πX∞ m=1 a2m ∞ X n=1 b2n12 where the constant factorπis the best possible

ARCHIVUM MATHEMATICUM (BRNO) Tomus 47 (2011), 229–243

HILBERT INEQUALITY FOR VECTOR VALUED FUNCTIONS

Namita Das and Srinibas Sahoo

Abstract. In this paper we consider a class of Hankel operators with operator valued symbols on the Hardy spaceH²_Ξ(T) where Ξ is a separable infinite dimensional Hilbert space and showed that these operators are unitarily equivalent to a class of integral operators inL²(0,∞)⊗Ξ.We then obtained a generalization of Hilbert inequality for vector valued functions. In the continuous case the corresponding integral operator has matrix valued kernels and in the discrete case the sum involves inner product of vectors in the Hilbert space Ξ.

1. Introduction Ifam, bn≥0 satisfy 0<P∞

m=1a²_m<∞and 0<P∞

n=1b²_n<∞, then (1.1)

∞

X

m=1

∞

X

n=1

ambn

m+n < πX^∞

m=1

a²_m

∞

X

n=1

b²_n¹₂

where the constant factorπis the best possible.

The integral version of the inequality (1.1) is as follows:

Iff,g≥0 andf,g∈L²(0,∞), then (1.2)

Z ∞ 0

Z ∞ 0

f(x)g(y)

x+y dx dy < πZ ∞ 0

f²(x)dx Z ∞

0

g²(x)dx¹₂ where the constant factorπis the best possible.

The inequalities (1.1) and (1.2) are the well-known Hilbert’s inequality (see Hardy et. al [4], Ch-9). Hardy and Riesz [3] gave the following generalizations of (1.1) and (1.2) for conjugate parameters.

Let ¹_p + ¹_q = 1, 1 < p < ∞, am, bn ≥ 0 satisfy 0 < P∞

m=1a^p_m < ∞ and 0<P∞

n=1b^q_n<∞. Then (1.3)

∞

X

m=1

∞

X

n=1

a_mb_n

m+n < π sin(π/p)

X^∞

m=1

a^p_m¹_pX^∞

n=1

b^q_n¹_q

where the constant factorπ/sin(π/p) is the best possible.

The integral version of the inequality (1.3) is as follows:

2010Mathematics Subject Classification: primary 26D15.

Key words and phrases: Hardy-Hilbert’s integral inequality,β-function, Hölder’s inequality.

Received November 24, 2010, revised March 2011. Editor O. Došlý.

Let ¹_p+¹_q = 1, 1< p <∞,f ∈L^p(0,∞),g∈L^q(0,∞). Then (1.4)

Z ∞ 0

Z ∞ 0

f(x)g(y)

x+y dx dy < π sin(π/p)

Z ∞ 0

f^p(x)dx¹_pZ ∞ 0

g^q(x)dx¹_q where the constant factorπ/sin(π/p) is the best possible.

The inequalities (1.3) and (1.4) are well-known as Hardy-Hilbert’s inequality.

These inequalities are important in analysis and its applications (see [6, Ch-5], [4, Ch-9]).

LetL²(T) denote the Hilbert space of square integrable, Lebesgue measurable complex valued functions on the unit circleT={z∈C| |z|= 1}, with pointwise algebraic operations and inner product

hf, gi= 1 2π

Z π

−π

f(e^iθ)g(e^iθ)dθ .

LetL^∞(T) denote the Banach space of essentially bounded, Lebesgue measurable, complex valued functions onTwith pointwise algebraic operations and essential supremum norm

kfk_∞= ess sup

|z|=1

|f(z)|.

Forp= 2 or∞, letH^p(T) be the closed subspace{f ∈L^p(T) :f(n) = 0 forb n <

0}ofL^p(T), with the restriction of the norm ofL^p. Heref(n) denote theb nth Fourier coefficient off. The spacesH²(T) andH^∞(T) are called Hardy spaces. The space H²(T) is a Hilbert space andH^∞(T) is a Banach space. Clearly, H^∞(T)⊂ H²(T).

Forϕ∈L^∞(T), the Hankel operatorSϕwith symbolϕ, fromH²(T) into itself is defined by Sϕf = P J(ϕf) where P is the orthogonal projection from L²(T) onto H²(T) and J: L²(T) → L²(T) is defined by J f(e^it) = f(e^−it). There are some useful unitary equivalences between Hankel operators and Hankel integral operators as we discuss in the following examples.

Example 1.1. Consider the function

ϕ(e^iθ) =−i(π−θ), 0≤θ <2π . Thenϕ∈L^∞(T) and if

ϕ(e^iθ) =

∞

X

n=−∞

a_ne^inθ

then

an=

( 0 if n= 0 ;

−_n¹ if n6= 0.

Hence the matrix of the Hankel operatorS_zϕ with respect to the standard ortho- normal basis of H²(T) is the Hilbert’s matrix

Γ =





















1 ¹₂ ¹₃ · · · ·

1 2

1 3

1

4 · · · ·

1 3

1 4

1

5 · · · ·

· · · ·



















 .

It is not difficult to see that the Hilbert’s matrix Γ as an operator on l²(Z+) is unitarily equivalent to the integral operator

(1.5)

K e^h

f (x) =

Z ∞ 0

eh(x+y)f(y)dy, f∈L²(0,∞),

whereeh(x) = ^e−x_x . This integral operator is known as Hankel integral operator. For more details see [8].

Example 1.2. Consider the classical singular integral operator, the Carleman’s operator defined onL²(0,∞) by

(1.6) (Khf) (x) =

Z ∞ 0

h(x+y)f(y)dy,

where the kernel function is h(x) = ¹_x. It is easy to verify (see [8]) that the Carleman’s operatorKh is unitarily equivalent to the Hankel operator defined on H²(T) whose matrix with respect to the standard orthonormal basis is

(1.7) S= 2





































1 0 ¹₃ 0 ¹₅ · · ·

0 ¹₃ 0 ¹₅ · · · ·

1

3 0 ¹₅ · · · · 0 ¹₅ · · · ·

1

5 · · · ·

· · · ·



































 .

Such integral operators are widely studied in the literature (see [7, 8]). It is not difficult to see thatkSzϕk=kK

e^h

k=kKhk=π.

Example 1.3. Letψ(z) =χ i^1−z_1+z

,|z|= 1,z6=−1 where χ(t) =

(1, t∈[−1,1]

0, t /∈[−1,1].

Thenψis the characteristic function of the set{z∈C:|z|= 1,Rez≥0}. In the Hilbert space l²(Z+) introduce the Hankel operator Γ_ψ with symbolψdefined by

(1.8) (Γ_ψx)_n =

∞

X

k=0

c_n+k+1x_k,

for x = (x0, x1, x2,· · ·) ∈ l²(Z+), where ck are the Fourier coefficients of the functionψ,

c_k(ψ) = 1 2π

Z 2π 0

e^ikθψ(e^iθ)dθ= 2

πksinπk 2

, k∈N.

The operator Γψ defined in (1.8) is unitarily equivalent to the Hankel integral operator K1

2 defined onL²(0,∞) as K1

2f (x) =

Z ∞ 0

2 π

sin(x+y)

x+y f(y)dy . For details see [5].

In this paper we observe that certain Hankel operators onH²_Ξ(T) are unitarily equivalent to a class of integral operators on L²_Ξ(0,∞), where Ξ is an infinite dimensional separable Hilbert space and using the unitary equivalence of these operators generalize the Hilbert inequality for vector valued functions. In §22, we deal with the Hankel integral operatorK

e^h

defined in (1.5). We show that the norm ofK

e^h

as an operator onL^p(0,∞) is equal toπ/sin(π/p), 1< p <∞and derive the associated integral inequality. As a consequence of this we have obtained an integral inequality involving the kernel [cosh(t−s)]⁻¹inL²(−∞,∞). In §33, we obtain the discrete and integral version of Hilbert’s inequality forCⁿ−valued functions and show that in the continuous case the corresponding integral operator has matrix valued kernel and in the discrete case the sum involves the inner product of vectors in the Hilbert spaceCⁿ. In §44, we consider the case ofH²_Ξ(T) and generalize the results of §33. Further, we also generalize the discrete version of Hilbert inequality for sequences in a Hilbert spaceH.

2. Norm of the Hankel integral operator K e^h and the associated inequalities

In this section we find the norm of K e^h

as an operator fromL^p(0,∞) into itself, 1< p <∞. But we establish first the discrete version of a Hilbert type inequality.

Theorem 2.1. Ifam, bn∈Csatisfy 0<P∞

m=0|am|²<∞and0<P∞

n=0|bn|²<

∞, then

(2.1)

∞

X

m, n=0 m+neven

a_m¯b_n m+n+ 1

≤π

2 X^∞

m=0

|am|²¹₂X^∞

n=0

|bn|²¹₂

and the constant factor ^π₂ is the best possible.

Proof. Letψ(z) =P∞ n=0

2

2n+1z^−(2n+1). Then ψ(−k) =ˆ

(₂

k if k is odd, k >0;

0, otherwise.

Hence for m, n= 0,1,2,· · ·,

cmn=hSzψz^m, zⁿi=hP J(zψz^m), zⁿi=hJ(ψz^m+1), zⁿi

=hψz^m+1, z⁻ⁿi=hψ, z^−(m+n+1)i

= ( ₂

m+n+1 if m+n+ 1 is odd;

0, otherwise.

Thus the matrix S given in (1.7) is the matrix ofS_zψ. Letf(z) =P∞

m=0a_mz^m andg(z) =P∞

n=0bnzⁿ and supposef, g∈ H²(T). Now as we have mentioned in example 1.2 that kSzψk =kKhk. It follows from the Hilbert integral inequality (1.2) thatkKhk=π. HencekSzψk=π. It follows from Cauchy-Schwarz inequality that

|hSzψf, gi| ≤ kSzψk kfk kgk ≤πkfk kgk. But

|hS_zψf, gi|= D

S_zψX^∞

m=0

a_mz^m ,X^∞

n=0

b_nzⁿE

=

∞

X

m,n=0

am¯bnhSzψz^m, zⁿi

= 2

∞

X

m,n=0 m+n even

a_m¯b_n m+n+ 1

.

Thus

∞

X

m,n=0 m+n even

a_m¯b_n m+n+ 1

≤ π

2 nX^∞

m=0

|am|²o¹₂nX^∞

n=0

|bn|²o¹₂ .

We now proceed to show that the norm of the operatorK

e^h

as an operator from L^p(0,∞) into itself is equal to _sin(π/p)^π , if 1< p <∞. It also gives us the following Hardy-Hilbert type integral inequality.

Theorem 2.2. Let ¹_p+¹_q = 1,1< p <∞,f ∈L^p(0,∞),g∈L^q(0,∞). Then Z ∞

0

Z ∞ 0

e^−(x+y)

x+y f(x)g(y)dx dy≤ π sin^π_p

Z ∞ 0

f^p(x)dx¹_pZ ∞ 0

g^q(y)dy¹_q and the constant factor _sin^ππ

p

is the best possible.

Proof. It follows from Hardy-Hilbert’s integral inequality (1.4), that Z ∞

0

Z ∞ 0

e^−(x+y)

x+y f(x)g(y)dx dy≤ π sin^π_p

Z ∞ 0

e^−pxf^p(x)dx¹_pZ ∞ 0

e^−qyg^q(y)dy¹_q

≤ π

sin^π_p Z ∞

0

f^p(x)dx_p¹Z ∞ 0

g^q(y)dy¹_q as e^−pt≤1 for t∈(0,∞).

It remains to show that the constant factor 1 in the inequality (2.2)

Z ∞ 0

e^−pxf^p(x)dx≤ Z ∞

0

f^p(x)dx is the best possible.

Suppose there exists a constantk, 0< k <1 such that (2.3)

Z ∞ 0

e^−pxf^p(x)dx < k Z ∞

0

f^p(x)dx for allf ∈L^p(0,∞).

Setting

fe(x) =

(1, 0≤x≤ ¹_plog¹_k 0, x > ¹_plog¹_k. we have

Z ∞ 0

fe^p(x)dx=

Z ¹_plog_k¹ 0

dx= 1 plog1

k; hencefe∈L^p(0,∞). Now

(2.4)

Z ∞ 0

e^−px−k

fe^p(x)dx= 1 p+k

plog k

e

.

Consider the function g(t) =−e^−pt+1−kpt, t∈[0,∞). Then g⁰(t) =pe^−pt−kp= 0 for t= ¹_plog¹_k andg⁰⁰(t) =−p²e^−pt<0 fort= ¹_plog¹_k. Hence g(t)> g(0) for t=_p¹log_k¹. Therefore 1 +klog ^k_e

>0. Now from (2.4) we get Z ∞

0

e^−px−k

fe^p(x)dx >0.

This is a contradiction to the assumption (2.3), which shows that the constant factor 1 in the inequality (2.2) is the best possible. Again the constant factor _sin^ππ

p

is the best possible in the Hardy-Hilbert’s integral inequality (1.4). Hence the result

follows.

Remark 2.3. It follows from Theorem 2.2 that kKhk=kK

e^h

k= π

sin(π/p).

As a consequence of Theorem 2.2, we obtain the following integral inequality involving the kernel [cosh(t−s)]⁻¹ in L²(−∞,∞).

Corollary 2.4. If f, g∈L²(−∞,∞)then

Z ∞

−∞

[cosh(t−s)]⁻¹f(s)g(t)ds dt

≤πkfk_L2(−∞,∞)kgk_L2(−∞,∞). Proof. Consider the map W: L²(0,∞)→L²(−∞,∞) defined by

W f(t) =√

2e^tf(e^2t).

The operatorW is a unitary operator. Letf be a continuous function with compact support in (0,∞) andh(x+y) =_x+y¹ , x=e^2t, y=e^2s. Then

(K_hf)(x) = Z ∞

0

f(y) x+ydy=

Z ∞

−∞

f(e^2s)2e^2s

e^2t+e^2s ds= 1

√2e^t Z ∞

−∞

2e^se^t

e^2t+e^2sW f(s)ds

= 1

√ 2e^t

Z ∞

−∞

[cosh(t−s)]⁻¹W f(s)ds= (W^∗CW f)(x) since ifg∈L²(−∞,∞) then ^√g(t)

2e^t = ^√1

2xg(¹₂logx) =W^∗g(x). ThusK_h=W^∗CW whereC is the convolution with (cosht)⁻¹. That is,

(Cf)(t) = Z ∞

−∞

[cosh(t−s)]⁻¹f(s)ds . SinceK_h andC are unitarily equivalent hencekCk=πand

|hCf, gi| ≤πkfk_L2(−∞,∞)kgk_L2(−∞,∞). Thus

Z ∞

−∞

[cosh(t−s)]⁻¹f(s)g(t)ds dt

≤πkfk_L2(−∞,∞)kgk_L2(−∞,∞).

3. Hilbert inequality for vector valued functions

In this section we generalize the discrete version of the Hilbert inequality (1.1) and here the sum involves the inner product of vectors in a Hilbert spaceH. Let L(H) denote the set of all bounded linear operators from the Hilbert spaceHinto itself.

Theorem 3.1. Let(xn)and(yn)be two sequences inHsuch that0<P∞

n=1kxnk²<

∞ and0<P∞

n=1kynk²<∞. Then (3.1)

∞

X

m=1

∞

X

n=1

|hxm, yni|

m+n < πnX^∞

m=1

kxmk²o¹₂nX^∞

n=1

kynk²o¹₂ ,

where the constant factorπ is the best possible.

Proof. Let H6={0}be a Hilbert space andE be an orthonormal basis forH. The set{e∈ E| hz, ei 6= 0 for somez =x_mory_n} is countable, let us enumerate this set as the sequence (e₁, e₂, e₃, . . .). Then every x_m andy_n can be expressed as

x_m=

∞

X

k=1

a_mke_k; y_n=

∞

X

k=1

b_nke_k,

whereamk=hxm, eki,bnk=hyn, eki. Then hxm, yni=

∞

X

k=1

amkbnk. By Parseval relationkx_mk²=P∞

k=1|a_mk|², for everymandky_nk²=P∞

k=1|b_nk|², for everyn. So, we have|amk| ≤ kxmkfor allmand|bnk| ≤ kynkfor alln. Hence for eachk,P∞

m=1|amk|²<∞andP∞

n=1|bnk|²<∞. Now using Hilbert’s inequality, we have for eachk,

∞

X

m=1

∞

X

n=1

|amk| |bnk|

m+n < πnX^∞

m=1

|a_mk|²o¹₂nX^∞

n=1

|b_nk|²o¹₂ .

Taking summation overkfrom 1 top, and using Cauchy-Schwartz inequality, we get

p

X

k=1

∞

X

m=1

∞

X

n=1

|amk| |bnk|

m+n < πnX^p

k=1

∞

X

m=1

|amk|²o¹₂nX^p

k=1

∞

X

n=1

|bnk|²o¹₂

=πnX^∞

m=1 p

X

k=1

|amk|²o¹₂nX^∞

n=1 p

X

k=1

|bnk|²o¹₂ .

It follows therefore that for everyp≥1, (3.2)

p

X

k=1

∞

X

m=1

∞

X

n=1

|a_mk| |b_nk|

m+n < πnX^∞

m=1

kxmk²o¹₂nX^∞

n=1

kynk²o¹₂ .

Notice that

|hxm, yni|=

∞

X

k=1

amkbnk

≤

∞

X

k=1

|amk| |bnk|.

It follows from the relation|amk| |bnk| ≤ ¹₂(|amk|²+|bnk|²) and the convergence of the series P∞

k=1|a_mk|² andP∞

k=1|b_nk|². Thus lettingp→ ∞ in (3.2), we obtain (3.1). In particular for the Hilbert space H= R, (3.1) reduces to the Hilbert’s inequality (1.1). Since the constant factor πin (1.1) is the best possible, so we conclude that the constant factorπin (3.1) is the best possible.

We shall now present the integral version of the inequality (3.1) and derive some related inequalities using tensor products.

Let L²_Cn(T) denote the Hilbert space of Cⁿ-valued, norm-square integrable, measurable functions onTandH²_Cn(T) the corresponding Hardy space of functions in L²_Cn(T) with vanishing negative Fourier coefficients. We note thatL²_Cn(T) =

L²(T)⊗Cⁿ andH²_Cn(T) =H²(T)⊗Cⁿ where the Hilbert space tensor product is used. When endowed with the inner product defined by

hf, gi_L2 Cn(T)=

Z

T

hf(z), g(z)i_Cⁿdz , for f, g∈ L²_Cn(T),

the spacesL²_Cn(T) andH²_Cn(T) become separable Hilbert spaces. Here the measures dz denotes the normalized Lebesgue measure onT. If Φ is a bounded, measurable Mn=Mn(C)-valued function (the algebra ofn×nmatrices with complex entries) inL^∞_M

n(T) =L^∞(T)⊗Mn, thenSΦdenotes the Hankel operator defined onH²_Cn(T) by

S_Φf =PeJe(Φf) for f ∈ H²_Cn(T),

wherePe is the orthogonal projection ofL²_Cn(T) ontoH²_Cn(T) and Je:L²_Cn(T)→ L²_Cn(T) is defined byJ Fe (e^it) =F(e^−it) and (Φf)(e^it) = Φ(e^it)f(e^it).

Let Φ∈ L^∞_M

n(T) and

Φ =











φ₁₁ 0 · · · 0

0 φ₂₂ · · · 0

... ... . .. ...

0 0 · · · φnn









 .

Then each entryφij of Φ is inL^∞(T) and

S_Φ=











Sφ₁₁ 0 · · · 0

0 Sφ₂₂ · · · 0 ... ... . .. ...

0 0 · · · Sφ_nn









 .

This is so asH²_Cn(T) =H²(T)⊕ H²(T)⊕ · · · ⊕ H²(T)

| {z }

n−times

.

LetL²_Cn(0,∞) =L²(0,∞)⊗Cⁿ =L²(0,∞)⊕L²(0,∞)⊕ · · · ⊕L²(0,∞). ForF, G∈ L²_Cn(0,∞), the norm is defined by

kFk_L2

Cn =Z ∞ 0

kF(x)k²_Cndx¹₂ and the inner product is defined by

hF, Gi= Z ∞

0

hF(x), G(x)i_Cⁿdx .

With the above inner productL²_Cn(0,∞) is a Hilbert space. For detail see [1]. Let

H(x+y) =









e^−(x+y)

x+y 0 · · · 0

... ... . .. ... 0 0 · · · ^e−(x+y)_x+y









n×n

.

Define B_H:L²_Cn(0,∞)→ L²_Cn(0,∞) by (B_HF) (x) =

Z ∞ 0

H(x+y)F(y)dy . The mapB_H is well-defined, linear and for G∈ L²

Cⁿ(0,∞), hBHF, Gi=

Z ∞ 0

Z ∞ 0

G^∗(x)H(x+y)F(y)dy dx , whereG^∗(x) denotes the adjoint ofG(x). Notice that

BH=











 K

e^h11

0 · · · 0

0 K

e^h22

· · · 0 ... ... . .. ...

0 0 · · · K

e^hnn













whereehij(x) = ^e−x_x for alli,j= 1,2, . . . , n.

Lemma 3.2. The operator BH: L²_Cn(0,∞)→ L²_Cn(0,∞) is a bounded operator andkBHk=π.

Proof. LetF = (f₁, f₂, . . . , f_n)^T, wheref_i∈L²(0,∞) for alli= 1,2, . . . , n. Then G=B_HF= (g₁, g₂, . . . , g_n)^T andg_i∈L²(0,∞) for all i= 1,2, . . . , n.

Now

kBHFk²= Z ∞

0

k(BHF)(x)k²_Cndx= Z ∞

0

kG(x)k²_Cndx

= Z ∞

0

Xⁿ

j=1

|gj(x)|² dx=

n

X

j=1

Z ∞ 0

|gj(x)|²dx

=

n

X

j=1

Z ∞ 0

K

e^hjj fj

(x)

2dx

=

n

X

j=1

kK e^hjj

fjk²≤

n

X

j=1

kK e^hjj

k²kfjk²≤

n

X

j=1

π²kfjk²

=π²

n

X

j=1

Z ∞ 0

|fj(x)|²dx=π² Z ∞

0

Xⁿ

j=1

|fj(x)|² dx

=π² Z ∞

0

kF(x)k²_Cndx=π²kFk². ThuskBHk ≤π.

Now it remains to show that thatkBHk ≥π.

Letf ∈L²(0,∞) andF = (f,0,· · ·,0)^T. ThenkFk=kfk. So,

|hK e^h11

f, fi|=|hBHF, Fi| ≤ kBHkkFk²=kBHkkfk² givesπ=kK

e^h11

k ≤ kBHk asK e^h11

is self-adjoint. Hence kBHk=π.

Now we generalize the Theorem 2.2, for the case p=q= 2, to vector-valued functions.

Theorem 3.3. If F, G∈ L²_Cn(0,∞), then

Z ∞ 0

Z ∞ 0

G^∗(x)H(x+y)F(y)dx dy

≤πZ ∞ 0

kF(x)k²dx¹₂Z ∞ 0

kG(y)k²dy¹₂ ,

where the constant factorπ is the best possible.

Proof. SincekB_Hk=π, so, the result follows from the fact that

|hBHF, Gi| ≤πkFk_L²

CnkGk_L²

Cn, for all F, G∈ L²_Cn(0,∞).

Now letφlj(e^iθ) =−i(π−θ)e^iθ, 0≤θ <2π, 1≤l,j≤nand

Φ =











φ11 0 · · · 0

0 φ22 · · · 0

... ... . .. ...

0 0 · · · φnn









 .

It is not difficult to see that

SΦ=











Sφ₁₁ 0 · · · 0

0 Sφ₂₂ · · · 0 ... ... . .. ...

0 0 · · · Sφ_nn











is unitarily equivalent to

BH =











 K

e^h11

0 · · · 0

0 K

e^h22

· · · 0 ... ... . .. ...

0 0 · · · K

e^hnn











 ,

whereehij(x) = ^e−x_x , 1≤i,j≤n. HencekSΦk=π.

Let ek = (0,0, . . . ,0,1,0, . . . ,0) with 1 in the k^th place and γkl = e^ilt⊗ek, k= 1,2, . . . , n,l= 0,1,2, . . .. Then{ek}ⁿ_k=1form an orthonormal basis forCⁿand {γkl}k=1,2,...,n;l=0,1,...,∞ form an orthonormal basis forH²_Cn(T) =H²(T)⊗Cⁿ. Theorem 3.4. Let Fe=f ⊗x∈ H²_Cn(T) andGe=g⊗y∈ H²_Cn(T). Then

∞

X

l, l⁰=0 n

X

k=1

hf⊗x, e^ilt⊗ekihg⊗y, e^il0t⊗eki l+l⁰+ 1

≤πkf⊗xk kg⊗yk. Proof. Notice that

hF , γe kli=hf⊗x, e^ilt⊗eki=hf, e^iltihx, eki and

hG, γe _ml⁰i=hg⊗y, e^il0t⊗e_mi=hg, e^il0tihy, emi.

Hence

hSΦF ,e Gie =

n

X

k, m=1

∞

X

l, l⁰=0

hF , γe klihG, γe ml⁰ihSΦ(γkl), γml⁰i

=

n

X

k, m=1

∞

X

l, l⁰=0

hF , γe klihG, γe ml⁰ih(Sφ⊗I_Cn)(e^ilt⊗ek), e^il0t⊗emi

=

n

X

k, m=1

∞

X

l, l⁰=0

hf, e^iltihx, e_kihg, e^il0ti hy, e_mi hS_φe^ilt⊗e_k, e^il0t⊗e_mi

=

n

X

k, m=1

∞

X

l, l⁰=0

hf, e^iltihx, ekihg, e^il0ti hy, emi hSφe^ilt, e^il0tihek, e_mi

=

n

X

k=1

∞

X

l, l⁰=0

hf, e^iltihx, ekihg, e^il0ti hy, eki hSφe^ilt, e^il0ti. Thus

hSΦF ,e Gie =

∞

X

l, l⁰=0 n

X

k=1

hf ⊗x, e^ilt⊗e_kihg⊗y, e^il0t⊗e_ki l+l⁰+ 1

and sinceSΦis a bounded linear operator inH²_Cn(T) andkSΦk=π, we obtain hSΦF ,e Gie

≤πkFek_H²

Cn(T)kGke _H²

Cn(T)=πkf⊗xk kg⊗yk.

The result follows.

Corollary 3.5. If Pn k=1

P∞

l=0|akl|²<∞andPn k=1

P∞

l⁰=0|bkl⁰|²<∞, then

∞

X

l,l⁰=0 n

X

k=1

aklbkl⁰

l+l⁰+ 1

≤πXⁿ

k=1

∞

X

l=0

|akl|²¹₂Xⁿ

k=1

∞

X

l⁰=0

|bkl⁰|²¹₂ and the constant πis best possible.

Proof. It is possible to findxk,yk,k= 1,2, . . . , n, and sequences (cl)^∞_l=0, (cl⁰)^∞_l0=0

such that akl = xkcl, bkl⁰ = ykcl⁰, P∞

l=0|cl|² < ∞ and P∞

l⁰=0|cl⁰|² < ∞. Let f(e^it) =P∞

l=0cle^ilt andg(e^it) = P∞

l⁰=0cl⁰e^il0t. Then f, g ∈ H²(T). So, for x= (xk)ⁿ_k=1,y= (yk)ⁿ_k=1∈Cⁿ, we havef⊗x,g⊗y∈ H²_Cn(T). Now

kf⊗xk²=kfk²kxk²=

∞

X

l=0

|cl|²

n

X

k=1

|xk|²=

n

X

k=1

∞

X

l=0

|cl|²|xk|²=

n

X

k=1

∞

X

l=0

|akl|². Similarly,

kg⊗yk²=

n

X

k=1

∞

X

l⁰=0

|b_kl⁰|².

On the other hand,hf⊗x, e^ilt⊗eki=hf, e^iltihx, eki=x_kc_l=a_klandhg⊗y, e^il0t⊗ eki=hg, e^il0tihy, eki=ykcl⁰ =bkl⁰. Hence the results follows from Theorem 3.4.

SincekSΦk=π, the constantπis the best possible.

4. Hankel operators with operator valued symbols

Let Ξ be a separable infinite dimensional Hilbert space. The measurem will denote the normalised Lebesgue measure on T.The spaceL²_Ξis defined to be the set of all (equivalence classes of) measurable, norm-square integrable, Ξ-valued functions defined on T. When endowed with the inner product defined by the equation

hf, gi= Z

T

hf(z), g(z)iΞdm , f, g∈L²_Ξ,

the spaceL²_Ξ becomes a separable Hilbert space. The subspace ofL²_Ξ consisting of those functions with vanishing negative Fourier coefficients will be denoted by H_Ξ². Each function inH_Ξ² admits a natural analytic continuation into D.

A function Φ fromTintoL(Ξ) is called weakly measurable in case the complex-va- lued functionz7→ hΦ(z)x, yiis Lebesgue measurable for everyxandy in Ξ. If Φ is weakly measurable then the real-valued function z→ kΦ(z)kis measurable and the space of all (equivalence classes of) weakly measurable, essentially bounded, L(Ξ)-valued functions onTwill be denoted byL^∞_L(Ξ)(T).

The space L^∞_L(Ξ)(T) is a C^∗− algebra with the algebraic operations defined pointwise and norm defined by the equation

kΦk_∞= ess sup

z∈T

kΦ(z)k, Φ∈L^∞_L(Ξ)(T) wherekΦ(z)k= sup

n

sup

m

|hΦ(z)e_n, e_mi|, z∈T,{e_n}^∞_n=0 is the orthonormal basis for Ξ and involution is defined by the equation Φ^∗(z) = (Φ(z))^∗. The mapping ζ→Φ(ζ)f,ζ∈Tare measurable forf ∈Ξ. This follows from the Pettis theorem (see [1]) as Ξ is separable.

For a function Φ∈L^∞_L(Ξ)(T) we define the Fourier coefficients Cn(Φ) = 1

2π Z 2π

0

e^−intΦ(e^it)dt , n∈Z. The integral is understood in the strong sense, i.e.,

C_n(Φ)f = 1 2π

Z 2π 0

e^−intΦ(e^it)f dt, f ∈Ξ.

We have clearlykC_n(Φ)k ≤ kΦk_∞ for all integers n.The space H_L(Ξ)^∞ (T) is the subspace of L^∞_L(Ξ)(T) consisting of those functions Φ whose Fourier coefficients C_n(Φ) vanish if n < 0. For Φ ∈ L^∞_L(Ξ)(T), we define the Hankel operator S_Φ fromH_Ξ²(T) into itself asSΦf =Q(J(Φf)) whereQis the orthogonal projection fromL²_Ξ(T) ontoH_Ξ²(T) and the symbol Φf denote the function onTdefined by (Φf)(e^it) = Φ(e^it)f(e^it) andJ:L²_Ξ(T)→L²_Ξ(T) is defined byJF(e^it) =F(e^−it).

In the following theorem we extend Theorem 3.3 for Ξ-valued functions.

Theorem 4.1. Let H(x) = ^e−x_x ⊗IΞ where IΞ is the identity operator from the Hilbert space Ξ into itself. Let L²_Ξ(0,∞) = L²(0,∞)⊗Ξ and define K_H :

L²_Ξ(0,∞)→L²_Ξ(0,∞)by

(KHF)(x) = Z ∞

0

H(x+y)F(y)dy . Then for F, G∈L²_Ξ(0,∞),

Z ∞ 0

(K_HF)(x), G(x)

Ξdx

≤πkFk_L2

Ξ(0,∞)kGk_L2 Ξ(0,∞). Proof. Leteh(x) = ^e−x_x and defineK

e^h

∈ L(L²(0,∞)) by (K

e^h

f)(x) = Z ∞

0

e^−(x+y)

x+y f(y)dy .

It is not difficult to see that the operatorKH is well-defined and sinceL²_Ξ(0,∞) = L²(0,∞)⊗Ξ we haveKH=P∞

n=0⊕K e^h

=K e^h

⊗IΞwhere (K e^h

⊗IΞ)(f⊗z) =K ehf⊗z if f ∈ L²(0,∞) and z ∈Ξ. Now kKHk =

P∞

n=0⊕K e^h

=kK

e^h

k =π. Thus by Cauchy-Schwarz inequality it follows that

hK_HF, Gi

≤ kK_Hk kFk_L2

Ξ(0,∞)kGk_L2 Ξ(0,∞)

=πkFk_L²

Ξ(0,∞)kGk_L²

Ξ(0,∞). Hence

Z ∞ 0

(K_HF)(x), G(x)

Ξdx

≤πkFk_L2

Ξ(0,∞)kGk_L2 Ξ(0,∞).

Theorem 4.2. If Fe=f⊗x,Ge=g⊗y∈ H²_Ξ(T) =H²(T)⊗Ξ, then

∞

X

l, l⁰=0

∞

X

k=0

hf⊗x, e^ilt⊗e_kihg⊗y, e^il0t⊗e_ki l+l⁰+ 1

≤πkf⊗xk kg⊗yk.

Proof. Letφ(e^iθ) =−i(π−θ)e^iθ, 0≤θ <2πand Φ =φ⊗I_Ξ. Then Φ∈L^∞_L(Ξ)(T).

LetS_Φbe the Hankel operator from H²_Ξ(T) into itself with symbol Φ. Notice that since H²_Ξ(T) =H²(T)⊗Ξ, we have S_Φ=S_φ⊗I_Ξ. Thus kS_Φk=kS_φk=π. Let Υkl =e^ilt ⊗ek, k = 0,1,2, . . . and l = 0,1,2, . . .. The sequence {Υkl} form an orthonormal basis forH²_Ξ(T). Then

hS_ΦF ,e Gie =

∞

X

l, l⁰=0

∞

X

k=0

hf ⊗x, e^ilt⊗ekihg⊗y, e^il0t⊗eki

l+l⁰+ 1 .

Since

hSΦF ,e Gie

≤ kSΦk kFk ke Gke =πkf⊗xk kg⊗yk,

the result follows.

Corollary 4.3. LetFe=f⊗xandGe=g⊗ywheref,g∈ H²(T)andx,y∈Ξ. Let cl(f)and cl⁰(g)denote the l^th and l^0th Fourier coefficients off and g respectively.

Then

∞

X

l, l⁰=0

hcl(f)x, cl⁰(g)yiΞ

l+l⁰+ 1

≤πkFek_H2

Ξ(T)kGke _H2 Ξ(T).