On Lorentz and Orlicz–Lorentz subspaces of bounded families and approximation type operators

RACSAMDOI 10.1007/s13398-013-0137-3

ORIGINAL PAPER

On Lorentz and Orlicz–Lorentz subspaces of boundedfamilies and approximation type operators

Manjul Gupta · Antara Bhar

Received: 8 August 2012 / Accepted: 8 July 2013© Springer-Verlag Italia 2013

Abstract For an arbitrary Banach space X and an arbitrary index set I,we denote by l∞,I (X),the Banach space of all bounded families {xi }i∈I in X, equipped with the sup norm; and byl p,q,M,I (X) and l p,q,r,I (X), subspaces of l∞,I (X), where p,q,r are positive reals and M isan Orlicz function. In case, X is a real Banach space which is also a σ -Dedekind completeBanach lattice, it is shown that l p,q,M,I (X) is σ -Dedekind complete Banach lattice containinga subspace order isometric to l∞ when 1/p − 1/q < −1. In this paper, we study theirstructural properties and characterize their elements. For X = K, the symbols l p,q,M (I )and l p,q,r (I ) are being used for the subspaces l p,q,M,I (X) and l p,q,r,I (X) respectively.Besides investigating relationships amongst the spaces l p,q,r (I ) for different positive indicesp,q and r, we consider their product. Using generalized approximation numbers of boundedlinear operators and these spaces, we consider operators of generalized approximation typel p,q,r and represent them as an infinite series of finite rank operators. We also establishthe quasi-Banach ideal structure of the class of all such operators. Finally we prove resultspreserving various set theoretic inclusion relations amongst these operator ideals. Theseresults generalize some of the earlier results proved for Lorentz spaces by A. Pietsch.

Keywords Approximation numbers of operators · Bounded families · Lorentz sequencespaces · Operator ideals · Orlicz sequence spaces

Mathematics Subject Classification (2000) 46A45 · 47B10 · 47L20 · 47B06

M. Gupta (B) · A. BharDepartment of Mathematics and Statistics, Indian Institute of Technology, 208016 Kanpur, Indiae-mail: [email protected]

A. Bhare-mail: [email protected]

M. Gupta, A. Bhar

1 Introduction

This paper, though independent in nature, is in continuation of our earlier work [8] which dealswith the vector-valued Lorentz sequence spaces defined corresponding to an Orlicz function.The origin of the present work essentially lies in the contributions of Hilbert [10] and Schmidt[28] who considered the sequence space l2 for their study on linear equations involvinginfinitely many unknowns. The sequence space l2 was extended to l p , for 1 < p < ∞by Riesz [27] in 1913, and further generalizations to Lorentz sequence spaces l p,q , for0 < p, q < ∞, appeared in a paper by Hardy and Littlewood [9] in 1931. Replacing theset of natural numbers N by an arbitrary index set I, the collection of bounded families, nullfamilies and absolutely summable families were shown to be Banach spaces in [24]. Alsousing the Lorentz sequence spaces and decreasing rearrangement of a bounded family, thefamilies l p,q(I ) were studied in [24]. In this paper, we introduce families l p,q,M,I (X) for0 < p, q ≤ ∞ corresponding to an Orlicz function M and l p,q,r,I (X) for 0 < p, q, r ≤ ∞,which envelope some of our results of [8] for Banach space X and I = N. Characterizing theelements of these spaces in the fifth section, we show that the operator ideals L(a)

p,q,r formedby operators of approximation type l p,q,r i.e the operators whose sequence of approximationnumbers belong to the space l p,q,r , are quasi-Banach operator ideals. After having provedthe representation of such operators as infinite series of finite rank operators, we derive thatsuch operator ideals are approximative. We also show that the product of operator idealsof the type L(a)

p,q,r is an operator ideal of similar type. In the final section of this paper weconsider results regarding the operators of generalized approximation type l p,q,r , which arethe generalization of approximation numbers defined with the help of a given quasi-normedoperator ideal. The results proved in this section generalize some of the results of previoussection. We also prove results preserving various set theoretic inclusion relations amongstthese classes of operators.

2 Preliminaries

An Orlicz function M is a continuous, convex function defined from [0,∞) to itself suchthat M(0) = 0, M(x) > 0 for x > 0. Such function M always has the integral representation

M(x) =x∫

0

p(t)dt

where p(t), known as the kernel of M, is right continuous, non-decreasing function for t > 0.Let us note that an Orlicz function M is always increasing and M(x) → ∞ as x → ∞. Alsotp(t) → ∞ as t → ∞ and tp(t) = 0 for t = 0, cf. [16], p. 139. However p(t) > 0 fort = 0 is equivalent to the fact that the Orlicz sequence space lM (see the definition below) isisomorphic to l1, cf. [14], p. 309. Therefore, we assume here that the kernel p(t) has value 0for t = 0 and obviously p(t) → ∞ as t → ∞.

Let ω be the family of all real or complex sequences, which forms a vector space withusual pointwise addition and scalar multiplication. We write en (n ≥ 1) for the n-th unitvector in ω, i.e en = {δnj }∞j=1, where δnj is the Kronecker delta, and φ for the subspace ofω, spanned by e′

ns, n ≥ 1, i.e

φ = sp{en : n ≥ 1}.

On Lorentz and Orlicz–Lorentz subspaces of bounded families and approximation type operators

A sequence space λ is a subspace of ω containing φ. A sequence space λ is said to be (i)symmetric if ασ = {ασ(i)} ∈ λ whenever α = {αi } ∈ λ and σ ∈ �, where � is collectionof all permutations of the set of natural numbers N, (ii) normal if β = {βi } ∈ λ whenever|βi | ≤ |αi |, i ≥ 1, for some α = {αi } ∈ λ.

A Banach sequence space (λ, ‖.‖λ) is called a BK-space provided each of the projectionmaps Pi : λ → K, Pi (α) = αi is continuous, for i ≥ 1, where K is the field of scalars andα = {α1, α2, ...}. A BK-space (λ, ‖.‖λ) is called an AK-space if α(n) → α, for each α ∈ λ,where α(n) = {α1, α2, ..., αn, 0, 0, ...}, the n-th section of α.

The norm ‖.‖λ is said to be monotone if ‖α‖λ ≤ ‖β‖λ for α, β in λ with |αi | ≤ |βi |,∀ i ≥ 1.

Let us now recall some concepts regarding bounded families from [24]. For a given indexset I, we denote by l∞(I ), the class of all scalar-valued bounded families defined on I. It is aBanach space with respect to the sup norm ‖.‖∞ defined as ‖α‖∞ = sup{|αi | : i ∈ I } forα ∈ l∞(I ). The cardinality of an element α in l∞(I ), written as card(α), is defined as thecardinality of the set {i ∈ I : αi = 0}. A member of l∞(I ) is said to be finite if its cardinalityis finite.

For an indexing set I, the class of all bounded families which converge to 0, is denoted as

c0 (I )={α={αi } ∈ l∞ (I ) : ∀δ > 0, ∃ a f ini te subset A ⊆ I, such that |αi | < δ, ∀ i /∈ A} .

The non-increasing rearrangement of an element α = {αi } in l∞(I ) is a sequence, denotedby {α∗

n} and given by

α∗n = in f {c ≥ 0 : card{i ∈ I : |αi | > c} < n}.

For 0 < p, q ≤ ∞ and arbitrary index set I, the Lorentz space l p,q(I ) is defined as

l p,q(I ) = {α = {αi } : {n1/p−1/q α∗n} ∈ lq}.

These spaces are quasi-Banach spaces equipped with the norm ‖.‖p,q ,

‖α‖p,q ={[∑

n≥1 nq/p−1(α∗

n

)q]1/qfor q < ∞

supn n1/pα∗n for q = ∞

For I = N, Lorentz spaces are nothing but the Lorentz sequence spaces which are usuallydenoted by λq,x or d(x, q) and are defined as

λq,x = d(x, q) =

⎧⎪⎨⎪⎩α = {αn} : ‖α‖q,x =

⎛⎝∑

n≥1

(α∗n)qwn

⎞⎠

1/q

< ∞

⎫⎪⎬⎪⎭

where 0 < q < ∞ and x = {wn} is a non-negative (not identically zero) weight sequence. Itis well known that the functional ‖α‖q,x is a norm (and d(x,q) is a Banach space) if and onlyif 1 ≤ q < ∞ and {wn} is non-increasing. Moreover, one can check that for 0 < q < ∞, thefunctional ‖α‖q,x is a quasi-norm (and d(x,q) is a quasi-Banach space) if and only if thereexists a constant K > 0 such that

∑2ni=1 wi ≤ K

∑ni=1 wi , for any n ∈ N.

Clearly, l p,q(I ) = d(x, q), where wn = nq/p−1.For detailed study of sequence spaces and Lorentz sequence spaces one is referred to

[3,4,12–16].A sequence M = {Mn} of Orlicz functions known as Musielak–Orlicz function has been

studied in [6,19]. Let pn be the kernel of Mn for n ∈ N. M is said to satisfy (i) L1 condition

M. Gupta, A. Bhar

if pn(u) ≥ pn+1(u), ∀ u ∈ [0,∞) and (ii) L2 condition if∑

n≥1 Mn(u) = ∞ for all u > 0.Given a Musielak–Orlicz function M , a convex modular M on ω is defined as

M ({αn}) = supσ∈�

∞∑n=1

Mn(ασ(n)).

In case M satisfies L1 condition, the following equivalence has been proved by Foralewskiin [5], p. 348 (cf. [6], p. 184).

Theorem 2.1 A Musielak–Orlicz function M = {Mn} satisfies L1 condition if and only ifM ({αn}) = ∑

n≥1 Mn(α∗n), ∀ α ∈ ω.

Corresponding to a convex modular M , we have modular sequence space defined in theliterature as

λM = {α = {αn} ∈ ω : M (βα) < ∞ f or some β > 0}.This space becomes a normed space under the Luxemburg norm

‖α‖ = inf

{β > 0 : M

(α

β

)≤ 1

}.

Observe that a modular sequence space λM is a symmetric sequence space.Coming to the vector-valued families/sequence spaces, for an arbitrary index set I and a

Banach space X, the spaces l∞,I (X) and c0,I (X) are defined as

l∞,I (X) = {x = {xi }i∈I in X : {‖xi‖} ∈ l∞(I )}and

c0,I (X) = {x = {xi }i∈I in X : {‖xi‖} ∈ c0(I )}.respectively. For x = {xi } ∈ l∞,I (X), if α = {‖xi‖}, we write

x∗n = (x)∗n = α∗

n .

If λ is a symmetric, normal, AK-BK sequence space equipped with the monotone norm ‖.‖λ

and X, a Banach space with its topological dual X∗ endowed with the operator norm topologygenerated by ‖.‖, the vector valued sequence spaces λs(X) and λw(X) defined below, havebeen introduced and studied earlier in [25,26], under different notations. Indeed, we have,

λs(X) = {x = {xn} in X : {‖xn‖} ∈ λ}and

λw(X) = {x = {xn} in X : { f (xn)} ∈ λ, ∀ f ∈ X∗}.The spaces λs(X) and λw(X) are Banach spaces when they are equipped respectively withthe norm defined as

‖x‖sλ = ‖{‖xn‖}‖λ, x = {xn} ∈ λs(X)

and

‖x‖wλ = sup{‖{ f (xn)}‖λ, f ∈ X∗, ‖ f ‖ ≤ 1}, x = {xn} ∈ λw(X).


For non-negative real-valued sequences {αn}, {βn}, the symbol {αn} ≺ {βn} means thatthere exists a constant C > 0, such that αn ≤ Cβn , ∀n ∈ N. If {αn} ≺ {βn} and {βn} ≺ {αn},it is written as {αn} � {βn}. Let us recall the following relation from [24], p. 36

n∑m=1

1

m1+α(1 + log m)β� 1

n1+α(1 + log n)β, f or α < 0 and − ∞<β <∞. (2.1)

A real Banach space X equipped with the partial order ≤ which is compatible with thevector space structure (i.e x ≤ y implies x + z ≤ y + z, for x, y, z ∈ X and ax ≥ 0, forevery positive element x in X and every non-negative real a) is said to be a Banach latticeif ‖x‖ ≤ ‖y‖ whenever |x | ≤ |y|, for x, y ∈ X where |x | = x

∨(−x), the absolute value

of x. A Banach lattice X is said to be σ -Dedekind complete (σ -DC) if any non-negativeorder-bounded sequence {xn} in X has supremum in X. A mapping T from a Banach lattice Xonto a closed subspace of a Banach lattice Y is said to be an order isometry if it is an orderisomorphism which is also a linear isometry. For the theory of Banach lattices, we refer to[2,17].

The following result of Hudzic [11] will be used in the sequel

Theorem 2.2 A σ -DC Banach lattice X contains an order isometric copy of l∞ if and onlyif there exists a sequence {xn} of positive elements in X such that xn

∧xm = 0, n = m,

‖xn‖ = 1, ∀ n ∈ N, supn xn exists in X and ‖ supn xn‖ = 1.

Using the above result, Foralewski et. al [6] proved

Theorem 2.3 If a Musielak–Orlicz function M does not satisfy L2-condition, then λM con-tains an order isometric copy of l∞.

We denote by X and Y, the Banach spaces defined over the same field K of real or complexnumbers. The notations L(X, Y ) and L are respectively used for the class of bounded linearoperators between X and Y and the class of all bounded linear operators between any pair ofBanach spaces.

Let 0 < r ≤ ∞, 1 ≤ p, p′, q, q ′ ≤ ∞, 1/p + 1/p′ = 1, 1/q + 1/q ′ = 1 and1 + 1/r ≥ 1/p + 1/q . An operator T ∈ L(X, Y ) is said to be (r,p,q)-nuclear if T has thefollowing representation

T x =∑n≥1

αn fn(x) yn

where {αn} ∈ lr , { fn} ∈ lwq ′(X∗) and {yn} ∈ lwp′(Y ).The class of all such operators is denoted as Nr,p,q . For T ∈ Nr,p,q(X, Y ) the quasi-norm

‖.‖Nr,p,q is defined as

‖T ‖Nr,p,q = in f ‖{αn}‖r ‖{ fn}‖w

q ′ ‖{yn}‖wp′

where infimum is taken over all possible representations of T, cf. [21].Let T ∈ L(X, Y ) with rank(T ) = n. Then there exists a finite representation

T x =n∑

i=1

αi fi (x) yi (2.2)

such that |αi | ≤ ‖T ‖, ‖ fi‖ = 1 and ‖yi‖ = 1, where fi ∈ X∗ and yi ∈ Y , fori = 1, 2, . . . , n, cf. [21], p. 256.

M. Gupta, A. Bhar

The n-th approximation number of an operator T ∈ L(X, Y ) is defined as follows:

an(T ) = in f {‖T − A‖ : A ∈ L(X, Y ), rank(A) < n}.An operator ideal A is a subclass of L containing all finite rank operators such that the

components A(X, Y ) = A∩L(X, Y ) for any pair of Banach spaces X,Y satisfy the followingconditions:

(I) If T1, T2 ∈ A(X, Y ), then T1 + T2 ∈ A(X, Y ).(II) If R ∈ L(X0, X), S ∈ A(X, Y ), T ∈ L(Y, Y0), then T S R ∈ A(X0, Y0), where X0, Y0,

X, Y are Banach spaces.

Let A be an operator ideal. Then a map ‖.‖A : A → R+ is said to be a quasi-norm if it

satisfies the following conditions:

(I) ‖IX‖A = 1, where X is a one-dimensional Banach space and IX is the identity operatorof X.

(II) There exists constant C ≥ 1, such that for any pair of Banach spaces X, Y and T1, T2 ∈A(X, Y ), we have

‖T1 + T2‖A ≤ C[‖T1‖A + ‖T2‖A].(III) If R ∈ L(X0, X), S ∈ A(X, Y ), T ∈ L(Y, Y0), then,

‖T S R‖A ≤ ‖T ‖‖S‖A‖R‖.A quasi-norm ‖.‖A on the operator ideal A is said to be a p-norm ( 0 < p ≤ 1) if

condition (II) of quasi-norm is replaced by the following p-triangle inequality

‖T1 + T2‖pA ≤ ‖T1‖p

A + ‖T2‖pA, f or T1, T2 ∈ A(X, Y ).

A quasi-normed operator ideal is an operator ideal equipped with an ideal quasi-normand a quasi-Banach operator ideal is a quasi-normed operator ideal of which each compo-nent is complete with respect to the ideal quasi-norm. For two quasi-Banach operator ideals(A1,‖.‖A1 ) and (A2,‖.‖A2 ), A1 ⊆ A2 means that the inclusion map from A1 to A2 is con-tinuous. An operator ideal A is said to be a p-Banach operator ideal if A is equipped witha p-norm such that all the components A(X, Y ) are complete. It has been shown in [21] that( Nr,p,q , ‖.‖N

r,p,q ) is a s-Banach operator ideal, where 1/s = 1/r + 1/p′ + 1/q ′, where p′,q ′ are conjugate exponents of p and q respectively.

Let A1 and A2 be two operator ideals. Then the product A1 ◦ A2 of A1 and A2 containsall those operators T in L(X, Y ) which can be factorized suitably through components ofA1 and A2 i.e if T ∈ L is in L(X,Y), then T ∈ A1 ◦ A2 if and only if T = RS, whereS ∈ A2(X, Z) and R ∈ A1(Z , Y ), for Banach spaces X, Y and Z.

A quasi-Banach operator ideal A is said to be approximative if the set F(X, Y ) of finiterank operators, is dense in every component A(X, Y ), w.r.to the quasi-norm of the operatorideal. For the theory of operator ideals, we refer to [4,21,24].

Let A be any quasi-Banach operator ideal. The n-th A -approximation number ofT ∈ A(X, Y ), cf. [24], is defined by

an(T |A) = in f { ‖T − S‖A, S ∈ L(X, Y ), rank(S) < n}.The sequence {an(T |A)} satisfies the following properties

(I) ‖T ‖A = a1(T |A) ≥ a2(T |A) ≥ ..... ≥ 0, for T ∈ A(X, Y ).(II) am+n−1((T1 + T2)|A) ≤ C [am(T1|A) + an(T2|A)], for T1, T2 ∈ A(X, Y ).


(III) an((RST )|A) ≤ ‖R‖ an(S|A) ‖T ‖, where R ∈ L(Y, Y0), S ∈ A(X, Y ), T ∈L(X0, X).

(IV) If rank(T ) < n, then an(T |A) = 0.(V) For T ∈ A2(X, Y ) and S ∈ A1(Y, Z), am+n−1((ST )|A1 ◦A2) ≤ am(S|A1) an(T |A2)

3 The families l p,q,M,I (X), 0 < p, q ≤ ∞

Let M be an Orlicz function, I be an arbitrary index set and X be a Banach space. Using thedecreasing rearrangement of x = {xi }i∈I ⊂ X , let us introduce,

l p,q,M,I (X) =⎧⎨⎩{xi }i∈I ∈ l∞,I (X) :

∑n≥1

M

(n1/p−1/q x∗

n

ρ

)< ∞, f or some ρ > 0

⎫⎬⎭

and

l p,∞,M,I (X) =⎧⎨⎩{xi }i∈I ∈ l∞,I (X) :

∑n≥1

M

(n1/p x∗

n

ρ

)< ∞, f or some ρ > 0

⎫⎬⎭ .

For x ∈ l p,q,M,I (X), define

‖x‖p,q,M,I = in f

⎧⎨⎩ρ > 0 :

∑n≥1

M

(n1/p−1/q x∗

n

ρ

)≤ 1

⎫⎬⎭ , f or q < ∞

and

‖x‖p,∞,M,I = in f {ρ > 0 :∑n≥1

M

(n1/p x∗

n

ρ

)≤ 1}, f or q = ∞.

Let us note that for I = N, we get the spaces l p,q,M (X) which have been shown to bequasi-Banach spaces for p < q and Banach spaces for p ≥ q in [8].

In case of arbitrary index set I = N, we would like to point out that support x = {i ∈I : ‖xi‖ = 0} for x ∈ l p,q,M,I,(X) is not necessarily countable; for example, consider theindex set I = R(the set of all reals), Banach space X = R and the family α = {αi }i∈R

defined as

αi ={

1 for i ∈ [−1, 1]1/n for i ∈ [−n, n] \ [− (n − 1) , n − 1] , n ≥ 2

Then α∗n = 1, ∀n ∈ N. Note that support α is not countable. If 1/p − 1/q + 1 < 0, we

have ∑n≥1

M(n1/p−1/qα∗n) ≤

∑n≥1

n1/p−1/q M(1) < ∞,

so α ∈ l p,q,M (I ).For 1/p − 1/q = −1 and M(x) = xr , where r > 1, we have

∑n≥1

M(n1/p−1/qα∗n) =

∑n≥1

1

nr< ∞.

So α ∈ l p,q,M (I ) which is the same as l p,q,r (I ), cf. next section.

M. Gupta, A. Bhar

Indeed, choosing M suitably for −1 < 1/p − 1/q < 0, one can show that α ∈ l p,q,M (I ),but support α is not countable.

Remark In case l p,q,M,I (X) ⊆ c0,I (X), support x is at most countable, for each x ∈l p,q,M,I (X) and for all p and q > 0.

In case p ≤ q , we prove

Proposition 3.1 For an arbitrary index set I = N and p ≤ q, any element x of l p,q,M,I (X)

has at most countable support.

Proof Since p ≤ q and M(u) > 0 for any u > 0, we get {x∗n } ∈ lM ⊂ c0, cf. [14], p. 91. Let

x = {xi } ∈ l p,q,M,I (X). By definition of x∗n , for each n ∈ N we can find cn > 0 such that

cn < 2x∗n and card{i ∈ I : ‖xi‖ > cn} < n.

Hence cardinality of the set A = ⋃n An , where An = {i ∈ I : ‖xi‖ > cn}, is finite or

countably infinite. Also for i ∈ I\A, ‖xi‖ ≤ cn < 2x∗n → 0 as n → ∞. Therefore ‖xi‖ = 0,

for i ∈ I\A. This proves the required result.ALITER:Since limn→∞ x∗

n = 0, we have

card(An) = card({i ∈ I : 1/(n + 1) ≤ ‖xi‖ < 1/n}) < ∞for any n ∈ N. Obviously

card(A0) = card({i ∈ I : ‖xi‖ ≥ 1}) < ∞.

Since the support of x is equal to A = ⋃∞n=0 An , it is at most countable. ��

Concerning the structural properties of these families, we have,

Theorem 3.2 The spaces lp,q,M,I (X) equipped with ‖.‖p,q,M,I are Banach spaces if q ≤ pand they are quasi-Banach spaces for q > p.

Proof For given 0 < p, q < ∞, the space l p,q,M,I (X) is a vector space with usual pointwiseaddition and scalar multiplication. Indeed, if x = {xi }, y = {yi } are in l p,q,M,I (X), we canfind positive reals ρ1 and ρ2 such that

∑n≥1

M

(n1/p−1/q x∗

n

ρ1

)< ∞

and

∑n≥1

M

(n1/p−1/q y∗

n

ρ2

)< ∞.

Then, for 1/p − 1/q > 0, we have

∑n≥1

M

(n1/p−1/q (x + y)∗n21/p−1/q(ρ1 + ρ2)

)=∑n≥1

M

((2n)1/p−1/q (x + y)∗2n

21/p−1/q(ρ1 + ρ2)

)

+∑n≥1

M

((2n − 1)1/p−1/q (x + y)∗2n−1

21/p−1/q(ρ1 + ρ2)

)


≤ 2∑n≥1

M

(n1/p−1/q (x∗

n + y∗n )

(ρ1 + ρ2)

)

≤ 2

⎡⎣ ρ1

ρ1 + ρ2

∑n≥1

M

(n1/p−1/q x∗

n

ρ1

)

+ ρ2

ρ1 + ρ2

∑n≥1

M

(n1/p−1/q y∗

n

ρ2

)⎤⎦ < ∞. ........(∗)

Hence x + y ∈ l p,q,M,I (X). However, for 1/p −1/q ≤ 0, x + y ∈ l p,q,M,I (X) by convexityof the modular ρM defined below.

For scalar multiplication, let us note that (λz)∗n = |λ|z∗n for any scalar λ and z ∈

l p,q,M,I (X).For proving the norm character of ‖.‖p,q,M,I , we note that the sequence of functions

M = {Mn} where for n ∈ N

Mn(u) = M(n1/p−1/qu), u ≥ 0

is a Musielak–Orlicz function. We have also the equality

pn(u) = n1/p−1/q . p (n1/p−1/qu) f or n ∈ N,

where pn and p denote the right derivatives of Mn and M, respectively. Obviously, if q ≤ p,the sequence {pn} is non-increasing for any u ≥ 0, or equivalently, the Musielak–Orliczfunction {Mn} satisfies the L1 condition. Hence, by Theorem 2.1 the modular

M (x) =∞∑

n=1

M(n1/p−1/q x∗n )

is convex and, in consequence, the functional ‖.‖p,q,M,I is a norm.To prove the quasi-norm character of the space l p,q,M,I (X) for q > p, let us consider

x, y ∈ l p,q,M,I (X). Given any ε > 0, there exist ρ1, ρ2 > 0 such that

ρ1 < ‖x‖p,q,M,I + ε/2;∑n≥1

M

(n1/p−1/q x∗

n

ρ1

)≤ 1

and

ρ2 < ‖y‖p,q,M,I + ε/2;∑n≥1

M

(n1/p−1/q y∗

n

ρ2

)≤ 1.

From (*) we have

∑n≥1

M

(n1/p−1/q (x + y)∗n

21/p−1/q+1(ρ1 + ρ2)

)≤ 1, f or 1/p − 1/q > 0.

Hence, ‖x + y‖p,q,M,I ≤ 21/p−1/q+1(‖x‖p,q,M,I + ‖y‖p,q,M,I ).The proof of completeness of these spaces is analogous to ones given in [8] and so

omitted. ��Next we prove

M. Gupta, A. Bhar

Theorem 3.3 For x ∈ l∞,I (X), we have

(I) x ∈ l p,q,M,I (X) ⇒ {x∗n+m} ∈ l p,q,M , ∀m = 0, 1, 2, ..... and, conversely

(II) {x∗2n−1} ∈ l p,q,M ⇒ x ∈ l p,q,M,I (X).

Proof For x ∈ l p,q,M,I (X), we have x∗n+m ≤ x∗

n for each m = 0, 1, 2, ....... As M isincreasing, the result follows.

For proving (II), consider x ∈ l∞,I (X) such that {x∗2n−1} ∈ l p,q,M . Then

∑n≥1

M

(n1/p−1/q x∗

2n−1

ρ

)< ∞, f or some ρ > 0.

Now, for 1/p − 1/q ≤ 0

∑n≥1

M

(n1/p−1/q x∗

n

ρ

)≤ 2

∑n≥1

M

(n1/p−1/q x∗

2n−1

ρ

)< ∞;

and, for 1/p − 1/q > 0

∑n≥1

M

(n1/p−1/q x∗

n

21/p−1/qρ

)≤ 2

∑n≥1

M

(n1/p−1/q x∗

2n−1

ρ

)< ∞.

Thus (II) follows. ��The above result immediately leads to:

Corollary 3.4 A bounded family x is in l p,q,M,I (X) if and only if {x∗2n−1} ∈ l p,q,M .

Remarks In view of the above corollary, it suffices to consider the sequence {x∗2n−1} for

checking whether a family x is in l p,q,M,I (X) or not. It may be mentioned here that all theabove results are valid for the spaces l p,∞,M,I (X), which can be proved analogously.

If X is a real Banach space we prove

Proposition 3.5 Let X be a σ -DC Banach lattice and q ≤ p. Then lp,q,M,I (X) becomes aσ -DC Banach lattice.

Proof Since q ≤ p, it has been shown in Theorem 3.2 that l p,q,M,I (X) is a Banach space.For proving that l p,q,M,I (X) is a Banach lattice, let us consider x, y ∈ l p,q,M,I (X), such

that |xi | ≤ |yi | for each i ∈ I . As X is a Banach lattice, |xi | ≤ |yi | ⇒ ‖xi‖ ≤ ‖yi‖. Sinceq ≤ p, the Musielak–Orlicz function M = {Mn} satisfies L1 condition, where Mn(u) =M(n1/p−1/qu), u > 0 and hence

M (x) =∑n≥1

Mn(x∗n ) ≤

∑n≥1

Mn(y∗n ) = M (y).

This would imply ‖x‖p,q,M,I ≤ ‖y‖p,q,M,I . Thus l p,q,M,I (X) is a Banach lattice.To show the σ -Dedekind completeness of the space l p,q,M,I (X), consider a positive

sequence {xn} from l p,q,M,I (X) such that the sequence is bounded above by y ∈ l p,q,M,I (X),i.e. for each i ∈ I , xn

i ≤ yi , ∀ n ∈ N. As X is σ -DC, there exists {xi } ⊂ X such thatsupn xn

i = xi and hence xi ≤ yi for each i. Consequently, x = {xi } ∈ l p,q,M,I (X), sinceM (x) ≤ M (y). This completes the proof. ��

As a consequence of the above proposition and Theorem 2.2, we have

Proposition 3.6 If 1/p − 1/q < −1, the space lp,q,M,I (X) contains an order linearlyisometric copy of l∞.


4 The spaces l p,q,r,I (X), for 0 < p, q, r ≤ ∞

For 0 < p, q, r ≤ ∞, an arbitrary index set I and Banach space X, we define,

l p,q,r,I (X) = {x = {xi }i∈I ∈ l∞,I (X) : {n1/p−1/q x∗n } ∈ lr }

For x ∈ l p,q,r,I (X),write

‖x‖p,q,r,I ={ [∑n≥1(n

1/p−1/q x∗n )r ]1/r for r < ∞

supn n1/p−1/q x∗n for r = ∞

Let us note that we get l p,q,r,I (X) as particular case of l p,q,M,I (X) when M(x) = xr for1 ≤ r < ∞. In case I = N, the set of natural numbers, we write l p,q,r (X) for l p,q,r,I (X).Further for X = K and q = r , the spaces l p,q,r (I ) are Lorentz spaces l p,q(I ) studied in[24].

Further, the spaces l p,q,r (I ) are particular cases of Lorentz spaces λx,r = d(x, r) forx = {wn} = {nr(1/p−1/q)}. Obviously,

∑2ni=1 wi ≤ 2(

∑ni=1 wi ), whenever 0 < r < ∞

and 1/p − 1/q < 0. Also, for 0 < r < ∞ and 1/p − 1/q ≥ 0, we have w2i−1 + w2i =(2i − 1)r(1/p−1/q) + (2i)r(1/p−1/q) ≤ 21+r(1/p−1/q)ir(1/p−1/q). Therefore

2n∑i=1

wi ≤ 21+r(1/p−1/q)n∑

i=1

wi .

Hence we get immediately that for any 0 < r < ∞, the functional ‖.‖p,q,r,I is a quasi-normand the space l p,q,r,I (X) is a quasi-Banach space. Further, ‖.‖p,q,r,I is a norm and l p,q,r,I (X)

is a Banach space if and only if q ≤ p and 1 ≤ r < ∞ cf. [18], p. 413.Also, contains an order linearly isometric copy of l∞ for r/p − r/q ≤ −1, by Proposi-

tion 3.6.A characterization of members of the space l p,q,r,I (X) in terms of the sequence {sk},

where s is a fixed integer in the interval [2,∞), is contained in,

Proposition 4.1 A bounded family x = {xi : i ∈ I } is in l p,q,r,I (X) if and only if{sk(1/p−1/q+1/r) x∗

sk }∞k=0 ∈ lr , where s is a fixed integer in [2,∞).

Proof For k = 0, 1, 2, ......, write

Uk = {n ∈ N : sk ≤ n < sk+1}.

Vk = {n ∈ N : sk < n ≤ sk+1}Then, for p ≤ q ,

∑n∈Uk

nr/p−r/q =sk+1−1∑

sk

nr(1/p−1/q) < s(k+1)(r/p−r/q)(sk+1 − sk)

= (s − 1)sr(1/p−1/q)skr(1/p−1/q+1/r)

= C1skr(1/p−1/q+1/r), C1 = (s − 1)sr(1/p−1/q)

and

∑n∈Uk


sk

nr(1/p−1/q) ≥ skr(1/p−1/q)(sk+1 − sk)

= (s − 1)skr(1/p−1/q+1/r)

= C2skr(1/p−1/q+1/r), C2 = s − 1.

M. Gupta, A. Bhar

Also, for q < p

∑n∈Uk


sk

nr(1/p−1/q) ≤ C2skr(1/p−1/q+1/r)

and

∑n∈Uk


sk

nr(1/p−1/q) ≥ C1skr(1/p−1/q+1/r).

Therefore ∑n∈Uk

nr/p−r/q � skr(1/p−1/q+1/r).

Similarly, ∑n∈Vk

nr/p−r/q � s(k+1)r(1/p−1/q+1/r).

Now [∑n≥1

(n1/p−1/q x∗n )r

]1/r

=[∑

k≥0

∑n∈Uk

(n1/p−1/q x∗n )r

]1/r

≤ C

[∑k≥0

skr(1/p−1/q+1/r) (x∗sk )

r

]1/r (4.1)

for some constant C > 0.Hence x ∈ l p,q,r,I (X) if {sk(1/p−1/q+1/r) x∗

sk } ∈ lr .Assume now x ∈ l p,q,r,I (X) and consider⎡⎣∑

k≥0

(sk(1/p−1/q+1/r) x∗sk )

r

⎤⎦

1/r

≤ C0

⎡⎣(x∗

1 )r +∑k≥0

(x∗sk+1)

r∑n∈Vk

nr/p−r/q

⎤⎦

1/r

≤ C0[(x∗1 )r +

∑k≥0

∑n∈Vk

nr/p−r/q (x∗n )r ]1/r

= C0

⎡⎣∑

n≥1

(n1/p−1/q x∗n )r

⎤⎦

1/r

< ∞. (4.2)

Therefore, {sk(1/p−1/q+1/r) x∗sk } ∈ lr .

This completes the proof. ��Note The above result is being used in the next section for representing an operator ofapproximation type l p,q,r , in terms of finite rank operators.

For a Banach algebra X, we define the product of two families coordinatewise as

l p1,q1,r1,I (X) ◦ l p2,q2,r2,I (X) = {z : z = x y, x = {xi } ∈ l p1,q1,r1,I (X) and y

= {yi } ∈ l p2,q2,r2,I (Y )}for 0 < p1, q1, r1, p2, q2, r2 ≤ ∞. For X = K, we show that the above product is a spaceof l p,q,r (I ) type.


Theorem 4.2 If p, p1, p2; q, q1, q2; r, r1, r2 are positive reals satisfying the relations,

1/p1 + 1/p2 = 1/p ; 1/q1 + 1/q2 = 1/q ; 1/r1 + 1/r2 = 1/r,

then

lp1,q1,r1(I ) ◦ l p2,q2,r2(I ) = l p,q,r (I ).

Proof For proving that l p1,q1,r1(I )◦ l p2,q2,r2(I ) ⊂ l p,q,r (I ), let us consider β ∈ l p1,q1,r1(I )◦l p2,q2,r2(I ), i.e. β = αγ , where α ∈ l p1,q1,r1(I ) and γ ∈ l p2,q2,r2(I ).

For p ≤ q ,[∑

n≥1(n1/p−1/qβ∗

n )r

]1/r

=[∑

n≥1((2n)1/p−1/q (αγ )∗2n)r + ∑

n≥1((2n − 1)1/p−1/q (αγ )∗2n−1)

r

]1/r

≤ 21/p−1/q+1/r

[∑n≥1

(n1/p1−1/q1α∗n )r (n1/p2−1/q2γ ∗

n )r

]1/r

≤ 21/p−1/q+1/r

(∑n≥1

(n1/p1−1/q1α∗n )r1

)1/r1(∑

n≥1(n1/p2−1/q2γ ∗

n )r2

)1/r2

< ∞and for q ≤ p[∑

n≥1(n1/p−1/qβ∗

n )r

]1/r

=[∑

n≥1((2n)1/p−1/q (αγ )∗2n)r + ∑

n≥1((2n − 1)1/p−1/q (αγ )∗2n−1)

r

]1/r

≤ 21/r

(∑n≥1

(n1/p1−1/q1α∗n )r1

)1/r1(∑

n≥1(n1/p2−1/q2γ ∗

n )r2

)1/r2

< ∞.

Hence β ∈ l p,q,r (I ) and l p1,q1,r1(I ) ◦ l p2,q2,r2(I ) ⊂ l p,q,r (I ) holds.For proving l p1,q1,r1(I ) ◦ l p2,q2,r2(I ) ⊃ l p,q,r (I ), let us consider α ∈ l p,q,r (I ) and write

γn = α∗n , and c = {γn}. Then c ∈ l p,q,r . Denoting by dk , the characteristic sequence of the

set {n ∈ N, 2k ≤ n < 2k+1}, k = 0, 1, 2, ...., i.e d0 = {1, 0, 0, ....}, d1 = {0, 1, 1, 0, 0, .....}etc., we have, c = ∑

k≥0 ck , where ck = cdk . Clearly each ck is a finitely non-zero sequence.For k = 0, 1, 2....., define

ak = 2k[(1/p−1/q+1/r)r/r1−(1/p1−1/q1+1/r1)] ‖ck‖−r/r2 ck

and

bk = 2k[(1/p−1/q+1/r)r/r2−(1/p2−1/q2 + 1/r2)] ‖ck‖r/r2 dk

where ‖ck‖ is the sup norm of ck . Note that

a0 = ‖c0‖−r/r2 c0 = γ−r/r21 {γ1, 0, 0, ....} = {γ r/r1

1 , 0, 0, ...}

a1 = 2[(1/p−1/q+1/r)r/r1−(1/p1−1/q1+1/r1)] γ−r/r22 {0, γ2, γ3, 0, 0, ...}

and so on........ If

a =∑k≥0

ak

M. Gupta, A. Bhar

then a ∈ l p1,q1,r1 . Similarly, if

b =∑k≥0

bk,

then b ∈ l p2,q2,r2 . Also c = ab.Thus c ∈ l p1,q1,r1 ◦ l p2,q2,r2 . Consequently,

l p1,q1,r1(I ) ◦ l p2,q2,r2(I ) ⊃ l p,q,r (I ).

This completes the proof. ��Remarks Let p, p1, p2; q, q1, q2; r, r1, r2 be positive reals satisfying the relations asmentioned in Theorem 4.2 and also q < p and r ≥ 1. Since the space l p,q,r (X)

is perfect in this case, cf. Proposition 4.2 of [8], as a consequence of Theorem 4.2, itfollows that l p1,q1,r1,I (X) ◦ l p2,q2,r2,I (X) ⊆ l p,q,r,I (X), where X is a Banach alge-bra and I = N. However, for an arbitrary index set I and Banach algebra X withan = ‖xn‖, bn = ‖yn‖, where {xn} ∈ l p1,q1,r1(X), {yn} ∈ l p2,q2,r2(X) one can estab-lish l p,q,r (X) ⊂ l p1, q1, r1 (X) ◦ l p2,q2,r2(X) by taking cn = z∗

n = ‖xn‖‖yn‖, for somez ∈ l p,q,r (X), in the proof of Theorem 4.2.

We now consider the interrelationships amongst the spaces l p,q,r (I ) for various positiveindices p,q,r. These are summarized in the form of the following

Theorem 4.3 (1) l p,q,r1(I ) ⊂ l p,q,r2(I ), for r1 ≤ r2, 0 < p, q ≤ ∞;(2) l p1,q,r (I ) ⊂ l p2,q,r (I ), for p1 ≤ p2, 0 < q, r ≤ ∞;(3) l p,q1,r (I ) ⊂ l p,q2,r (I ), for q2 ≤ q1, 0 < p, r ≤ ∞;(4) l p,q,∞(I ) ⊂ l p′,q,r (I ), for 1/p′ < 1/p − 1/r , 0 < q, r ≤ ∞;(5) l p,q,∞(I ) ⊂ l p,q ′,r (I ), for 1/q ′ > 1/q + 1/r , 0 < p, r ≤ ∞.

All the inclusions are proper.

Proof One can easily establish the inclusion in all the above cases (1) to (5). However, forshowing proper inclusion with specific values of p, q, r, we consider the following examples:

(1) For q = ∞, r1 = 1, and r2 = ∞, we have, l p,∞,1 is proper subset of l p,∞,∞, e.g,consider the sequence {1/n2}. Then {1/n2} is not in l p,∞,1, but belongs to l p,∞,∞, forp = 1.

(2) In this case, consider the sequence {1/n} for the space l1,1,1 and l2,1,1 i.e p1 = 1,p2 = 2, q = 1, r = 1. Then {1/n} ∈ l2,1,1; but {1/n} is not in l1,1,1 = l1.

(3) Here consider q1 = 2, q2 = 1, p = 2, r = 1, and again the sequence {1/n}. Then{1/n} ∈ l p,q2,r ; but /∈ l p,q1,r .

(4) In this case, let us take p = 1/4, p′ = 1, q = 1, r = 2 and the sequence {1/n}. Then{n1/p−1/q x∗

n } = {n2} /∈ l∞, but {n1/p′−1/q x∗n } = {1/n} ∈ l2. Thus {1/n} ∈ l p′,q,r ; but

/∈ l p,q,∞.(5) For the proper inclusion, consider p = 1/4, q = 1, q ′ = 1/4, r = 2 and the sequence

{1/n}. Then {n1/p−1/q ′x∗

n } = {1/n} ∈ l2, but {n1/p−1/q x∗n } = {n2} /∈ l∞. ��

Note Indeed, examples of bounded families α may be constructed in such a way that thesequence {α∗

n} turns out to be one of the above sequences.Finally, in this section we give some examples of null sequences which do not belong

to the Lorentz sequence spaces l p,q for different values of p and q. Also some relationshipsamongst the spaces l p,q,M (I ), l p,q(I ) and l p(I ) are illustrated for different values of p, q anddifferent Orlicz functions M. Indeed, we have


Example 4.4 Let I = N.

(i) For 0 < p < q , consider the sequence α = {αn}, α1 = 1, and αn = 1/ log n, n ≥ 2.Then {αn} is a decreasing sequence and

αn ≤ n1/p−1/q αn, ∀ n ≥ 1.

⇒ {αn} /∈ l p,q for every 0 < p < q . Thus α = {αn} /∈ ⋃p<q l p,q .(ii) For p > q , consider

βn = 1

i; 2i−1 + 1 ≤ n ≤ 2i , f or i = 1, 2, 3, .......

The sequence {βn} is a non-increasing sequence of rational numbers and

∑n≥1

(n1/p−1/q βn)q =∑n≥1

2n∑i=2n−1+1

iq/p−1

nq≥∑n≥1

2q/p−12n−1

nq= ∞.

Hence {βn} /∈ ⋃p>q l p,q .

Example 4.5 For fixed r > 1, let us consider the Orlicz function M(x) = xr (|log x | + 1),for x > 0 and M(0) = 0. Then, for any indexing set I, p ≤ q and α = {αi }i∈I ∈ l∞(I )

∑n≥1

(α∗n)r ≤

∑n≥1

(n1/p−1/q α∗n)r ≤

∑n≥1

(n1/p−1/q α∗n)r [|log(n1/p−1/qα∗

n)| + 1],

⇒ l p,q,M (I ) ⊆ l p,q,r (I ) ⊆ lr (I ).

However, these inclusions may be proper, e.g, consider p = 1, q = 2, r = 3 and I = N.The sequence {αn} = { 1

n5/6(log n)1/2 }n>1 is non-increasing and

∑n>1

(n1/p−1/q α∗n)r =

∑n>1

1

n(log n)3/2 < ∞.

But,

∑n>1

(n1/p−1/qα∗n)r [|log(n1/p−1/qα∗

n)| + 1] =∑n>1

[1/3 log n + 1/2 log(log n) + 1]n(log n)3/2

>∑n>1

1

3n(log n)1/2 = ∞.

⇒ α /∈ l1,2,M . Hence, l1,2,M is properly contained in l1,2,3. Also, for the proper containmentof l1,2,3 in l3, consider the sequence {n−5/6}.

Example 4.6 For the Orlicz function M(x) = x2

log(e+x), for x ≥ 0, lM (I ) = l2(I ) for any

indexing set I. Indeed,

limn→∞

(α∗n)2

(α∗n )2

log(e+tn(α))

= 1

for α = {αi } ∈ c0(I ).

M. Gupta, A. Bhar

5 Operators of the approximation type l p,q,r

In this section, we consider the representation of operators of the type l p,q,r in terms of finiterank operators. Recalling the definition of approximation numbers, let us consider

Definition 5.1 An operator T ∈ L(X, Y ) is said to be of approximation type l p,q,r if{an(T )} ∈ l p,q,r , for 0 < p, q, r ≤ ∞.

The collection of these operators between Banach spaces X and Y is written as

L(a)p,q,r (X, Y ) = { T ∈ L(X, Y ) : {an(T )} ∈ l p,q,r }.

The notation L(a)p,q,r is used for the collection of such operators between any pair of Banach

spaces.For T ∈ L(a)

p,q,r (X, Y ), we define,

‖T ‖p,q,r =⎡⎣∑

n≥1

(n1/p−1/q an(T ))r

⎤⎦

1/r

.

It is shown in [8], for r ≥ 1 the class L(a)p,q,r is a quasi-Banach operator ideal with respect

to ‖.‖p,q,r . However, for 0 < r < 1, proceeding on lines similar to the above case, one

can easily prove that L(a)p,q,r is a quasi-Banach operator ideal with respect to the quasi-norm

‖.‖p,q,r . The triangular inequality in the two cases assumes the form:

‖T1 + T2‖p,q,r ≤ max(21/p−1/q+1/r , 21/r )(‖T1‖p,q,r + ‖T2‖p,q,r ), f or r ≥ 1;and

‖T1 + T2‖p,q,r ≤ max(21/p−1/q+2/r−1, 22/r−1)(‖T1‖p,q,r + ‖T2‖p,q,r ), f or 0 < r < 1.

Remark For q < p and r ≥ 1, L(a)p,q,r becomes Banach operator ideal; cf. Note, above

Theorem 3.1.

Motivated by the results proved in [22,23], the representation of an operator of approxi-mation type l p,q,r is given in

Theorem 5.2 Let s be a fixed integer in [2,∞) and p, q, r be positive real numbers. If anoperator T ∈ L(X, Y ) is of approximation type lp,q,r , then

T =∞∑

k=0

Tk (5.1)

where the convergence of the series is considered in the operator norm topology of L(X, Y )

and Tk ∈ L(X, Y ) are finite rank operators with rank(Tk) < sk; {sk(1/p−1/q+1/r) ‖Tk‖} ∈lr , for k = 0, 1, 2, ...... The converse is true provided 1/p − 1/q + 1/r ≥ 0. Moreover theabove representation yields an equivalent quasi-norm on L(a)

p,q,r defined as

‖T ‖sp,q,r = in f ‖{sk(1/p−1/q+1/r) ‖Tk‖}‖, (5.2)

infimum being taken over all such representations of T in case 1/p − 1/q + 1/r ≥ 0.


Proof Let us fix an integer s ∈ [2,∞) and consider T ∈ L(a)p,q,r (X, Y ). Using the definition

of approximation numbers, we can find Lk ∈ L(X, Y ) with rank(Lk) < sk such that

‖T − Lk‖ < 2 ask (T ). (5.3)

If ask (T ) = 0, then rank(T ) < sk and in this case, T = Lk = Tk . Therefore, assume thatask (T ) > 0, for each k = 0, 1, 2, ...... Define

T0 = 0 , T1 = 0 , Tk+2 = Lk+1 − Lk , k = 0, 1, 2, .......

Then rank(Tk+2) < sk+2 and ‖Tk+2‖ ≤ 4 ask (T ) for k = 0, 1, 2, ...... Further, we have

s(k+2)(1/p−1/q+1/r) ‖Tk+2‖ ≤ K0 sk(1/p−1/q+1/r) ask (T ) , k = 0, 1, 2, ...... (5.4)

where K0 = 4 s2(1/p−1/q+1/r). Hence {sk(1/p−1/q+1/r)‖Tk‖} ∈ lr , by Proposition 4.1 andthe sequence space lr is normal.

Now from Eq. (5.3),∑n+1

k=0 Tk = Ln and {an(T )} ∈ c0, we get T = ∑∞k=0 Tk .

For proving the converse, we assume that T ∈ L(X, Y ) admits the given representationwith the properties stated in the hypothesis. Then rank(

∑h−1k=0 Tk) < sh . Let us fix w and θ

such that 0 < w < min(1, r) and 0 < θ < 1. Define u by the relation 1/w = 1/r + 1/u.Then for any n ≥ h,

n∑k=h

‖Tk‖ ≤[

n∑k=h

‖Tk‖w

]1/w

=[

n∑k=h

s−θ(1/p−1/q+1/r)kw sθ(1/p−1/q+1/r)kw ‖Tk‖w

]1/w

≤[

n∑k=h

s−θ(1/p−1/q+1/r)ku

]1/u [ ∞∑k=h

sθ(1/p−1/q+1/r)kr ‖Tk‖r

]1/r

≤ C0 s−θ(1/p−1/q+1/r)h

[ ∞∑k=h


]1/r

(5.5)

< ∞

where C0 =[ ∞∑

k=1s−θ(1/p−1/q+1/r)ku

]1/u

is a finite constant.

Hence∞∑

k=h‖Tk‖ < ∞ and ash (T ) ≤ ∑∞

k=h ‖Tk‖. Consequently, by (5.5),for any n ∈ N,

n∑h=0

sh(1/p−1/q+1/r)r arsh (T ) ≤

n∑h=0

sh(1/p−1/q+1/r)r Cr0 s−θ(1/p−1/q+1/r)hr

×[ ∞∑

k=h


],

= Cr0

[ n∑k=0

(

k∑h=0

s(1−θ)(1/p−1/q+1/r)hr )

M. Gupta, A. Bhar

× sθ(1/p−1/q+1/r)kr ‖Tk‖r +∞∑

k=n+1

(

n∑h=0

s(1−θ)(1/p−1/q+1/r)hr )

× sθ(1/p−1/q+1/r)kr ‖Tk‖r]

≤ C2

[ ∞∑k=0

sk(1/p−1/q+1/r)r ‖Tk‖r

](5.6)

where C2 = Cr0 A, A = 1

s(1−θ)(1/p−1/q+1/r)r −1, a constant which we get in the inequality∑k

h=0 s(1−θ)(1/p−1/q+1/r)hr ≤ A.s(1−θ)(1/p−1/q+1/r)kr . Hence T ∈ L(a)p,q,r (X, Y ), by

Proposition 4.1.The equivalence of two quasi-norms, follows from Eqs. (4.1), (4.2), (5.4) and (5.6).This completes the proof. ��As an immediate consequence of the above theorem, we get

Proposition 5.3 The quasi-Banach operator ideal L(a)p,q,r is approximative.

Proof Immediate. ��Theorem 5.4 Let 0 < r ≤ 1 and 0 < p, q < ∞. If A is any r-Banach operator ideal suchthat

‖T ‖A ≤ c n1/p−1/q+1/r ‖T ‖, whenever rank(T ) ≤ n

where c > 0 is a constant, then L(a)p,q,r ⊆ A.

Proof Let T ∈ L(a)p,q,r (X, Y ). By Theorem (5.2) we can choose {Tk} ∈ L(X, Y ) such

that T = ∑k≥1 Tk , rank(Tk) < 2k , for each k and {2k(1/p−1/q+1/r)‖Tk‖} ∈ lr . Then

{‖Tk‖A} ∈ lr since

‖Tk‖A ≤ c 2k(1/p−1/q+1/r) ‖Tk‖, f or each k.

As A is r-Banach operator ideal and ‖.‖A is finer than operator norm ‖.‖, cf. [21], p. 90, wehave T ∈ A(X, Y ). ��Proposition 5.5 Let 0 < r ≤ ∞, 1 ≤ p, p′, q, q ′ ≤ ∞ such that 1/p + 1/p′ = 1,1/q + 1/q ′ = 1 and 1 + 1/r ≥ 1/p + 1/q. For T ∈ L(X, Y ), we have

‖T ‖Nr,p,q ≤ n1/r+1/p′+1/q ′ ‖T ‖

whenever rank(T ) ≤ n.

Proof Let T ∈ L(X, Y ) with rank(T ) ≤ n and Y0 = T (X). Then dim(Y0) ≤ n. IfIY0 : Y0 → Y0 is the identity map on Y0, then by (2.1) we have

‖IY0‖Nr,p,q ≤ n1/r+1/p′+1/q ′

.

Since rank(T ) ≤ n, we consider the factorization of T as T = J YY0

IY0 T0, where T0 is the

onto operator from X to Y0 such that T x = T0x with ‖T ‖ = ‖T0‖ and J YY0

is the injectionmap from Y0 to Y. Hence

‖T ‖Nr,p,q ≤ ‖J Y

Y0‖ ‖IY0‖N

r,p,q ‖T0‖ ≤ n1/r+1/p′+1/q ′ ‖T ‖.This completes the proof. ��


Proposition 5.6 Let 0 < r ≤ ∞, 1 ≤ p, p′, q, q ′ ≤ ∞ with 1/p+1/p′ = 1, 1/q+1/q ′ =1 and 1 + 1/r ≥ 1/p + 1/q. Then

L(a)p1,q,s ⊆ Nr,p,q

where 1/s = 1/p + 1/p′ + 1/q ′ and p1 = p2p−1 .

Proof Let T ∈ Nr,p,q(X, Y ) and rank(T ) ≤ n. Then

‖T ‖Nr,p,q ≤ n1/r+1/p′+1/q ′ ≤ n1/p1−1/q+1/s ‖T ‖.

Hence from Theorem 5.4 we have L(a)p1,q,s ⊆ Nr,p,q . ��

Concerning the product of two operator ideals of the type L(a)p,q,r , we have

Theorem 5.7 If p1, q1, r1, p2, q2, r2, p, q, r are positive reals satisfying the relations1/pi − 1/qi + 1/ri ≥ 0, for i = 1, 2 and 1/p1 + 1/p2 = 1/p, 1/q1 + 1/q2 = 1/qand 1/r1 + 1/r2 = 1/r , then

L(a)p1,q1,r1

◦ L(a)p2,q2,r2

= L(a)p,q,r .

Proof Let us note that the inclusion L(a)p1,q1,r1 ◦ L(a)

p2,q2,r2 ⊂ L(a)p,q,r is immediate from the

definition of the product of two operator ideals and the Hölder’s inequality.For proving the converse, let us consider T ∈ L(a)

p,q,r (X, Z).Then by Theorem (5.2), we canfind {Tk} ∈ L(X, Z) such that T = ∑∞

k=0 Tk , rank(Tk) ≤ 2k, {2k(1/p−1/q+1/r) ‖Tk‖} ∈lr , f or k = 0, 1, 2, ..... Since rank(Tk) ≤ 2k , we find the factorization of Tk as Tk = Rk Sk

satisfying the following properties:Rk ∈ L(Yk, Z), Sk ∈ L(X, Yk) and ‖Tk‖ = ‖Rk‖ ‖Sk‖ where dim(Yk) ≤ 2k with the

arrangement

2k(1/p1−1/q1+1/r1) ‖Rk‖ = (2k(1/p−1/q+1/r) ‖Tk‖)r/r1

and

2k(1/p2−1/q2+1/r2) ‖Sk‖ = (2k(1/p−1/q+1/r) ‖Tk‖)r/r2 .

Corresponding to the sequence {Yk} of finite dimensional Banach spaces, consider the Banachspace l2({Yk}) defined as

Y = l2({Yk}) = {{xk} : xk ∈ Yk, f or each k and {‖xk‖} ∈ l2},(one may refer to [1,7,20,29] for the detailed study of vector-valued sequence spaces.)

For k ∈ N, let Pk ∈ L(Y, Yk) and Ik ∈ L(Yk, Y ) be the usual projection and inclusionmaps given by

Pk({xn}) = xk for {xn} ∈ Y and Ik(x) = δxk = {0, 0, ...., x, 0, 0, ..} , x ∈ Yk being

placed at the kth coordinate.Then we have

rank(Rk Pk) ≤ 2k ; {2k(1/p1−1/q1+1/r1) ‖Rk Pk‖} ∈ lr1

and

rank(Ik Sk) ≤ 2k ; {2k(1/p2−1/q2+1/r2) ‖Ik Sk‖} ∈ lr2 .

M. Gupta, A. Bhar

Thus the operators R and S defined as

R =∞∑

k=0

Rk Pk , S =∞∑

k=0

Ik Sk

are respectively in L(a)p1,q1,r1(Y, Z) and L(a)

p2,q2,r2(X, Y ) by Theorem (5.2). Clearly T = RS

and so T ∈ L(a)p1,q1,r1 ◦ L(a)

p2,q2,r2(X, Z).

Hence L(a)p1,q1,r1 ◦ L(a)

p2,q2,r2 ⊃ L(a)p,q,r and the result follows. ��

Remarks The restriction 1/p − 1/q + 1/r > 0 on the indices p, q, r in the definition ofthe spaces l p,q,r (I ) yields that the spaces l p,q,r (I ) are ls,r (I ) where 1/s = 1/p − 1/q+ 1/r .

6 Operators of generalized approximation type l p,q,r

In the final section of this paper we define operators of generalized approximationtype l p,q,r and discuss their characterization through finite rank operators. Let us beginwith

Definition 6.1 Let A be any quasi-Banach operator ideal. An operator T ∈ A(X, Y ) is saidto be of A-approximation type l p,q,r if {an(T |A)} ∈ l p,q,r .

The set of all these operators is denoted by A(ga)p,q,r (X, Y ).

For T ∈ A(ga)p,q,r (X, Y ) we define

‖T ‖Ap,q,r = ‖{an(T |A)}‖p,q,r .

This is easy to verify that (A(ga)p,q,r , ‖.‖A

p,q,r ) is quasi-Banach operator ideal.The results preserving the various set theoretic inclusion relations are proved in

Proposition 6.2 Let A1 and A2 be two quasi-Banach operator ideals such that A1 ⊆ A2.Then

(A1)(ga)p,q,r ⊆ (A2)

(ga)p,q,r .

Proof Since A1 ⊆ A2, we can find some positive constant c such that ‖T ‖A2 ≤ c ‖T ‖A1 ,for every T ∈ A1. This would imply an(T |A2) ≤ c an(T |A1).

Hence the result. ��

Proposition 6.3 Let A1 and A2 be two quasi-Banach operator ideals. Then

(I) A1 ◦ (A2)(ga)p,q,r ⊆ (A1 ◦ A2)

(ga)p,q,r

(II) (A1)(ga)p,q,r ◦ A2 ⊆ (A1 ◦ A2)

(ga)p,q,r

(III) (A1)(ga)p1,q1,r1 ◦(A2)

(ga)p2,q2,r2 ⊆ (A1◦A2)

(ga)p,q,r , where 1/p1+1/p2 = 1/p, 1/q1+1/q2 =

1/q and 1/r1 + 1/r2 = 1/r .

Proof Immediate by using Property (V) of generalized approximation numbers. ��


Theorem 6.4 Let s be a fixed integer in [2,∞) and p, q, r be positive real numbers. Ifan operator T ∈ A(X, Y ) is of A-approximation type lp,q,r then T has the followingrepresentation

T =∞∑

k=0

Tk (6.1)

where Tk ∈ A(X, Y ) are finite rank operators with rank(Tk) < sk; {sk(1/p−1/q+1/r)

‖Tk‖A} ∈ lr , for k = 0, 1, 2, ...... The converse is true when 1/p − 1/q + 1/r ≥ 0.

Proof Proof is analogous to Theorem 5.2 and hence omitted. ��Theorem 6.5 Let 0 < p1, p2, p, q1, q2, q, r1, r2, r ≤ ∞ with 1/pi − 1/qi + 1/ri ≥ 0 ,for i = 1, 2 and 1/p1 + 1/p2 = 1/p; 1/q1 + 1/q2 = 1/q; 1/r1 + 1/r2 = 1/r . If A1

and A2 are two quasi-Banach operator ideals such that

‖T ‖A1 ≤ C n1/p1−1/q1+1/r1 ‖T ‖A2

whenever rank(T ) ≤ n, then

(A2)(ga)p,q,r ⊆ (A1)

(ga)p2,q2,r2 .

Proof Let T ∈ (A2)(ga)p,q,r (X, Y ). According to Theorem 6.4, we can find {Tk} ⊆ A2(X, Y )

such that T = ∑k≥1 Tk , where rank(Tk) < 2k , for every k and {2k(1/p−1/q+1/r) ‖Tk‖A2} ∈

lr ⊆ lr2 . Since rank(Tk) < 2k by hypothesis, we have

‖Tk‖A1 ≤ C 2k(1/p1−1/q1+1/r1) ‖Tk‖A2

⇒ 2k(1/p2−1/q2+1/r2)‖Tk‖A1 ≤ C 2k(1/p−1/q+1/r) ‖Tk‖A2 and hence {2k(1/p2−1/q2+1/r2)

‖Tk‖A1} ∈ lr2 .Thus T ∈ (A1)

(a)p2,q2,r2(X, Y ). This completes the proof. ��

Lemma 6.6 For 0 < p, q < ∞, 0 < r < ∞ and T ∈ L(X, Y )

‖T ‖Ap,q,r ≤ C n1/p−1/q+1/r ‖T ‖A

for some positive constant C, whenever rank(T ) ≤ n.

Proof Since(

n∑k=1

kr/p−r/q

)1/r

≤ C n1/p−1/q+1/r

for p, q < ∞ by Eq. (2.1), we have

‖T ‖Ap,q,r =

⎛⎝∑

k≥1

(k1/p−1/q an(T |A))r

⎞⎠

1/r

≤ C n1/p−1/q+1/r‖T ‖A.

for the operator T ∈ L(X, Y ) with rank(T ) ≤ n. This completes the proof. ��Lemma 6.7 Let A be a quasi-Banach operator ideal. Then for any T ∈ A(ga)

p,q,∞(X, Y )

n1/p−1/q a2n−1(T |A) ≤ an(T |A(ga)p,q,∞).

M. Gupta, A. Bhar

Proof Given any ε > 0, we can choose T1, T2 ∈ L(X, Y ) such that rank(T1), rank(T2) <

n and

‖T − T1‖Ap,q,∞ < (1 + ε) an(T |A(ga)

p,q,∞)

and

‖T − T1 − T2‖A < (1 + ε) an(T − T1|A).

Since rank(T1 + T2) < 2n − 1, we have

n1/p−1/q a2n−1(T |A) ≤ n1/p−1/q ‖T − (T1 + T2)‖A< (1 + ε) n1/p−1/q an(T − T1|A)

≤ (1 + ε) ‖T − T1‖Ap,q,∞

< (1 + ε)2 an(T |A(ga)p,q,∞).

As ε > 0 is arbitrary, we get n1/p−1/q a2n−1(T |A) ≤ an(T |A(ga)p,q,∞). ��

As consequences of the above results, we prove

Proposition 6.8 LetAbe a quasi Banach operator ideal and 0< p1, p2, p, q1, q2, q, r1, r2, r≤ ∞ be scalars which satisfy the following conditions 1/p2 − 1/q2 + 1/r2 > 0;1/p1 + 1/p2 = 1/p; 1/q1 + 1/q2 = 1/q and 1/r1 + 1/r2 = 1/r . Then

A(ga)p,q,r ⊆ (A(ga)

p1,q1,r1)(ga)p2,q2,r2 .

Proof Let T ∈ L(X, Y ) with rank(T ) ≤ n. Then

‖T ‖Ap1,q1,r1

≤ C n1/p1−1/q1+1/r1‖T ‖A

by Lemma 6.6. Now apply Theorem 6.5 to get the required result. ��Proposition 6.9 LetAbe a quasi Banach operator ideal and 0< p1, p2, p, q1, q2, q, r1, r2, r≤ ∞ be scalars which satisfy the following conditions 1/p1 +1/p2 = 1/p; 1/q1 +1/q2 =1/q and 1/r1 + 1/r2 = 1/r . Then

(A(ga)p1,q1,r1)

(ga)p2,q2,r2 ⊆ (A(ga)

p1,q1,∞)(ga)p2,q2,r2 ⊆ A(ga)

p,q,r2 .

Proof First inclusion is a consequence of Proposition 6.2. Therefore, consider T ∈ L(X, Y )

which is of A(ga)p1,q1,∞-approximation type l p2,q2,r2 . Then

n1/p1−1/q1 a2n−1(T |A) ≤ an(T |A(ga)p1,q1,∞),

by Lemma 6.7. Consequently,

n1/p−1/q a2n−1(T |A) ≤ n1/p2−1/q2 an(T |A(ga)p1,q1,∞)

Hence (A(ga)p1,q1,r1)

(ga)p2,q2,r2 ⊆ (A(ga)

p1,q1,∞)(ga)p2,q2,r2 ⊆ A(ga)

p,q,r2 . ��Acknowledgments The authors express their gratitude and thank the anonymous referee for pointing outthe results on order structure which helped us to include Propositions 3.5 and 3.6; and also the addition ofexamples on countable/uncountable support of families in the beginning of the Sect. 3. The authors are alsothankful to the referee for providing the alternative proof (ALITER) of Proposition 3.1 and the proof for thenorm character of ‖.‖p,q,M,I for q ≤ p in Theorem 3.2.


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On Lorentz and Orlicz–Lorentz subspaces of bounded families and approximation type operators