INEQUALITIES IN MATRIX SPACES

on the occasion of his 70th birthday

INEQUALITIES IN MATRIX SPACES

IRINA POPA and NICOLAE POPA

We extend an inequality of A. Shields from 1983, proved for the Schatten classC1, toCpfor 1≤p≤2. We discuss also some more inequalities in Bergman-Schatten classes, extending results from [6].

AMS 2000 Subject Classification: 47B10.

Key words: Hardy-Littlewood inequalities, Schatten class, Bergman-Schatten classes.

1. INTRODUCTION

It is well-known that the Hardy-Littlewood inequalities

∞

X

n=0

(n+ 1)^p−2|a_n|^p

!1/p

≤C(p)kfk_H^p, 0< p≤2 hold for the functions f(t) =P

n≥0a_ne^int belonging to Hardy spaces H^p(T).

See [5].

The similarity between functions and infinite matrices was remarked for the first time by Arazy in [1] and exploited further by Shields in [12].

More precisely, for an infinite matrix A = (a_ij)i,j≥1, we consider the diagonal matrixA_k ^def= (a⁰_ij)i,j≥1,wherea⁰_ij =

a_ij ifj−i=k

0 otherwise ,k∈Z,and we identify the later to the kth Fourier coefficientof the matrixA.

On the other hand, we have enough reasons to consider the Schatten class of orderp, denoted byC_p,to be the similar notion of the Lebesgue space L^p(T),0< p <∞.See [12]. (We note that the above analogy is quite imperfect since for 0 < p₁ < p₂ < ∞,it follows that L^p2(T) ⊂L^p1(T) contrarily to the known inclusion Cp1 ⊂Cp2.)

Then the analogue of Riesz projection P₀(f) ^def= P

n≥0a_ne^int, where f is a trigonometric polynomial, is the triangular projection P_T(A) = (a⁰_ij)i,j≥1,

MATH. REPORTS12(62),2 (2010), 169–180

where

a⁰_ij =

aij if 0≤j−i≤k

0 otherwise ,

A= (aij)i,j≥1 and aij = 0 if |i−j|> kfor some fixed k∈N.

Therefore, the analogue of the Hardy space H^p(T), 0 < p < ∞, is the space T_p ^def= {A|A upper triangular matrix;A∈C_p},with the normkAk_Tp= kAk_Cp.

On the other hand, if f(t) = ane^int, it follows that kfk_H^p = |a_n|, thus the corresponding object to|a_n|would bekA_nk_Cp =k(a_i,i+n)_ik_`p,whereA_n= (ai,i+n)i≥1, n∈N.

Then we may expect that the inequalities (1)

∞

X

n=0

(n+ 1)^p−2kA_nk^p_C

p ≤K(p)kAk^p_T

p

hold for all A∈T_p,1≤p≤2,or, equivalently, (2)

∞

X

n=0

(n+ 1)^p−2

∞

X

i=1

|a_i,i+n|^p

!

≤K(p)kAk^p_T

p, 1≤p≤2.

Forp= 1 the inequality (2) with K(1) =π, that is,

∞

X

n=0

(n+ 1)⁻¹

∞

X

i=1

|a_i,i+n|

!

≤πkAk_T1, A∈T1

was proved by Shields in 1983. See [12].

For another, more general, proof see [4, Theorem 2.2 a)].

Now we describe the content of the paper. In Section 2 we consider first the case 1< p ≤2 and we prove (only for 1< p <2) a weaker result as (1).

We conjecture that (1) holds also in the case 1 < p <2.Of course, (1) is easy to prove in the casep= 2 withK(2) = 1.Also we get a result dual to the first inequality. In Section 3 we introduce the Bergman-Schatten classes and prove the similar inequalities to those proved by Nakamura, Ohya and Watanabe for Bergman function spaces. See [6, pp. 83–84].

2. INEQUALITIES IN SCHATTEN CLASSES

In order to state and prove the result we have to recall some notation from the paper [10].

Let A be an upper triangular matrix and let (Ak)k≥0 be the sequence of its diagonal matrices. We denote by Tp(`²_R) the completion of the space of all finite sequences (A_k)ⁿ_k=0 with respect to the norm k(A_k)ⁿ_k=0k_T

p(`²_R) def=

k(Pn

k=0A^∗_kAk)^1/2k_Cp, where 1≤p ≤2.Here A^∗_k is the adjoint matrix of Ak. It is clear that Tp(`²_R) is a space of upper triangular matrices.

Similarly, Tp(`²_C) is the completion of the space of all finite sequences (A_k)ⁿ_k=0 with respect to the norm k(A_k)ⁿ_k=0k_T

p(`²_C)

def= k(Pn

k=0A_kA^∗_k)^1/2k_Cp. Tp(`²_C) is a space of upper triangular matrices too.

Now, let us denote by Tp(`<sup>2</sup><sub>R</sub>) +Tp(`²_C) the space of all upper triangular matrices A such that there exist A⁰ ∈ Tp(`<sup>2</sup><sub>R</sub>) and A<sup>00</sup> ∈ Tp(`²_C) with A = A⁰+A⁰⁰.We introduce on this space the norm

kAk_T

p(`<sup>2</sup><sub>R</sub>)+Tp(`²_C)

def= inf

Ak=A⁰_k+A⁰⁰_k

(

X

k≥0

A^0∗_kA⁰_k

!1/2 Cp

+

X

k≥0

A⁰⁰_kA^00∗_k

!1/2 Cp

) ,

1≤p≤2.Let us remark that

^∞ X

k=0

A^∗_kA_k 1/2

Cp

=

" _∞ X

j=1

^∞ X

i=1

|a_ij|²

p/2#1/p

, 1≤p≤2, and similarly

∞

X

k=0

AkA^∗_k 1/2

Cp

=

" _∞ X

i=1

∞

X

j=i

|a_ij|²

!p/2#1/p

,

so that kAk_T

p(`<sup>2</sup><sub>R</sub>)+Tp(`²_C) = inf

Ak=A⁰_k+A⁰⁰_k, k≥0

(" _∞ X

j=1

∞

X

i=1

|a_ij|²

!p/2#1/p

+

+

" _∞ X

i=1

∞

X

j=i

|a_ij|²

!p/2#1/p) .

Moreover, the relation (I.13) in [10] implies that

(3) X

k≥0

kA_kk²_C

p

!1/2

≤ kAk_T

p(`<sup>2</sup><sub>R</sub>)+Tp(`²_C). It is useful to remark that

(4) kA_nk_Tp(`2

R)+Tp(`²_C) =kA_nk_Cp = X

i≥1

|a_i,i+n|^p

!1/p

, 1≤p≤2.

Now we state and prove the following result.

Theorem 1. Let 1< p <2 andA∈Tp(`<sup>2</sup><sub>R</sub>) +Tp(`²_C). Then there exists a positive constant K(p) such that

(5)

∞

X

n=0

(n+ 1)^p−2kA_nk^p_C

p ≤K(p)kAk_T

p(`<sup>2</sup><sub>R</sub>)+Tp(`²_C).

Proof. We follow the idea of the proof in [5, pp. 95–97].

LetA be an upper triangular matrix such thatA=Pn

k=0Ak,and letµ be the measure onN defined by

µ(n) = 1

(n+ 1)², n= 0,1,2, . . . . Let us consider A(t) = P∞

n=0Ane^int = A∗Et, where t ∈ T, ∗ means the Schur product of matrices and Et = (e_kj)k,j≥1 is the Toeplitz matrix, where e_kj = e^i(j−k)t,for all j, k≥1.Therefore, kA(t)k_Cp ≤ kAk_Tp.

Now, denote (n+ 1)Anby Aen and let us fixs >0.ThenA(t) = Φs(t) + Ψ_s(t), t∈T,where

Φ_s(t) =

( A(t) ifkA(t)k_Tp > s, 0 ifkA(t)k_Tp ≤s, and Ψs(t) =A(t)−Φs(t), t∈T.

It is clear that we have [A(t)]_n = [Φ_s(t)]_n+ [Ψ_s(t)]_n, n = 0,1,2, . . .; therefore, An∗[Et]n = [Φs]n ∗[Et]n+ [Ψs]n∗[Et]n, where [Φs]n = An and [Ψ_s]_n= 0 if kAk_Tp = sup

t∈T

kA(t)k_Tp > s. The same holds for Ψ_s. Then

X

n≥0

(n+ 1)^p−2kA_nk^p_C

p=X

n≥0

kA˜_nk^p_C

pµ(n)≤ (6)

≤2^p

X

n≥0

k[ ˜Φs]nk^p_C

pµ(n) +X

n≥0

k[ ˜Ψs]nk^p_C

pµ(n)

.

Put nowα(s) =µ{n;k[ ˜Φs]nk_Cp > s} and β(s) =µ{n;k[ ˜Ψs]nk_Cp > s}= µ(E_s).Then

(7) s²β(s)≤X

n≥0

[Ψ_s]_n

2 Cp.

Let us denote by Fs the set {n≥ 0;k[ ˜Φs]nk_Cp > s}.Since kA_nk_Cp ≤ kAk_Cp, n∈N,(see [7]) we have

µ(Fs) =µ{n; (n+ 1)k[Φ_s]nk_Cp > s} ≤µ{n; (n+ 1)kΦ_sk_Cp > s}=

= X

{n;kΦsk_Cp>_n+1^s }

1

(n+ 1)² ≤4 X

{n;kΦsk_Cp>_n+1^s }

Z n+1 n

1

x²dx≤ 4 n₀+ 1, where n0= min{n;kΦ_sk_Cp > _n+1^s }.

Thus

(8) α(s) =µ(F_s)≤ 4kΦ_sk_Cp

s , s >0.

By (9) we have

∞

X

n=0

k[ ˜Φ_s]_nk^p_C

pµ(n) =− Z ∞

0

s^pdα(s) =p Z ∞

0

s^p−1α(s)ds≤

≤4p Z ∞

0

s^p−1kΦ_sk_Cp

s ds≤4p

Z kAk_Cp 0

kAk_Cps^p−2ds≤ 4p p−1kAk^p_T

p, for all t∈T.

By (8) it follows

∞

X

n=0

k[ ˜Ψ_s]_nk^p_C

pµ(n) =p Z ∞

0

s^p−1β(s)ds≤p Z ∞

0

s^p−3

∞

X

n=0

k[Ψ_s]_nk²_C

pds=

=p Z ∞

kAk_Cp

∞

X

n=0

kA_nk²_Cps^p−3ds≤ 2p

2−pkAk^p−2_C

p

∞

X

n=0

kA_nk²_Cp ≤

(by (4))

≤ p

2−pkAk^p−2_T

p kAk²_T

p(`<sup>2</sup><sub>R</sub>)+Tp(`²_C). Using (7) we get

(9)

∞

X

n=0

(n+ 1)^p−2kA_nk^p_C

p ≤K(p) h

kAk^p_T

p+kAk^p−2_T

p kAk²_T

p(`<sup>2</sup><sub>R</sub>)+Tp(`²_C)

i .

But in [8] it is proved that kAk_Cp ≤ kAk_Tp(`2

R)+Tp(`²_C), consequently we get the inequality of Hardy-Littlewood type

(10) X

n≥0

(n+ 1)^p−2kA_nk^p_C

p ≤K(p)kAk^p

Tp(`<sup>2</sup><sub>R</sub>)+Tp(`²_C), 1< p <2.

SinceT_p(`<sup>2</sup><sub>R</sub>)+T<sub>p</sub>(`²_C) is the completion of the space of all finite sequences, Theorem 1 is proved.

Remark. Theorem 1 is weaker than thestrong version of Hardy-Littlewood inequality

(11) X

n≥0

(n+ 1)^p−2kA_nk^p_C

p ≤K(p)kAk^p_T

p,

for all upper triangular matrices A. Indeed, in [8] it is proved that there is an upper triangular matrix A ∈ C_p such that A 6∈ T_p(`<sup>2</sup><sub>R</sub>) +T<sub>p</sub>(`²_C), 1 < p <2.

Then it is natural to ask the following:

Question. Does Theorem 1 hold for 1< p <2 with (12) instead of (6)?

Using the duality betweenTp(`<sup>2</sup><sub>R</sub>) +Tp(`²_C) and Tp(`<sup>2</sup><sub>R</sub>)∩Tp(`²_C) (see [9]) given by

hA, Bi=

∞

X

k=0

tr(AkB_k^∗), where A=P

k≥0A_k, B =P

k≥0B_k,we have Theorem 2. Let 2 ≤ q < ∞ and A = P

k≥0Ak such that P

n≥0(n+ 1)^q−2kA_nk^q_C

q <∞. ThenA∈Tq(`<sup>2</sup><sub>R</sub>)∩Tq(`²_C) and kAk_T

q(`<sup>2</sup><sub>R</sub>)∩Tq(`²_C)=

def= max ^∞

X

i=1

^∞ X

j=i

|a_i,j|²

q/21/q

, ^∞

X

j=1

^j X

i=1

|a_i,j|²

q/21/q!

≤

≤C(q) X

n≥0

(n+ 1)^q−2 ∞

X

i=1

|a_i,i+n|^q !1/q

=C(q)

X

n≥0

(n+ 1)^q−2kA_nk^q_C

q

1/q

.

Proof. Let p = q/(q −1) and G = Pn

k=0Gk be a finite type band matrix with

kGk_T

p(`<sup>2</sup><sub>R</sub>)∩Tp(`²_C)≤1.

Let nowSn(A) =Pn

k=0A_k.Then

|hG, S_n(A)i|=

n

X

k=0

tr(G_kA^∗_k) ≤

n

X

k=0

∞

X

i=1

g_i,k+ia_i,k+i ≤

(by H¨older inequality)

≤

n

X

k=0

X

i

|g_i,k+i|^p 1/p

X

i

|a_i,k+i|^q 1/q

=

=

n

X

k=0

kG_kk_CpkA_kk_Cq (again by H¨older inequality)

≤

n

X

k=0

kG_kk^p_C

p k+ 1p−21/p

n

X

k=0

kA_kk^q_C

q(k+ 1)^q−2 1/q

≤

(by the previous theorem)

≤ C(p)kGk_Tp(`2

R)+Tp(`²_C) n

X

k=0

kA_kk^q_C

q(k+ 1)^q−2

!1/q

≤

≤C(p)

n

X

k=0

kA_kk^q_C

q(k+ 1)^q−2

!1/q

.

Hence

kS_n(A)k_T

q(`<sup>2</sup><sub>R</sub>)∩Tq(`²_C) = sup

kGkTp(`2 R)+Tp(`2

C)≤1

|hG, S_n(A)i| ≤

≤C(p)

n

X

k=0

kA_kk^q_C

q(k+ 1)^q−2

!1/q

,

for all n and the theorem is proved.

Now we discuss an inequality of Hausdorff-Young type, which is probably known. We include here for the convenience of the reader.

Theorem 3(Hausdorff-Young Theorem). For 1 ≤p ≤ ∞, let q be the conjugate index, with ¹_p +¹_q = 1.

(i)If 1≤p≤2 thenA∈Tp implies P∞

n=0kA_nk^q_T

p

1/q

≤ kAk_Tp. (ii)If2≤p≤ ∞then{kA_nk_Tp} ∈`_qimplieskAk_Tp≤

P∞

n=0kA_nk^q_T

p

1/q

. Proof. In case (ii), ifp= 2 we havep=q= 2 and if P∞

n=0

P∞

l=1|a_l,n+l|²1/2

< ∞ then, clearly, A= (aij)∈ T2 and kAk_T2 ≤ P∞ n=0

P∞

l=1|a_l,n+l|²1/2

, in other words the map T({A_n}_n≥0) =P

n≥0A_n has the norm less than 1 from the space `2(`2) into T2. Here `p(`q) means the space of all matrices (aij)i,j

such that P∞ n=1

P∞

i=1|a_i,i+n|² <∞.

Ifp=∞, q = 1 and the mapT has the norm less than 1 from`<sub>1</sub>(`∞) into T∞, since, clearly, kAk_T∞ ≤ P∞

n=0kA_nk_T∞. Using the complex interpolation we get the conclusion.

Case (i) results by duality, since`<sub>p</sub>(`_q)^∗ =`<sub>q</sub>(`_p),and (T_p)^∗ =T_q.

3. INEQUALITIES IN BERGMAN-SCHATTEN CLASSES

Let 0≤p <∞.We denote L^p(D, `2) :=

n

r→A(r) a strong measurableCp-valued function defined on [0,1); such that kA(r)k_Lp(D,`2):=

2

Z 1 0

kA(r)k^p_C

prdr 1/p

<∞o , where C_p is the Schatten class of orderp,and we denote

L˜^p_a(D, `<sub>2</sub>) ={A(r) :=A∗P(r); A upper triangular matrices | kAk<sub>L</sub>p(D,`2)<∞}, which is a subspace of L^p(D, `₂).

By L^p_a(D, `₂) we mean the space of all upper triangular matrices such thatkA(r)k_L^p

a(D,`2) <∞, whereA(r) =P(r)∗A,r∈[0,1).We call the spaces L<sup>p</sup>a(D, `2),the Bergman-Schatten classes.

We can state and prove the matrix version of Theorem 3, [6, p. 83]:

Theorem 4.1.Ifp= 1andA∈L¹_a(D, `₂), thenP∞

n=1n⁻²kA_nk_C1 <∞.

2. If 1< p≤2 and A∈L^p,unc_a (D, `₂) =n

A| kAk_p,unc^def= Z 1

0

Z 1 0

X

k≥0

_k(t)A_kr^k

p Cp

rdrdt 1/p

<∞o ,

where _k means thekth Rademacher function, then we have

∞

X

n=1

n^p−3kA_nk^p_C

p <∞.

3. If 2≤p <∞ and P∞

n=1n^p−3kA_nk^p_C

p <∞, thenA∈L^p,unca (D, `2).

4. If 2≤p <∞ and A∈L^pa(D, `₂),then P∞

n=1n⁻¹kA_nk_Cp <∞.

5. If 1< p≤2 and P

n≥0 1

n+1kA_nk^p_C

p <∞,thenA∈L^pa(D, `2).

Proof. 1. Let A ∈ L¹_a(D, `₂). We use the results of Shields [11] applied to the function z→A(rz),with a fixed 0< r <1.Then we have

∞

X

n=0

(n+ 1)⁻¹kA_nrⁿk_C1 ≤C{M_1,1(r, A)}, where M_1,1(r, A) = _2π¹ R2π

0 kP

n≥0A_nrⁿe^inθk₁dθ. Multiplying the above ine- quality by r and integrating on (0,1) we have

∞

X

n=0

(n+ 1)⁻¹kA_nk_C1 Z 1

0

rⁿ⁺¹dr≤ kAk_L1 a,

or ∞

X

n=0

(n+ 1)⁻²kA_nk_C1 ≤CkAk_L1 a.

2. Let nowA∈A^p,unc and 1< p≤2.By Theorem 1 applied forA_r(z) = A(rz), 0< r <1,with Ar∈Tp(`<sup>2</sup><sub>R</sub>) +Tp(`²_C),we have

∞

X

n=0

(n+ 1)^p−2kA_nrⁿk^p_C

p ≤C(p) Z 1

0

∞

X

k=0

k(t)Akr^k

p Cp

dt.

Proceeding as in 1. we get

∞

X

n=0

(n+ 1)^p−2kA_nk^p_C

p

Z 1 0

r^np+1dr ≤C(p)kAk^p_p,unc,

or ∞

X

n=0

(n+ 1)^p−3kA_nk^p_C

p≤C(p)kAk^p_p,unc.

3. By Theorem 3 applied toAr(z) =A(rz),for 0< r <1,we have kA_rk^p_p,unc≤C(p)

∞

X

n=0

(n+ 1)^p−2kA_nrⁿk^p_C

p

and, consequently, Z 1

0

∞

X

k=0

_k(t)A_kr^k

p Cp

dt≤C(p)

∞

X

n=0

(n+ 1)^p−2kA_nrⁿk^p_C

p. Hence

Z 1 0

Z 1 0

∞

X

k=0

_k(t)A_kr^k

p Cp

rdr

dt≤C(p)

∞

X

n=0

(n+ 1)^p−2kA_nk^p_C

p

Z 1 0

r^np+1dr and finally

kAk^p_p,unc≤C(p)

∞

X

n=0

(n+ 1)^p−3kA_nk^p_C

p.

4. We have the following interpolation theorem (see [11]): Let1< p <∞ and 1−θ= 1/p.Then L^p_a(D, `<sub>2</sub>) = [L<sup>1</sup><sub>a</sub>(D, `₂),B(`₂)]_θ with equivalent norms.

In the same manner we get L^pa(D, `2) = [L<sup>2</sup><sub>a</sub>(D, `2),B(`<sub>2</sub>)]θ with 2/p = 1 −θ. Now, let T : B(`₂) → `∞(`∞); given by T(A) = (A_n)n≥0. T maps continuouslyL²_a(D, `2) into`2(`2, w),wherew(n) = 1/(n+1),for alln≥0,and (`₂, w) is the space of all sequences x= (x_n)n≥0 such thatP

n≥0|x_n|²w(n)<

∞,with natural norm. Indeed, similarly to Theorem at page 84 in [6], we get kT(A)k² =X

n≥0

1

n+ 1kA_nk²_∞≤C(2)kAk²_L2 a(D,`2).

Hence T :L^pa(D, `2) →[`2(`2, w), `∞(`∞)]θ =`p(`p, w), whereθ = 1−2/p, is a bounded operator, see [13] for the last equation, and



 X

n≥0

1

n+ 1kA_nk^p_C

p





1/p

≤C(p)kAk_L^p

a(D,`2).

5. We have [L^p_a(D, `<sub>2</sub>)]<sup>∗</sup> = L<sup>q</sup><sub>a</sub>(D, `₂), 1/p+ 1/q = 1. See [11]. On the other hand [`p(`p, w)]^∗ = `q(`q, w) for 1/p+ 1/q = 1. Then the conclusion follows from 4.

Now, we give the Hausdorff-Young Theorem for Bergman-Schatten classes extending Theorem 2 [6, pages 81–82].

Theorem 5. Let 1≤p≤ ∞ and letq = _p−1^p be its conjugate exponent.

(i)If 1≤p≤2 thenA∈L^pa(D, `2) implies that



 X

n≥0

(n+ 1)^1−qkA_nk^q_T

p





1/q

≤ kAk_Tp.

(ii)If 2≤p≤ ∞then P

n≥0(n+ 1)^1−qkA_nk^q_T

p <∞ implies that kAk_Tp ≤

∞

X

n=0

(n+ 1)^1−qkA_nk^q_T

p

!1/q

.

Proof. In case (ii), let µ be the discrete measure on the set N which assigns the massµ(n) =n+1 to the integern= 0,1,2, . . .. Consider the linear operatorT that maps the sequence{_n+1¹ A_n}to the formal seriesP

n≥0A_nzⁿ. We want to show that T is bounded as an operator from L^q(N,dµ;Tp) to L^p(D,dσ;T_p), with norm kTk ≤ 1, where L^p(D,dσ;T_p) is the space of all p-Lebesgue Bochner integrable Tp-valued functions defined on the unit disk D with respect to the area measure dσ on D. For p = 2 this follows from the relation

kT({A_n/(n+ 1)})k²_L2(D,dσ;T2)= 2X

n≥0

kA_nk²_C

2(n+ 1)⁻¹=

=k{A_n/(n+ 1)}k²_L2(N,dµ;T2). For p=∞ it is trivial that sup_|z|≤1kP

n≥0 A_nzⁿk_T∞ ≤P

n≥0kA_nk_T∞. We use now the complex interpolation of vector valuedL^p spaces.

Case (i). Let A(z) ∈ L^p(D,dσ;T_p). Define the linear operator T(A) = {b_n}, wherebn =R

DA(z)zⁿdσ. IfA∈L^pa(D, `2) thenbn =An/(n+ 1).With the measureµdefined as before, we want to show thatT is a bounded operator from L^p(D,dσ;Tp) toL^q(N,dµ;Tp),with norm kTk ≤1.Forp= 1 this is the

trivial inequality kb_nk_T1 ≤ kAk_L1(D,dσ;T1) for all n ∈ N. For p = 2 it follows from the relation

∞

X

n=0

(n+ 1)kb_nk²_T

2 =

∞

X

n=0

(n+ 1) Z

D

A(z)zⁿdσ

2

=

= Z

D

Z

D

A(z)A(ζ)

∞

X

n=0

(n+ 1)(zζ)ⁿdσ(z)dσ(ζ) =

= Z

D

Z

D

A(z)A(ζ)

(1−zζ)²dσ(z)dσ(ζ) =hP A(·), A(·)i, where P denotes the Bergman projection. Since

|hP A(·), A(·)i| ≤ kP A(·)k₂kA(·)k₂≤ kA(·)k²₂, this gives the desired inequality for p= 2.

One can conclude by interpolation as before that

∞

X

n=0

(n+ 1)kb_nk^q_T

p

!1/q

≤ Z

D

kA(z)k^p_T

pdσ 1/p

, A(·)∈L^p(D,dσ;T_p).

Specializing toA∈L^pa(D, `2) and recalling thatbn=An/(n+ 1),we arrive at the desired result.

REFERENCES

[1] J. Arazy,Some remarks on interpolation theorems and the boundness of the triangular projection in unitary matrix spaces.Integral Eq. Op. Th.1(1978), 453–495.

[2] A.B. Aleksandrov and V.V. Peller,Hankel and Toeplitz-Schur multipliers. Math. Ann.

324(2002), 277–327.

[3] S. Barza, L.E. Persson and N. Popa, A matriceal analogue of Fejer’s theory. Math.

Nachr.260 (2003), 14–20.

[4] O. Blasco and A. Pelczynski, Theorems of Hardy and Paley for vector-valued analytic functions and related classes of Banach spaces. Trans. Amer. Math. Soc. 323(1991), 335–367.

[5] P.L. Duren,Theory ofH^pSpaces. Academic Press, New York, 1970.

[6] P.L. Duren and A. Schuster,Bergman Spaces. Mathematical Surveys and Monographs, Vol.100. Amer. Math. Soc., Providence, RI, 2004.

[7] I.C. Gohberg and M.G. Krein,Introduction to the Theory of Linear Nonselfadjoint Op- erators.Translations of Mathematical Monographs, Vol.18. Amer. Math. Soc., Provi- dence, RI, 1969.

[8] A. Harcharras, Fourier analysis, Schur multipliers on S^p and non-commutative Λ(p)- sets. Studia Math.137(1999), 203–260.

[9] F. Lust-Piquard,In´egalit´es de Khintchine dans Cp,1< p <∞. C. R. Acad. Sci. Paris 303(1986), 289–292.

[10] F. Lust-Piquard and G. Pisier, Non-commutative Khintchine and Paley inequalities.

Ark. Mat.29(1991), 241–260.

[11] N. Popa, Matriceal Bloch and Bergman-Schatten spaces. Rev. Roumaine Math. Pures Appl.52(2007), 459–478.

[12] A.L. Shields, An analogue of a Hardy-Littlewood-Fejer inequality for upper triangular trace class operators.Math. Z.182(1983), 473–484.

[13] H. Triebel,Interpolation Theory, Function Spaces, Differential Operators. Johann Am- brosius Barth, Heidelberg, 1995.

Received 25 September 2009 Technical University of Civil Engineering Department of Mathematics and Informatics

020396, Bucharest, Romania [email protected]

and Romanian Academy

“Simion Stoilow” Institute of Mathematics P.O. Box 1-764

014700 Bucharest, Romania [email protected]