolation

We follow the notation used in [2] and we refer the reader to [13] and [5] for de- tails on the complex interpolation method. For completeness, we recall the classical Calderón-Lions theorem.

Theorem 2.1. Let X andY be two compatible couples. Assume thatT is a linear operator fromX_j toY_j bounded byM_j,j = 0,1. Then fort∈[0,1]

kTk_X

[t]→Y_[t] ≤M₀^1−tM₁^t.

Here and subsequently, let1≤p < ∞, n∈N, s≥1,a∈Gl(H)⁺and B_p⁽ⁿ⁾ ={A= (A₁, . . . , A_n)^t :A_i ∈B_p},

(where withtwe denote the transpose of then-tuple) endowed with the norm kAk_p,a;s = (kA₁k^s_p,a+· · ·+kA_nk^s_p,a)^1/s,

andCⁿendowed with the norm

|(a₀, . . . , an−1)|_s = (|a₀|^s+· · ·+|an−1|^s)^1/s.

From now on, we denote withBp,a;s⁽ⁿ⁾ the spaceBp⁽ⁿ⁾endowed with the normk(·, . . . ,·)k_p,a;s. From Calderón-Lions interpolation theory we get that forp₀, p₁, s₀, s₁ ∈[1,∞) (2.1)

B_p⁽ⁿ⁾

0,1;s0, B_p⁽ⁿ⁾

1,1;s1

[t]

=B_p⁽ⁿ⁾_t_,1;s_t, where

1 pt

= 1−t p0

+ t p1

and 1

st

= 1−t s0

+ t s1

.

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Note that forp= 2, (1.1) and (1.2) both reduce to the parallelogram law 2(kAk²₂+kBk²₂) = kA−Bk²₂+kA+Bk²₂,

while for the casesp = 1,∞ these inequalities follow from the triangle inequality forB₁ andB(H) respectively. Then the inequalities (1.1) and (1.2) can be proved (forn= 2via Theorem2.1) by interpolation between the previous elementary cases with the linear operator T₂ : B_p,1;p⁽²⁾ −→ B_p,1;p⁽²⁾ , T₂(A) = (A₁ +A₂, A₁ −A₂)^t as observed by Klaus.

In this section, we generalize (2.1) for the Finsler normsk(·, . . . ,·)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

B_p,a;s⁽ⁿ⁾ , B_p,b;s⁽ⁿ⁾

[t]

=B⁽ⁿ⁾_p,γ

a,b(t);s, whereγ_a,b(t) = a^1/2(a^−1/2ba^−1/2)^ta^1/2.

Remark 1. Note that whena andb commute the curve is given byγ_a,b(t) = a^1−tb^t. The previous corollary tells us that the interpolating space,B_p,γ_a,b_(t);scan be regarded as a weightedp-Schatten space with weighta^1−tb^t(see [2, Th. 5.5.3]).

We observe that the curve γ_a,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]).

There is a natural action ofGl(H)onBp⁽ⁿ⁾,defined by

(2.2) l :Gl(H)×B_p⁽ⁿ⁾−→B_p⁽ⁿ⁾, l_g(A) = (gA₁g^∗, . . . , gA_ng^∗)^t.

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Proposition 2.3 ([7, Prop. 3.1]). The norm inBp,a;s⁽ⁿ⁾ is invariant for the action of the group of invertible elements. By this we mean that for eachA∈ Bp⁽ⁿ⁾, a ∈Gl(H)⁺ andg ∈Gl(H), we have

A

p,a;s = l_g(A)

p,gag^∗;s.

Now, we state the main result of this paper, the general case1≤ p₀, p₁, s₀, s₁ <

∞.

Theorem 2.4. Leta, b ∈ Gl(H)⁺,1 ≤ p₀, p₁, s₀, s₁ < ∞, n ∈ N andt ∈ (0,1).

Then

B_p⁽ⁿ⁾₀_,a;s₀, B_p⁽ⁿ⁾

1,b;s1

[t]

=B_p⁽ⁿ⁾

t,γa,b(t);st, where

1

p_t = 1−t p₀ + t

p₁ and 1

s_t = 1−t s₀ + t

s₁.

Proof. For the sake of simplicity, we will only consider the casen= 2and omit the transpose. The proof below works forn-tuples (n ≥3) with obvious modifications.

By the previous proposition,k(X₁, X₂)k_[t]is equal to the norm ofa^−1/2(X₁, X₂)a^−1/2 interpolated between the norms k(·,·)k_p₀_,1;s₀ and k(·,·)k_p₁_,c;s₁. Consequently it is sufficient to prove our statement for these two norms.

Lett∈(0,1)and(X1, X2)∈Bp⁽²⁾t such thatk(X1, X2)k_p_t_,c^t_;s_t = 1, and define g(z) =

U₁

c^z²c⁻^t²X₁c^−t² c^z²

λ(z)

, U₂

c^z²c⁻^t²X₂c⁻^t²c^z²

λ(z)

= (g₁(z), g₂(z)), where

λ(z) = p_t

1−z p₀ + z

p₁

s_t

1−z s₀ + z

s₁

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andX_i =U_i|X_i|is the polar decomposition ofX_ifori= 1,2.

Then for eachz ∈S,g(z)∈Bp⁽²⁾0 +Bp⁽²⁾1 and kg(iy)k^s_p⁰

0,1;s0 =

2

X

k=1

U_k

c^iy² c⁻^t²X_kc⁻²^tc^iy²

λ(iy)

s0

p0

!

≤

2

X

k=1

c^iy²c⁻^t²X_kc⁻²^tc^iy²

pt

!

≤

2

X

k=1

kX_kk^p_p^t

t,c^t

!

= 1

and

kg(1 +iy)k^s_p¹₁_,c;s₁ ≤

2

X

k=1

kX_kk^p_p^t

t,c^t

!

= 1.

Since g(t) = (X₁, X₂) and g = (g₁, g₂) ∈ F

B_p⁽²⁾₀_,1;s₀, Bp⁽²⁾1,c;s1

, we have k(X₁, X₂)k_[t]≤1. Thus we have shown that

k(X₁, X₂)k_[t]≤ k(X₁, X₂)k_p

t,c^t;st. To prove the converse inequality, letf = (f₁, f₂)∈ F

B_p⁽²⁾₀_,1;s₀, Bp⁽²⁾1,c;s1

; f(t) = (X₁, X₂)and Y₁, Y₂ ∈ B_0,0(H)(the set of finite-rank operators) with kY_kk_q_t ≤ 1, where q_t is the conjugate exponent for 1 < p_t < ∞ (or a compact operator and q=∞ifp= 1). Fork = 1,2, let

g_k(z) = c⁻^z²Y_kc⁻^z².

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Consider the functionh:S→ C²,|(·,·)|_s

t

,

h(z) = (tr(f1(z)g1(z)), tr(f2(z)g2(z))).

Sincef(z)is analytic in

◦

Sand bounded inS, thenhis analytic in

◦

Sand bounded in S, and

h(t) = tr

c⁻^t²X₁c⁻²^tY₁ , tr

c⁻²^tX₂c⁻²^tY₂

= (h₁(t), h₂(t)).

By Hadamard’s three line theorem applied tohand the Banach space(C²,|(·,·)|_s_t), we have

|h(t)|_s_t ≤max

sup

y∈R

|h(iy)|_s_t,sup

y∈R

|h(1 +iy)|_s_t

. Forj = 0,1,

sup

y∈R

|h(j+iy)|_s_t = sup

y∈R 2

X

k=1

|tr(f_k(j +iy)g_k(j+iy))|^s^t

!_st¹

= sup

y∈R 2

X

k=1

|tr(c^−j/2fk(j+iy)c^−j/2gk(iy))|^s^t

!_st¹

≤sup

y∈R 2

X

k=1

kf_k(j+iy)k^s_p,c^tj

!_st¹

≤ kfk_F

B_p⁽²⁾

0,1;s0,B⁽²⁾p1,c;s1

,

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then

kX₁k^s_p^t

t,c^t +kX₂k^s_p^t

t,c^t = sup

kY₁k_qt≤1,Y₁∈B₀₀(H) kY2k_qt≤1,Y2∈B00(H)

{|h₁(t)|^s^t +|h₂(t)|^s^t}

= sup

kY1k_qt≤1,Y1∈B00(H) kY₂k_qt≤1,Y₂∈B₀₀(H)

|h(t)|^s_s^t

t ≤ kfk^s^t

F B_p⁽²⁾

0,1;s0,B⁽²⁾p1,c;s1

.

Since the previous inequality is valid for each f ∈ F

B_p⁽²⁾₀_,1;s₀, Bp⁽²⁾1,c;s1

with f(t) = (X₁, X₂), we have

k(X₁, X₂)k_p_t_,c^t_;s_t ≤ k(X₁, X₂)k_[t].

In the special case thatp₀ =p₁ =pands₀ =s₁ =swe obtain Theorem2.2.

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3. Clarkson-Kissin Type Inequalities

Bhatia and Kittaneh [3] proved that if2≤p < ∞, then n²^p

n

X

j=1

kA_jk²_p ≤

n

X

j=1

kB_jk²_p ≤n²⁻²^p

n

X

j=1

kA_jk²_p.

n

X

j=1

kA_jk^p_p ≤

n

X

j=1

kB_jk^p_p ≤n^p−1

n

X

j=1

kA_jk^p_p. (for0 < p ≤ 2, these two inequalities are reversed) where B_j = Pn

k=1θ^j_kA_k with θ₁, . . . , θ_nthenroots of unity.

If we interpolate these inequalities we obtain that

n¹^p

n

X

j=1

kA_jk^s_p^t

!_st¹

≤

n

X

j=1

kB_jk^s_p^t

!_st¹

≤n(¹⁻¹_p) Xⁿ

j=1

kA_jk^s_p^t

!_st¹ , where

1 st

= 1−t 2 + t

p. Dividing byn^s^t, we obtain

n¹^p 1 n

n

X

j=1

kAjk^s_p^t

!_st¹

≤ 1 n

n

X

j=1

kBjk^s_p^t

!_st¹

≤n(¹⁻¹p) 1 n

n

X

j=1

kA_jk^s_p^t

!_st¹ .

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This inequality can be rephrased as follows, ifµ∈[2, p]then

n¹^p 1 n

n

X

j=1

kAjk^µ_p

!_µ¹

≤ 1 n

n

X

j=1

kBjk^µ_p

!¹_µ

≤n(¹⁻¹p) 1 n

n

X

j=1

kA_jk^µ_p

!_µ¹ .

In each of the following statementsR ∈Gl(Hⁿ)and we denote byTRthe linear operator

T_R :B_p⁽ⁿ⁾−→B_p⁽ⁿ⁾ T_R(A) = RA= (B₁, . . . , B_n)^t, with B_j = Pn

k=1R_jkA_k and α = kR⁻¹k, β = kRk (we use the same symbol to denote the norm inB(H)andB(Hⁿ)).

We observe that if the norm ofT_Ris at mostM when TR: B_p⁽ⁿ⁾,k(·, . . . ,·)kp,1,s

→ B_p⁽ⁿ⁾,k(·, . . . ,·)kp,1,r

,

then if we consider the operatorT_Rbetween the spaces T_R: B_p⁽ⁿ⁾,k(·, . . . ,·)k_p,a,s

→ B_p⁽ⁿ⁾,k(·, . . . ,·)k_p,b,r , its norm is at mostF(a, b)M with

F(a, b) =







min{kb⁻¹kkak,ka^1/2b⁻¹a^1/2kka⁻¹kkak} ifa 6=b,

ka⁻¹kkak if a =b.

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Remark 2. Ifa^−1/2 ∈Gl(H)commutes withR ∈B(Hⁿ), that is, ifa^−1/2commutes withR_jk for all1≤j, k ≤n, thenF is reduced to

F(a, b) =







min{kb⁻¹kkak,ka^1/2b⁻¹a^1/2k}=ka^1/2b⁻¹a^1/2k ifa6=b,

1 if a=b.

In [12], Kissin proved the following Clarkson type inequalities for then-tuples A∈Bp⁽ⁿ⁾. If2≤p <∞andλ, µ∈[2, p], or if0< p ≤2andλ, µ∈[p,2], then

n^−f(p)α⁻¹ 1 n

n

X

j=1

kA_jk^µ_p

!¹_µ

≤ 1 n

n

X

j=1

kB_jk^λ_p

!_λ¹ (3.1)

≤n^f^(p)β 1 n

n

X

j=1

kA_jk^µ_p

!_µ¹ ,

wheref(p) =

1 p −¹₂

.

Remark 3. This result extends the results of Bhatia and Kittaneh proved forµ=λ= 2orpandR = (R_jk)where

R_jk =e(ⁱ2π(j−1)(k−1)

n )1.

We use the inequalities (3.1) and the interpolation method to obtain the following inequalities.

Theorem 3.1. Leta, b∈Gl(H)⁺, A∈Bp⁽ⁿ⁾,1≤p <∞andt∈[0,1],then

(3.2) ˜k

n

X

j=1

kA_jk^µ_p,a

!¹_µ

≤

n

X

j=1

kB_jk^λ_p,γ

a,b(t)

!_λ¹

≤K˜

n

X

j=1

kA_jk^µ_p,a

!_µ¹

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where

˜k= ˜k(p, a, b, t) =F(a, a)^t−1F(b, a)^−tn^λ¹⁻^µ¹⁻|p¹−¹₂|α⁻¹, and

K˜ = ˜K(p, a, b, t) = F(a, a)^1−tF(a, b)^tn^λ¹⁻¹^µ⁺|¹_p⁻¹₂|β, if2≤pandλ, µ∈[2, p]or if1≤p≤2andλ, µ∈[p,2].

Proof. We will denote byγ(t) =γ_a,b(t), when no confusion can arise.

Consider the spaceB⁽ⁿ⁾p with the norm:

kAk_p,a;s = (kA₁k^s_p,a+· · ·+kA_nk^s_p,a)^1/s, wherea∈Gl(H)⁺.

By (3.1), the norm ofT_Ris at mostF(a, a)n¹^λ⁻¹^µ⁺|¹p−¹₂|βwhen T_R : B_p⁽ⁿ⁾,k(·, . . . ,·)k_p,a;µ

−→ B_p⁽ⁿ⁾,k(·, . . . ,·)k_p,a;λ , and the norm ofT_Ris at mostF(a, b)n^λ¹⁻¹^µ⁺|¹p−¹₂|βwhen

T_R: B_p⁽ⁿ⁾,k(·, . . . ,·)k_p,a;µ

−→ B_p⁽ⁿ⁾,k(·, . . . ,·)k_p,b;λ .

Therefore, using the complex interpolation, we obtain the following diagram of interpolation fort ∈[0,1]

(Bp⁽ⁿ⁾,k(·, . . . ,·)k_p,a;λ)

(Bp⁽ⁿ⁾,k(·, . . . ,·)k_p,a;µ) ^T^R ^//

TR

**U

UU UU UU UU UU UU UU U

TR

44i

ii ii ii ii ii ii ii ii

(Bp⁽ⁿ⁾,k(·, . . . ,·)k_p,γ_(t);λ)

(Bp⁽ⁿ⁾,k(·, . . . ,·)k_p,b;λ).

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By Theorem2.1,T_Rsatisfies

(3.3) kT_R(A)k_p,γ_(t);λ ≤F(a, a)^1−tF(a, b)^tn¹^λ⁻¹^µ⁺|¹p−¹₂|βkAk_p,a;µ. Now applying the Complex method to

(B⁽ⁿ⁾p ,k(·, . . . ,·)kp,a;λ)

T_R−1

**U

UU UU UU UU UU UU UU UU

(Bp⁽ⁿ⁾,k(·, . . . ,·)kp,γ(t);λ)^T^R⁻¹ ^//(Bp⁽ⁿ⁾,k(·, . . . ,·)k_p,a;µ)

(Bp⁽ⁿ⁾,k(·, . . . ,·)k_p,b;λ)

Ti_Ri−1iiiiiiiii44 ii

ii ii

one obtains

(3.4) kT_R⁻¹(A)k_p,a;µ≤F(a, a)^1−tF(b, a)^tn^µ¹⁻^λ¹⁺|¹_p⁻¹₂|αkAk_p,γ(t);λ. Replacing in (3.4)AbyRAwe obtain

(3.5) kAkp,a;µ ≤F(a, a)^1−tF(b, a)^tn¹^µ⁻¹^λ⁺|¹p−¹₂|αkRAkp,γ(t);λ, or equivalently

(3.6) F(a, a)^t−1F(b, a)^−tn^λ¹⁻^µ¹⁻|p¹−¹₂|α⁻¹kAk_p,a;µ ≤ kT_R(A)k_p,γ(t);λ. Finally, the inequalities (3.3) and (3.6) complete the proof.

We remark that the previous statement is a generalization of Th. 4.1 in [7] where T_n = T_R with R =

e⁽ⁱ2π(j−1)(k−1)

n )1

1≤j,k≤n and a^−1/2 commutes with R for all a∈Gl(H)⁺.

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On the other hand, it is well known that ifx₁, . . . , x_nare non-negative numbers, s ∈ R and we denoteM_s(x) = _n¹ Pn

i=1x^s_i1/s

then for 0 < s < s⁰, M_s(x) ≤ M_s⁰(x).

If we denote kBk = (kB₁k_p, . . . ,kB_nk_p) and we consider 1 < p ≤ 2, then it holds fort∈[0,1]and _s¹

t = ^1−t_p + _q^t that

M_s_t(kBk)≤ M_q(kBk)≤r²^p⁻¹β²^qn⁻¹^q

n

X

j=1

kA_jk^p_p

!¹_p ,

or equivalently

(3.7)

n

X

j=1

kB_jk^s_p^t

!¹

st

≤r²^p⁻¹β²^qn^st¹⁻¹^q

n

X

j=1

kA_jk^p_p

!¹_p .

Analogously, for2≤p < ∞we get

(3.8)

n

X

j=1

kA_jk^p_p

!¹_p

≤ρ¹⁻²^pα²^pn¹^q⁻^st¹

n

X

j=1

kB_jk^s_p^t

!_st¹

where _s¹

t = ^1−t_q +_p^t.

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 andt ∈

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[0,1],

(Bp⁽ⁿ⁾,k(·, . . . ,·)k_p,1;p)

(Bp⁽ⁿ⁾,k(·, . . . ,·)k_p,1;p) ^T^R ^//

TR

**T

TT TT TT TT TT TT TT T

TR

44i

ii ii ii ii ii ii ii i

(B⁽ⁿ⁾p ,k(·, . . . ,·)k_p,1;s_t)

(B⁽ⁿ⁾p ,k(·, . . . ,·)k_p,1;q).

By Theorem2.1and (3.1),T_Rsatisfies

(3.9) kT_R(A)k_p,1;s_t ≤ n^f(p)β^1−t

r²^p⁻¹β²^qt

kAk_p,1;p.

Finally, from the inequalities (3.7) and (3.9) we obtain

n

X

j=1

kB_jk^s_p^t

!_st¹

≤minn

r^p²⁻¹β²^qn^st¹⁻¹^q, n^f^(p)(1−t)β^1+t(²^q⁻¹⁾r⁽²^p^−1)to Xⁿ

j=1

kA_jk^p_p

!¹_p .

We can summarize the previous facts in the following statement.

Theorem 3.2. Let A ∈ Bp⁽ⁿ⁾ and B = RA, where R = (R_jk) is invertible. Let r = maxkR_jkk, ρ = maxk(R⁻¹)_jkkandq be the conjugate exponent of p. Then, fort ∈[0,1]we get

n

X

j=1

kA_jk^p_p

!¹_p

≤minn

ρ¹⁻²^pα²^pn¹^q⁻^st¹ , n^f^(p)tα^t+(1−t)²^pρ⁽¹⁻²^p^)(1−t)o Xⁿ

j=1

kB_jk^s_p^t

!¹_{s t}

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if2≤pand _s¹

t = ^1−t_q + ^t_p,or

n

X

j=1

kBjk^s_p^t

!_st¹

≤min n

r^p²⁻¹β²^qn^st¹⁻¹^q, n^f^(p)(1−t)β^1+t(²^q⁻¹⁾r⁽²^p^−1)t

o Xⁿ

j=1

kAjk^p_p

!¹_p , if1< p≤2and _s¹

t = ^1−t_p +_q^t.

Finally using the Finsler normk(·, . . . ,·)k_p,a;s, Calderón’s method and the previous inequalities we obtain:

Corollary 3.3. Leta, b ∈ Gl(H)⁺, A ∈ Bp⁽ⁿ⁾ and B = RA, whereR = (R_jk) is invertible. Letr= maxkR_jkk,ρ= maxk(R⁻¹)_jkkandqbe the conjugate exponent ofp. Then, fort, u ∈[0,1]we get

n

X

j=1

kA_jk^p_p,a

!¹_p

≤F(a, a)^1−uF(b, a)^uM₁

n

X

j=1

kB_jk^s_p,γ^t

a,b(u)

!_st¹ ,

if2≤p, _s¹

t = ^1−t_q + _p^t and

M₁ =M₁(R, p, t) = minn

ρ¹⁻²^pα²^pn¹^q⁻^st¹ , n^f(p)tα^t+(1−t)²^pρ⁽¹⁻^p²^)(1−t)o or

n

X

j=1

kB_jk^s_p,γ^t

a,b(u)

!_st¹

≤F(a, a)^1−uF(a, b)^uM₂

n

X

j=1

kA_jk^p_p

!¹_p , if1< p≤2, _s¹

t = ^1−t_p +_q^t and M₂ =M₂(R, p, t) = minn

r^p²⁻¹β²^qn^st¹⁻¹^q, n^f^(p)(1−t)β^1+t(²^q⁻¹⁾r⁽²^p^−1)to .

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References

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[4] R. BOAS, Some uniformly convex spaces, Bull. Amer. Math. Soc., 46 (1940), 304–311.

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[10] O. HIRZALLAHANDF. KITTANEH, Non-commutative Clarkson inequalities for unitarily invariants norms, Paciic J. Math., 202 (2002), 363–369.

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