Nuclearity of operators related to finite measure spaces

NUCLEARITY OF OPERATORS RELATED TO FINITE MEASURE SPACES

VISHVESH KUMAR AND SHYAM SWARUP MONDAL

Abstract. Let (S,B, m) be a finite measure space. The aim of this paper is to give necessary and sufficient conditions on symbols such that the correspondingZ-operator fromL^p1(Z) intoL^p2(Z) andS-operator fromL^p1(S) intoL^p2(S) to be nuclear for 1≤p1, p2 <∞. We show that the adjoint of the nuclearZ-operator fromL^p02 intoL^p01is again a nuclear operator. As applications, we get the symbol of the product of the nuclear operators with bounded operators.

1. Introduction

The theory of pseudo-differential operators is one of the most important tools in modern mathe- matics. Pseudo-differential operators are widely used in harmonic analysis, PDEs, geometry, math- ematical physics, time-frequency analysis, imaging and computations [23]. The study of pseudo- differential operators is first introduced by Kahn and Nirenberg [25] and later used by H¨ormander [23] for problems in partial differential equations. Pseudo-differential operators on different classes of groups are broadly studied by several authors with their applications [5, 6, 7, 8, 11, 12, 20, 26, 27, 28, 29, 37, 39, 40, 41, 42, 43].

In particular, the pseudo-differential operators on the latticeZⁿare suitable for solving difference equations onZⁿ. Discrete pseudo-differential operators appear early in the classic Fourier analysis as well as discrete counterparts of Caldero´n-Zygmund singular integral operators. A suitable theory of pseudo-differential operators on the latticeZⁿ has been developed by several authors but without symbolic calculus [3, 18, 31, 34, 35]. Very recently, a global symbolic calculus has been introduced and constructed by Botchway, Kabiti and Ruzhansky with many applications [3].

The idea of formulation a global definition of pseudo-differential operators on the unit circleS¹ using Fourier series was first proposed by Agranovich crediting Volevich [1]. Later, this theory was developed by Turunen and Vainikko [42], and Ruzhansky and Turunen [39]. Though, some interdependent results for this periodic theory is given in the works of Ruzhansky and Turunen [39, 40], Cardona [4, 8], Delgado [14] and Molahajloo and Wong [30, 32].

For the trace class operators on Hilbert spaces, in general the trace which is given by the inte- gration of its kernel over the diagonal is equal to the sum of its eigenvalues. However, this feature fails in Banach spaces. The importance of nuclear operator lies in the work of Grothendieck, who proved that for 2/3-nuclear operators, the trace in Banach spaces agrees with the sum of all the eigenvalues with multiplicities counted. Therefore, the notion of nuclear operators becomes useful.

Nuclear operators on Banach spaces as generalizations of trace class operators can be traced at least to Grothendieck [21, 22]. The initiative of finding the necessary and sufficient conditions for a pseudo-differential operator defined on a group to be nuclear has been started by Delgado and Wong [18]. The nuclearity of pseudo-differential operators onRⁿ has been studied by Aoki [2] and Rempala [36]. Characterizations of nuclear operators in terms of decomposition of symbol through

Date: June 27, 2021.

2010Mathematics Subject Classification. Primary 47F05, 47G30; Secondary 43A85,

Key words and phrases. Pseudo-differential operators; Finite measure space; Fourier transform;S-operators;Z- operators.

Fourier transform were investigated by Ghaemi, Jamalpour Birgani and Wong forS¹[19]. Later they generalized their results on nuclearity to the pseudo-differential operators for any arbitrary compact group [20]. On the other hand characterizations of nuclear operators in terms of decomposition of symbol for Zwas studied by Jamalpour Birgani [24]. MoreoverL^p-boundedness and L^p-nuclearity of multilinear pseudo-differential operators onZⁿ and the toursTⁿ proved by D. Cardona and the first author in [7]. However, a more general result,L^p-nuclearity and its traces of pseudo-differential operators on compact Lie groups, compact homogeneous spaces and compact manifolds studied by Delgado and Ruzhansky [13, 15, 16, 40].

The motivation to studyS-operator and Z-operators comes from the Fourier analysis associated with the classical orthogonal polynomials expansions on bounded interval or on half real line. The pseudo-differential operator on the circle S¹ or on the lattice Z defined using the usual Fourier analysis on S¹ or onZ . In the same way we can define pseudo-differential operator with the help of the Fourier analysis associated with the classical orthogonal polynomials expansion on bounded interval or on half real line. Here, we work in a more general setting by considering a finite measure spacesS equipped with a measure msuch thatm(S)<∞such that L²(S) is a separable Hilbert space and define the pseudo-differential operators with the help of the orthonormal basis of separable Hilbert spaceL²(S).Such operators were first introduced by Catan˘a asS-operators andZ-operators related to the measure spaceS which are generalization of pseudo-differential operators onS¹ [30]

andZ[31] respectively [9, 10]. Properties such as L²-boundedness, compactness ofS-operator and Z-operator are given in [9, 10].

In this paper, we consider that{en}_n∈IwithI =ZorNis an orthonormal basis for the separable Hilbert spaceL²(S) =L²(S,B, m) with an extra property that for everyn∈ I, m{s∈S:en(s) = 0}= 0. Although, by following the notation in [9, 10] we prefer to writeI =Zin this paper but our results also true whenI=Nwhich is the case of many interesting examples discussed below other than circleS¹.Examples of such kind of finite measure spaces for which theirL²-Hilbert spaces are separable and the corresponding orthonormal basis owns this property are as follows: let us consider the triple (S,B, m), whereS is the interval (a, b),Bis the set of all Baire subsets in (a, b) andmis a Baire measure.

(1) When a = −1, b = 1 and m(ds) = ds is the standard Lebesgue measure, consider the orthogonal sequence of Legendre polynomials

P_n(x) = 1 2ⁿn!

dⁿ

dxⁿ x²−1ⁿ

n≥0

on (−1,1).

Then, the set

(

en = Pn

kPnk =

r2n+ 1 2 Pn

)

n≥0

is an orthonormal basis for L²(−1,1) with the standard inner product (·,·) and the norm k · k.

(2) Whena=−∞, b=∞andm(ds) =e^−s2ds, s∈S=R, consider the orthogonal sequence of Hermite polynomials

Hn(x) = (−1)ⁿe^x2 dⁿ dxⁿ

e^−x2

n≥0

onR. So,

(

en= Hn

kHnk = s

n!

2ⁿ√ πHn

)

n≥0 2

is an orthonormal basis forL²(R).

(3) When a = 0, b = ∞ and m(ds) = e^−sds, consider the sequence of orthogonal Laguerre polynomials

L_n(x) =e^x n!

dⁿ

dxⁿ xⁿe^−x

n≥0

on (0,∞), then the set

en= Ln

kLnk =Ln

n≥0

is an orthonormal basis forL²(0,∞).

The main aim of this paper is to study the nuclear Z-operators and S-operators related to a finite measure space with application to adjoints and products. We give necessary and sufficient conditions on the symbols such that the correspondingZ-operators andS-operator to be a nuclear operator. Further we provide necessary and sufficient conditions on the symbols to guarantee that the adjoint and the product of nuclearZ-operators are nuclear.

The schema of the paper is as follows: In section 2, we recallS-operators andZ-operators (also known as the generalized pseudo-differential operator onS andZ) related to a finite measure space (S,B, m) from [9, 10]. We begin section 3 by the definition of nuclear operators on Banach space and present a characterization of nuclearZ-operators. We also give necessary and sufficient conditions on the symbols to guarantee that the adjoints and products of nuclear Z-operators are nuclear.

Characterization of nuclear S-operator are given in Section 4. We end this paper by presenting a few examples of S-operator andZ-operator corresponding to some finite measure spaces for which the results in Section 3 and Section 4 are valid.

2. Preliminary

In this section, we recall some basic facts related to the Fourier analysis on finite measure space (S,B, m) developed by Catanˇa in [9, 10]. Let (S,B, m) be a finite measure space (i.e. the total measure of S, m(S) is finite) such that the corresponding Hilbert space L²(S) = L²(S,B, m) is separable. Forf, g∈L²(S), the inner product on Hilbert spaceL²(S) is given by

(f, g)_L2(S)= Z

S

f(s)g(s)dm(s).

Let{en}_n∈Zbe an orthonormal basis forL²(S). On the other handL²(Z) is a Hilbert space with respect to the inner product is given by

(a, b)_L2(Z)=

∞

X

n=−∞

anbn, a, b∈L²(Z).

We define the linear operatorsF_S :L²(S)→L²(Z) andF_Z:L²(Z)→L²(S), by (FSf) (n) = (m(S))^−1/2(f, en)_L2(S), n∈Z,

(2.1)

for allf in L²(S) and

(F_Za) (s) = (m(S))^1/2X

n∈Z

anen(s), s∈S, (2.2)

for all sequenceainL²(Z). The linear operatorsFS andF_Zare bijections and they are the inverses to each other. So,

F_SF_Z =I:L²(Z)→L²(Z)

and

F_ZF_S =I:L²(S)→L²(S),

where we denoted by the same capital letterI, the identity operator ofL²(Z) andL²(S) respectively.

Moreover, these operators also satisfy the following version of the Plancherel formula.

Theorem 2.1. For allf andg inL²(S), we have

(FSf,FSg)_L2(Z)= (m(S))⁻¹(f, g)L²(S)

and for allaandb in L²(Z), we have

(F_Za,F_Zb)_L2(S)=m(S)(a, b)_L2(Z).

Letσbe a measurable function onS×Z . Then for allf in L²(S), we define the functionT_σf formally by

(T_σf) (s) = (m(S))^1/2X

n∈Z

σ(s, n)(FSf)(n)e_n(s), s∈S,

where the linear operatorFSf is given in (2.1). We callTσtheS-operator or the generalized pseudo- differential operator on S [related to the measure space (S,B, m)] corresponding to the symbol σ, whenever the series is convergent for alls∈S.

Again, letτ:Z×S→Cbe a measurable function. Then for every sequenceainL²(Z), we define the sequenceTτaformally by

(Tτa) (n) = (m(S))^−1/2 Z

S

τ(n, s) (F_Za) (s)en(s)dm(s), n∈Z,

where the linear operatorF_Zais given in (2.2). We callTτ theZ-operator or the generalized pseudo- differential operator on Z [related to the measure space (S,B, m)] corresponding to the symbolτ, whenever the integral exists for alln∈Z.See [9, 10] for more details aboutS andZ-operators.

3. Z-operators

We first recall the basic notions of nuclear operators on Banach spaces. Let T be a bounded linear operator from a complex Banach spaceX into another complex Banach space Y such that there exist sequences{x⁰_n}^∞_n=1 in the dual space X⁰ ofX and{yn}^∞_n=1 inY such that

∞

X

n=1

kx⁰_nk_X0kynk_Y <∞ and

T x=

∞

X

n=1

x⁰_n(x)y_n, x∈X.

Then we callT :X→Y a nuclear operator and if X =Y,then its nuclear trace tr(T) is given by tr(T) =

∞

X

n=1

x⁰_n(yn).

It can be proved that the definition of a nuclear operator and the definition of the trace of a nuclear operator are independent of the choices of the sequences{x⁰_n}^∞_n=1and{y_n}^∞_n=1.

The following theorem proved by Delgado [13] present a characterization of nuclear operators in terms of the conditions on the kernel of the operator.

4

Theorem 3.1. Let(X1, µ1)and(X2, µ2)beσ-finite measure spaces. Thenabounded linear operator T : L^p1(X1, µ1) → L^p2(X2, µ2),1 ≤ p1, p2 < ∞, is nuclear if and only if there exist sequences {gn}^∞_n=1 inL^p01(X₁, µ₁)and{hn}^∞_n=1 in L^p2(X₂, µ₂)such that for all f ∈L^p1(X₁, µ₁)

(T f)(x) = Z

X1

K(x, y)f(y)dµ1(y), x∈X2, where

K(x, y) =

∞

X

n=1

hn(x)gn(y), x∈X2, y∈X1

and ∞

X

n=1

kgnk_Lp10

(X1,µ1)khnk_Lp2(X2,µ2)<∞.

The function K on X₂×X₁ in Theorem 3.1 is called the kernel of the nuclear operator T : L^p1(X₁, µ₁)→L^p2(X₂, µ₂).

Let (X, µ) be aσ-finite measure space. Let T : L^p(X, µ)→L^p(X, µ) 1 ≤p < ∞, be a nuclear operator. Then by Theorem 3.1, we can find sequences{gn}^∞_n=1inL^p0(X, µ) and{hn}^∞_n=1inL^p(X, µ) such that

∞

X

n=1

kgnk_Lp0

(X,µ)khnk_Lp(X,µ)<∞ and for allf ∈L^p(X, µ)

(T f)(x) = Z

X

K(x, y)f(y)dµ(y), x∈X, where

K(x, y) =

∞

X

n=1

h_n(x)g_n(y), x, y∈X and it satisfies

Z

X

|K(x, y)|dµ(y)≤

∞

X

n=1

kgnk_Lp0

(X,µ)khnk_Lp(X,µ). The trace tr(T) ofT :L^p(X, µ)→L^p(X, µ) is given by

tr(T) = Z

X

K(x, x)dµ(x).

(3.1)

Now, we present a characterization of nuclear Z-operators fromL^p1(Z) intoL^p2(Z). Remember that {e_n}_n∈

Z be an orthonormal basis for L²(S) such that for every n ∈ Z, m({s ∈ S : e_n(s) = 0}) = 0.

Theorem 3.2. Letτ be a measurable function onZ×S. Then theZ-operatorTτ:L^p1(Z)→L^p2(Z) is nuclear for1≤p1, p2<∞if and only if there exist sequences{gk}^∞_k=1 ∈L^p01(Z) and{hk}^∞_k=1 ∈ L^p2(Z)such that

∞

X

k=−∞

kgkk_Lp0

1(Z)khkk_Lp2(Z)<∞ and for everyn∈Z

τ(n, s) = (m(S))^−1/2(e_n(s))⁻¹

∞

X

k=−∞

h_k(n)(F_Zg_k) (s), s∈ {s∈S:e_n(s)6= 0}.

Proof. Suppose thatTτ:L^p1(Z)→L^p2(Z) is nuclear, where 1≤p1, p2<∞.Then by Theorem 3.1, there exist sequences{gk}^∞_k=1 in L^p01(Z) and{hk}^∞_k=1 inL^p2(Z) such that

∞

X

k=1

kgkk_Lp0

1(Z)khkk_Lp2(Z)<∞ and for allf ∈L^p1(Z),we have

(T_τf) (n) = (m(S))^−1/2 Z

S

τ(n, s) (F_Zf) (s)e_n(s)dm(s)

= Z

S

τ(n, s)

∞

X

l=−∞

f(l)el(s)

!

en(s)dm(s)

=

∞

X

l=−∞

Z

S

τ(n, s)e_l(s)e_n(s)dm(s)

f(l)

=

∞

X

l=−∞

∞

X

k=−∞

hk(n)gk(l)f(l).

(3.2)

Letm∈Zand the functionf is given by f(l) =fm(l) =

0 ifm6=l, 1 ifm=l.

Then from (3.2) we get Z

S

τ(n, s)e_m(s)e_n(s)dm(s) =

∞

X

k=−∞

h_k(n)g_k(m) (3.3)

and so

FS(τ(n,·)en(·))

(m) = (m(S))^−1/2

∞

X

k=−∞

hk(n)gk(m).

(3.4) Now,

τ(n, s)en(s) = (m(S))^1/2

∞

X

m=−∞

FS(τ(n,·)en(·))

(m)em(s)

=

∞

X

m=−∞

∞

X

k=−∞

hk(n)gk(m)

! em(s)

=

∞

X

k=−∞

hk(n)

∞

X

m=−∞

gk(m)em(s)

= (m(S))^−1/2

∞

X

k=−∞

h_k(n)(F_Zg_k)(s).

Therefore, for alln∈Z

τ(n, s) = (m(S))^−1/2(e_n(s))⁻¹

∞

X

k=−∞

h_k(n)(F_Zg_k)(s), s∈ {s∈S:e_n(s)6= 0}.

6

Conversely, suppose that there exist sequences {gk}^∞_k=1 in L^p01(Z) and {hk}^∞_k=1 in L^p2(Z) such that

∞

X

k=−∞

kg_kk

L^p01(Z)kh_kk_Lp2(Z)<∞ and for alln∈Z

τ(n, s) = (m(S))^−1/2(e_n(s))⁻¹

∞

X

k=−∞

h_k(n)(F_Zg_k) (s), s∈ {s∈S :e_n(s)6= 0}.

Then, for allf ∈L^p1(Z)

(Tτf) (n) = (m(S))^−1/2 Z

S

τ(n, s) (F_Zf) (s)en(s)dm(s)

= (m(S))⁻¹ Z

S

∞

X

k=−∞

hk(n)(F_Zgk)(s)

!

(F_Zf) (s)dm(s)

= (m(S))^−1/2 Z

S

∞

X

k=−∞

hk(n)

∞

X

m=−∞

gk(m)em(s)

!

(F_Zf) (s)dm(s)

= (m(S))^−1/2

∞

X

k=−∞

hk(n)

∞

X

m=−∞

gk(m) Z

S

(F_Zf) (s)em(s)dm(s)

=

∞

X

m=−∞

∞

X

k=−∞

h_k(n)g_k(m) (FSF_Zf) (m)

=

∞

X

m=−∞

∞

X

k=−∞

hk(n)gk(m)f(m)

for alln∈Z.Therefore by Theorem 3.1,Tτ:L^p1(Z)→L^p2(Z) is a nuclear operator.

In order to find the trace of a nuclearZ-operator fromL^p(Z) intoL^p(Z), in the next theorem, we give another characterization of nuclearZ-operator fromL^p1(Z) intoL^p2(Z),1≤p₁, p₂<∞.

Theorem 3.3. Let τ be a measurable function on Z×S. Then the Z-operator Tτ : L^p1(Z) → L^p2(Z),1 ≤ p₁, p₂ < ∞, is nuclear if and only if there exist sequences {g_k}^∞_k=1 ∈ L^p01(Z) and {h_k}^∞_k=1 ∈L^p2(Z)such that

∞

X

k=−∞

kgkk

L^p01(Z)khkk_Lp2(Z)<∞ and

Z

S

τ(n, s)em(s)en(s)dm(s) =

∞

X

k=−∞

hk(n)gk(m).

Proof. Let Tτ : L^p1(Z)→ L^p2(Z) is nuclear operator. From equation (3.3) there exist sequences {gk}^∞_k=1∈L^p01(Z) and{hk}^∞_k=1∈L^p2(Z) such that

∞

X

k=−∞

kgkk_Lp0

1(Z)khkk_Lp2(Z)<∞

and for everyn, m∈Z, Z

S

τ(n, s)em(s)en(s)dm(s) =

∞

X

k=−∞

hk(n)gk(m).

Conversely, if there exist sequences{g_k}^∞_k=1∈L^p01(Z) and{h_k}^∞_k=1∈L^p2(Z) such that

∞

X

k=−∞

kgkk_Lp0

1(Z)khkk_Lp2(Z)<∞ and for everyn, m∈Z,

Z

S

τ(n, s)em(s)en(s)dm(s) =

∞

X

k=−∞

hk(n)gk(m).

Now, for allf ∈L^p1(Z)

(Tτf) (n) = (m(S))^−1/2 Z

S

τ(n, s) (F_Zf) (s)en(s)dm(s)

= Z

S

τ(n, s)

∞

X

m=−∞

f(m)em(s)en(s)dm(s)

=

∞

X

m=−∞

Z

S

τ(n, s)em(s)en(s)dm(s)

f(m)

=

∞

X

m=−∞

∞

X

k=1

hk(n)gk(m)f(m)

for alln∈Z.Therefore by Theorem 3.1,T_τ:L^p1(Z)→L^p2(Z) is a nuclear operator.

An immediate consequence of Theorem 3.3 gives the trace of a nuclear Z-operator fromL^p(Z) intoL^p(Z),1≤p <∞. Indeed, we have the following result.

Corollary 3.4. Let Tτ : L^p(Z) → L^p(Z) be a nuclear operator for 1 ≤ p < ∞. Then the trace tr (Tτ)of Tτ is given by

tr (Tτ) =

∞

X

n=−∞

Z

S

τ(n, s)|en(s)|² dm(s).

Proof. Using trace formula (3.1) and Theorem 3.3 we have tr (Tτ) =

∞

X

n=−∞

∞

X

k=−∞

hk(n)gk(n)

=

∞

X

n=−∞

Z

S

τ(n, s)en(s)en(s)dm(s)

=

∞

X

n=−∞

Z

S

τ(n, s)|en(s)|² dm(s).

IfT_τ is a nuclear operator then as a consequence of Theorem 3.2 and Theorem 3.3 we have the following necessary conditions.

8

Theorem 3.5. Let τ be a measurable function on Z×S such that the coresponding Z-operator Tτ:L^p1(Z)→L^p2(Z)is a nuclear operator for2≤p1<∞,1≤p2<∞. Then

∞

X

n=−∞

kτ(n,·)en(·)k^p2

L^p01(S)<∞.

Proof. LetT_τ :L^p1(Z)→L^p2(Z) is a nuclear operator for 2≤p₁ <∞,1≤p₂ <∞. By Theorem 3.2, there exist sequences{gk}^∞_k=1∈L^p01(Z) and{hk}^∞_k=1∈L^p2(Z) such that

∞

X

k=−∞

kgkk

L^p01(Z)khkk_Lp2(Z)<∞

and for alln∈Z

τ(n, s) = (m(S))^−1/2(en(s))⁻¹

∞

X

k=−∞

hk(n)(F_Zgk)(s), s∈ {s∈S:en(s)6= 0}.

Using Minkowski’s inequality and plancherel formula we get

kτ(n,·)en(·)k_Lp0 1(S)=

Z

S

|τ(n, s)en(s)|^p01ds _p¹0

1

= (m(S))^−1/2



 Z

S

∞

X

k=−∞

h_k(n)(F_Zg_k)(s)

p⁰₁

ds





1 p0 1

≤(m(S))^−1/2

∞

X

k=−∞

|hk(n)|

Z

S

(F_Zgk)(s)

p⁰₁

ds _p¹0

1

≤(m(S))^−1/2

∞

X

k=−∞

|h_k(n)|

Z

S

(F_Zg_k)(s)

2

ds ¹₂

=

∞

X

k=−∞

|hk(n)|kgkk_L2(Z)

≤

∞

X

k=−∞

|hk(n)|kgkk

L^p01(Z)

and therefore

kτ(n,·)en(·)k^p2

L^p01(S)≤

∞

X

k=−∞

|hk(n)|kgkk_Lp0 1(Z)

!^p2

.

Using Minkowski’s inequality we have

∞

X

n=−∞

kτ(n,·)en(·)k^p2

L^p01(S)

!_p¹

2

≤

∞

X

n=−∞

∞

X

k=−∞

|hk(n)|kgkk_Lp0 1(Z)

!p2!p¹2

≤

∞

X

k=−∞

∞

X

n=−∞

|hk(n)|^p2

!_p¹

2

kgkk_Lp0 1(Z)

=

∞

X

k=−∞

kh_kk_Lp2(Z)kg_kk

L^p01(Z)<∞.

Theorem 3.6. Let τ be a measurable function on Z×S such that the corresponding Z-operator Tτ:L^p1(Z)→L^p2(Z)is a nuclear operator for1≤p1, p2<∞. Then

F_S(τ(•,·)e_•(·)) (m)

_Lp2(Z)

m∈Z

∈L^p01(Z).

Proof. LetTτ : L^p1(Z) →L^p2(Z) is a nuclear operator for 1≤ p1, p2 <∞. Then from equation (3.4), there exist sequences{gk}^∞_k=1∈L^p01(Z) and{hk}^∞_k=1∈L^p2(Z) such that for allm, n∈Z

FS(τ(n,·)en(·))

(m) = (m(S))^−1/2

∞

X

k=−∞

hk(n)gk(m).

Using Minkowski’s inequality we get

∞

X

n=−∞

FS(τ(n,·)en(·)) (m)

p2

!_p¹

2

=

∞

X

n=−∞

FS(τ(n,·)en(·)) (m)

p₂!_p¹

2

= (m(S))^−1/2

∞

X

n=−∞

∞

X

k=−∞

hk(n)gk(m)

p2!_p¹₂

≤(m(S))^−1/2

∞

X

k=−∞

∞

X

n=−∞

|hk(n)gk(m)|^p2

!_p¹

2

= (m(S))^−1/2

∞

X

k=−∞

|gk(m)|

∞

X

n=−∞

|hk(n)|^p2

!_p¹

2

= (m(S))^−1/2

∞

X

k=−∞

|gk(m)|khkkL^p2(Z) 10

and therefore







∞

X

m=−∞

∞

X

n=−∞

FS(τ(n,·)en(·)) (m)

p2

!

p0 1 p2







1 p0 1

≤(m(S))^−1/2





∞

X

m=−∞

∞

X

k=−∞

|gk(m)|khkk_L^p2(Z)

!^p0₁



1 p0 1

≤(m(S))^−1/2

∞

X

k=−∞

∞

X

n=−∞

|g_k(m)|^p01

!_p¹0 1

kh_kk_Lp2(Z)

= (m(S))^−1/2

∞

X

k=−∞

khkkL^p2(Z)kgkk_Lp0

1(Z)<∞.

Theorem 3.7. Let τ be a measurable function on Z×S such that the coresponding Z-operator Tτ:L^p1(Z)→L^p2(Z)is a nuclear operator for1≤p1, p2<∞. Then

FS(τ(n,·)en(·)) (•)

_Lp0

1(Z)

n∈Z

∈L^p2(Z).

Proof. LetTτ : L^p1(Z) →L^p2(Z) is a nuclear operator for 1≤ p1, p2 <∞. Then from equation (3.4), there exist sequences{g_k}^∞_k=1∈L^p01(Z) and{h_k}^∞_k=1∈L^p2(Z) such that for allm, n∈Z

FS(τ(n,·)en(·))

(m) = (m(S))^−1/2

∞

X

k=−∞

hk(n)gk(m).

Using Minkowski’s inequality we get

∞

X

m=−∞

FS(τ(n,·)en(·)) (m)

p⁰₁!_p¹0 1

=

∞

X

m=−∞

FS(τ(n,·)en(·)) (m)

p⁰₁!_p¹0 1

= (m(S))^−1/2





∞

X

m=−∞

∞

X

k=−∞

hk(n)gk(m)

p⁰₁



1 p0 1

≤(m(S))^−1/2

∞

X

k=−∞

∞

X

m=−∞

|hk(n)gk(m)|^p01

!_p¹0 1

= (m(S))^−1/2

∞

X

k=−∞

|hk(n)|

∞

X

m=−∞

|gk(m)|^p01

!_p¹0 1

= (m(S))^−1/2

∞

X

k=−∞

|hk(n)|kgkk_Lp0 1(Z)

and therefore





∞

X

n=−∞

∞

X

m=−∞

FS(τ(n,·)en(·)) (m)

p⁰₁!

p2 p0 1





1 p2

≤(m(S))^−1/2

∞

X

n=−∞

∞

X

k=−∞

|hk(n)|kgkk_Lp0 1(Z)

!^p2!p¹2

≤(m(S))^−1/2

∞

X

k=−∞

∞

X

n=−∞

|hk(n)|^p2

!_p¹₂

kgkk_Lp0 1(Z)

= (m(S))^−1/2

∞

X

k=−∞

khkk_L^p2(Z)kgkk_Lp0

1(Z)<∞.

3.1. Adjoints. LetT_τ:L^p1(Z)→L^p2(Z),1≤p₁, p₂<∞,be a nuclear operator. In this sub-section we show that the adjoint operatorT_τ^∗ :L^p02(Z)→L^p01(Z) of Tτ, is also nuclear. We also present a necessary and sufficient condition on the symbol τ so that the corresponding nuclear operator T_τ fromL²(Z) intoL²(Z) to be self-adjoint.

Theorem 3.8. Letτ be a measurable function onZ×Ssuch thatTτ:L^p1(Z)→L^p2(Z)is a nuclear operator for1≤p1, p2<∞.ThenT_τ^∗, the adjoint ofTτ,is also a nuclear operator fromL^p02(Z)into L^p01(Z)with symbolτ^∗ given by

τ^∗(n, s) = (m(S))^−1/2(en(s))⁻¹

∞

X

k=−∞

gk(n)(F_Zhk)(s), (n, s)∈Z× {s∈S :en(s)6= 0}, where{g_k}^∞_k=1 and{h_k}^∞_k=1 are two sequences in L^p01(Z)andL^p2(Z)respectively such that

∞

X

k=−∞

kg_kk

L^p01(Z)kh_kk_Lp2(Z)<∞.

Proof. For allf ∈L^p1(Z) andg∈L^p2(Z), we have

∞

X

i=−∞

(Tτf)(i)g(i) =

∞

X

i=−∞

f(i)T_τ^∗g(i).

Therefore,

∞

X

i=−∞

Z

S

τ(i, s) (F_Zf) (s)ei(s)dm(s)

g(i) =

∞

X

i=−∞

f(i) Z

S

τ^∗(i, s) (F_Zg) (s)ei(s)dm(s)

. (3.5)

Now, letm, n∈Zand for alli∈Z, f(i) =fn(i) =

0 ifi6=n,

1 ifi=n, g(i) =gm(i) =

0 ifi6=m, 1 ifi=m.

12

Therefore from (3.5) we have, Z

S

τ(m, s)en(s)em(s)dm(s) = Z

S

τ^∗(n, s)em(s)en(s)dm(s) and thus

FS(τ(m,·)em(·))

(n) =

FS(τ^∗(n,·)en(·)) (m).

(3.6)

SinceTτ :L^p1(Z)→L^p2(Z) is a nuclear, from Theorem 3.2, there exist sequences{gk}^∞_k=1∈L^p01(Z) and{hk}^∞_k=1∈L^p2(Z) such that

∞

X

k=−∞

kgkk_Lp0

1(Z)khkk_Lp2(Z)<∞ and for alln∈Z

τ(n, s) = (m(S))^−1/2(en(s))⁻¹

∞

X

k=−∞

hk(n)(F_Zgk)(s), s∈ {s∈S:en(s)6= 0}.

Using (3.6) and the nuclearity ofTτ, we have τ^∗(n, s)en(s) = (m(S))^1/2

∞

X

m=−∞

FS(τ^∗(n,·)en(·))

(m)em(s)

= (m(S))^1/2

∞

X

m=−∞

FS(τ(m,·)em(·))

(n)e_m(s)

=

∞

X

m=−∞

Z

S

τ(m, w)em(w)en(w)dm(w)

em(s)

= (m(S))^−1/2

∞

X

m=−∞

Z

S

∞

X

k=−∞

hk(m)(F_Zgk)(w)

!

en(w)dm(w)

! em(s)

= (m(S))^−1/2

∞

X

m=−∞

∞

X

k=−∞

hk(m) Z

S

(F_Zgk)(w)en(w)dm(w) !

em(s)

=

∞

X

m=−∞

∞

X

k=−∞

hk(m)(FSF_Zgk)(n)

! em(s)

=

∞

X

m=−∞

∞

X

k=−∞

h_k(m)g_k(n)e_m(s)

=

∞

X

k=−∞

gk(n)

∞

X

m=−∞

hk(m)em(s)

= (m(S))^−1/2

∞

X

k=−∞

gk(n)(F_Zhk)(s).

Therefore, for alln∈Z

τ^∗(n, s) = (m(S))^−1/2(e_n(s))⁻¹

∞

X

k=−∞

g_k(n) (F_Zh_k)(s), s∈ {s∈S:e_n(s)6= 0}.

As a consequence of the above theorem and the help of Theorem 3.2, we can give a criterion in terms of symbol such that the correspondingZ-operator to be self-adjoint.

Corollary 3.9. Let τ be a measurable function on Z×S such that Tτ:L²(Z)→L²(Z)is nuclear.

ThenTτ :L²(Z)→L²(Z)is self-adjoint if and only if there exist sequences{gk}^∞_k=1 and{hk}^∞_k=1 in L²(Z)such that

∞

X

k=−∞

kh_kk_L2(Z)kg_kk_L2(Z)<∞,

∞

X

k=−∞

hk(n)(F_Zgk)(s) =

∞

X

k=−∞

gk(n)(F_Zhk)(s), and

τ(n, s)en(s) = (m(S))^−1/2

∞

X

k=−∞

hk(n)(F_Zgk)(s), for alln∈Z, s∈ {s∈S:e_n(s)6= 0}.

We can give another formula for the adjoints of nuclearZ-operators in terms of symbols. Indeed, we have the following theorem.

Theorem 3.10. Let τ be a measurable function on Z×S such that the coresponding Z-operator Tτ :L^p1(Z)→L^p2(Z) is a nuclear operator for1 ≤p1, p2 <∞. Then for allm∈Zand w∈ {s∈ S:em(s)6= 0},

τ^∗(m, w) =m(S)(em(w))⁻¹

∞

X

n=−∞

en(w) Z

S

em(s)τ(n, s)en(s)dm(s).

Proof. LetTτ :L^p1(Z)→L^p2(Z) is a nuclear operator for 1≤p1, p2 <∞. By Theorem 3.2, there exist sequences{gk}^∞_k=1∈L^p01(Z) and{hk}^∞_k=1∈L^p2(Z) such that

∞

X

k=−∞

kgkk

L^p01(Z)khkk_Lp2(Z)<∞ and for all (n, s)∈Z× {s∈S:en(s)6= 0}

τ(n, s) = (m(S))^−1/2(en(s))⁻¹

∞

X

k=−∞

hk(n)F_Zgk(s).

Letm∈Z.Then

em(s)τ(n, s)en(s) = (m(S))^−1/2em(s)

∞

X

k=−∞

hk(n)F_Zgk(s)

14

and

Z

S

em(s)τ(n, s)en(s)dm(s) = (m(S))^−1/2

∞

X

k=−∞

hk(n) Z

S

F_Zgk(s)em(s)dm(s)

=

∞

X

k=−∞

h_k(n)g_k(m).

Thus, using Theorem 3.8 we get m(S)

∞

X

n=−∞

en(w) Z

S

em(s)τ(n, s)en(s)dm(s)

=m(S)

∞

X

n=−∞

en(w)

∞

X

k=−∞

hk(n)gk(m)

=m(S)

∞

X

k=−∞

∞

X

n=−∞

en(w)hk(n)

! gk(m)

= (m(S))^1/2

∞

X

k=−∞

(F_Zh_k)(w)g_k(m) =τ^∗(m, w)e_m(w).

Therefore for allm∈Z τ^∗(m, w) =m(S)(em(w))⁻¹

∞

X

n=−∞

en(w) Z

S

em(s)τ(n, s)en(s)dm(s), w∈ {s∈S:em(s)6= 0}.

3.2. Products. In the next subsection we show that the product of a nuclear Z-operator with a bounded operator is again a nuclear operator and as a consequence, we give a necessary and sufficient condition on symbol so that the corresponding nuclearZ-operator to be normal.

Theorem 3.11. Let τ and η be two measurable function on Z×S such that the corresponding Z-operatorsTτ :L^p(Z)→L^p(Z)andTη :L^p(Z)→L^p(Z)are nuclear operator and bounded operator respectively for1≤p <∞. ThenTηTτ :L^p(Z)→L^p(Z)is a nuclear operator Tλ:L^p(Z)→L^p(Z) with symbolλis given by

λ(m, s) = (m(S))^1/2(e_m(s))⁻¹

∞

X

l=−∞

τ(l, s)e_l(s)

FS(η^∗(l,·)el(·)) (m), for allm∈Z, s∈ {s∈S:em(s)6= 0}.

Proof. For allf, g∈L^p(Z), we have

∞

X

l=−∞

(T_λf)(l)g(l) =

∞

X

l=−∞

(T_ηT_τf)(l)g(l) =

∞

X

l=−∞

(T_τf)(l)T_η^∗g(l).

Therefore,

∞

X

l=−∞

Z

S

λ(l, s)(F_Zf)(s)e_l(s)dm(s)

g(l)

=

∞

X

l=−∞

Z

S

τ(l, s)(F_Zf)(s)el(s)dm(s)

(Tη^∗g)(l)

= (m(S))^−1/2

∞

X

l=−∞

Z

S

τ(l, s)(F_Zf)(s)el(s)dm(s) Z

S

η^∗(l, w)(F_Zg)(w)el(w)dm(w) (3.7)

Now, letm, n∈Zand for alll∈Z, f(l) =fn(l) =

0 ifl6=n,

1 ifl=n, g(l) =gm(l) =

0 ifl6=m, 1 ifl=m.

Therefore from (3.7) we have, Z

S

λ(m, s)e_n(s)e_m(s)dm(s)

=

∞

X

l=−∞

Z

S

τ(l, s)en(s)el(s)dm(s) Z

S

η^∗(l, w)em(w)el(w)dm(w) and

F_S(λ(m,·)e_m(·))

(n) = (m(S))^1/2

∞

X

l=−∞

F_S(τ(l,·)e_l(·)) (n)

F_S(η^∗(l,·)e_l(·)) (m) So,

λ(m, s)em(s) = (m(S))^1/2

∞

X

n=−∞

FS(λ(m,·)em(·))

(n)en(s)

=m(S)

∞

X

n=−∞

∞

X

l=−∞

FS(τ(l,·)el(·)) (n)

FS(η^∗(l,·)el(·))

(m)e_n(s)

=m(S)

∞

X

l=−∞

FS(η^∗(l,·)el(·)) (m)

∞

X

n=−∞

FS(τ(l,·)el(·))

(n)en(s)

= (m(S))^1/2

∞

X

l=−∞

τ(l, s)el(s)

FS(η^∗(l,·)el(·)) (m).

Thus, for allm∈Z

λ(m, s) = (m(S))^1/2(em(s))⁻¹

∞

X

l=−∞

τ(l, s)el(s)

FS(η^∗(l,·)el(·))

(m), s∈ {s∈S:em(s)6= 0}.

Since T_τ is nuclear operator and T_η is bounded operator, by applying Theorem 3.2, the proof will

be complete.

As a consequence of the above theorem, we can give a criterion in terms of symbol such that the correspondingZ-operator to be normal.

16