(an Orlicz function Φ is not necessarily convex) to a Banach space X. We provide the space L(L

Agnieszka Oelke ^∗

Topological properties of spaces of linear operators on non-locally convex Orlicz spaces

Abstract. We study the topological properties of the space L(L

, X) of all continu- ous linear operators from an Orlicz space L

(an Orlicz function Φ is not necessarily convex) to a Banach space X. We provide the space L(L

, X) with the Banach space structure. Moreover, we examine the space L

(L

, X) of all singular operators from L

to X.

2000 Mathematics Subject Classification: 46E30, 47B38.

Key words and phrases: Orlicz spaces, linear operators, singular operators.

1. Notation and terminology. For terminology concerning Riesz spaces and function spaces we refer to [AB], [KA]. Let (Ω, Σ, µ) be a σ-finite atomless measure space and let L ⁰ denote the set of µ-equivalence classes of all real valued measurable functions defined on Ω. Then L ⁰ is a super Dedekind complete Riesz space under the natural ordering. Let 1 A stand for the characteristic function of a set A ∈ Σ.

Let ℕ denote the set of all natural numbers.

Now we recall notations and terminology concerning Orlicz spaces ([M], [MO], [RR]). By an Orlicz function we mean here a mapping Φ : [0, ∞) → [0, ∞) that is non-decreasing, left continuous, continuous at 0, vanishing only at 0 and satisfying lim inf _t→∞ ^Φ(t) _t > 0 (see [Ro]).

An Orlicz function Φ determines a modular % Φ : L ⁰ → [0, ∞] by the formula % Φ (u) = R

Ω Φ(|u(ω)|)dµ. The Orlicz space L ^Φ is an ideal of L ⁰ defined by L ^Φ = {u ∈ L ⁰ : % _Φ (λu) < ∞ for some λ > 0} and equipped with the complete topology T Φ of the F -Riesz norm ||u || Φ := inf{λ > 0 : % Φ (u/λ) ¬ λ} for u ∈ L ^Φ . It is known that the space (L ^Φ , T Φ ) is locally convex if and only if Φ is equivalent to a convex Orlicz function (see [MO]).

Let E ^Φ = {u ∈ L ⁰ : % Φ (λu) < ∞ for all λ > 0}. Then E ^Φ is a closed ideal of L ^Φ and supp E ^Φ = Ω.

∗

Partly supported by EFS

Let L ^Φ ₀ = {u ∈ L ⁰ : % Φ (u) < ∞}. Then L ^Φ 0 (called an Orlicz class) is an absolutely convex, absorbing subset of L ^Φ , and let p Φ stand for the Minkowski functional of L ^Φ ₀ , i.e., p Φ (u) = inf{λ > 0 : % ^Φ (u/λ) < ∞}. Then p ^Φ (u) ¬ ||u || ^Φ for u ∈ L ^Φ and E ^Φ = ker p Φ .

2. The space of continuous linear operators on Orlicz spaces. We as- sume that (X, k·k ^X ) is a real Banach space. By L(L ^Φ , X) we will denote the space of all ( || · || ^Φ , k · k ^X )-continuous linear operators from L ^Φ into X. A ( || · || ^Φ , k · k ^X )- continuous linear operator T : L ^Φ → X is said to be singular if there exists an ideal M of L ^Φ with supp M = Ω such that T (u) = 0 for all u ∈ M. By L ^s (L ^Φ , X) we will denote the set of all singular operators from L ^Φ into X.

The space of continuous linear functionals on an Orlicz space L ^Φ (Φ is not supposed to be convex) was studied by W. Orlicz (see [O]). In this section we extend the results of [O] to the setting of linear operators T : L ^Φ → X.

For T ∈ L(L ^Φ , X) let kT k ^L

→X := sup 

kT (u)k ^X : u ∈ L ^Φ , % Φ (u) ¬ 1 .

One can show that kT k L

→X < ∞ and k · k L

→X is a norm on L(L ^Φ , X) (see [CMO, Theorem 2.3]).

Proposition 2.1 Let T ∈ L(L ^Φ , X) . Then for u ∈ L ^Φ the following inequality holds:

kT (u)k ^X ¬ kT k L

→X (% Φ (u) + 1).

Proof Let % Φ (u) < ∞. Then % ^Φ (u) = n 0 + r 0 for some n 0 ∈ ℕ and 0 ¬ r ⁰ <

1. For A ∈ Σ let m ^u (A) = % Φ ( 1 A u) = R

A Φ(|u(ω)|)dµ. The countably additive measure m u : Σ → ℝ is atomless because (Ω, Σ, µ) is atomless. Hence there exist pairwise disjoint sets A i ∈ Σ, i = 1, . . . , n ⁰ + 1 such that u = Σ ⁿ _i=1

⁺¹ 1 A

u and

% Φ (1 A

u) = 1 for i = 1, . . . , n 0 and % Φ (1 A

u) = r 0 . Thus

kT (u)k ^X ¬

n X

+1 i=1

kT (1 ^A

u) k ^X ¬ kT k L

→X (n 0 + 1) ¬ kT k L

→X (% Φ (u) + 1).

Theorem 2.2 (L(L ^Φ , X), k · k ^L

→X ) is a Banach space.

Proof Assume that (T n ) is a Cauchy sequence in (L(L ^Φ , X), k · k L

→X ). Let u ∈ L ^Φ , i.e., % Φ (λu) < ∞ for some λ > 0. First, we shall show that (T ⁿ (u)) is a Cauchy sequence in (X, k · k ^X ). Indeed, let ε > 0 be given. There exists k ε ∈ ℕ such that

kT ⁿ − T ^m k L

→X ¬ λε

% Φ (λu) + 1 for all m, n ­ k ^ε . Hence

kT ⁿ (u) − T ^m (u)k ^X = 1

λ k(T ⁿ − T ^m )(λu)k ^X ¬ 1

λ kT ⁿ − T ^m k L

→X (% Φ (λu) + 1) ¬ ε

for any m, n ­ k ^ε . It means that (T n (u)) is a Cauchy sequence in (X, k · k ^X ).

Define T (u) := lim _n→∞ T n (u) in (X, k · k ^X ). Now, we shall show that T ∈ L(L ^Φ , X). It is clear that T is a linear operator. Let ||u || Φ < 1. Then % Φ (u) ¬

||u || Φ . Take ε > 0. There exists n ε ∈ ℕ such that kT ⁿ − T ^m k ^L

→X ¬ 2 ^ε for all

m, n ­ n ^ε . Fix n ­ n ε . Then

kT ⁿ (u) − T ^m (u)k ^X ¬ 2 kT ⁿ − T ^m k L

→X ¬ ε for any m ­ n ^ε . Hence we have

k(T ⁿ − T )(u)k ^X ¬ kT ⁿ (u) − T ^m (u)k ^X + kT ^m (u) − T (u)k ^X ¬ ε + kT ^m (u) − T (u)k ^X for any m ­ n ^ε . Consequently

k(T ⁿ − T )(u)k ^X ¬ ε + lim _{m →∞} kT ^m (u) − T (u)k ^X = ε.

It means that the operator T n − T : L ^Φ → X is ( || · || Φ , k · k ^X )-continuous. Thus the operator T = T n

− (T ⁿ

− T ) is also ( || · || Φ , k · k ^X )-continuous.

It remains to check that kT ⁿ − T k L

→X → 0. Let % ^Φ (u) ¬ 1 and let ε > 0 be given. There exists k ε ∈ ℕ such that kT ⁿ − T ^m k L

→X ¬ ε for m, n ­ k ^ε . We fix n ­ k ^ε . Note that for all m ­ k ^ε we have kT ⁿ (u) − T ^m (u)k ^X ¬ ε. Hence for n ­ k ^ε we have

k(T ⁿ − T )(u)k ^X ¬ kT ⁿ (u) − T ^m (u)k ^X + kT ^m (u) − T (u)k ^X ¬ ε + kT ^m (u) − T (u)k ^X . Therefore

k(T ⁿ − T )(u)k ^X ¬ ε + lim _m→∞ kT ^m (u) − T (u)k ^X = ε.

This means that kT ⁿ − T k L

→X ¬ ε for all n ­ k ^ε , and the proof

is complete. _■

3. The space of singular operators on Orlicz spaces. Singular linear functionals on Orlicz spaces L ^Φ has been studied by T. Ando [A] and M. Rao [R]

(Φ is supposed to be convex) and by M. Nowak [N] in the general case. In this section we describe the space L ^s (L ^Φ , X).

Proposition 3.1 For T ∈ L ^s (L ^Φ , X) we have kT k ^L

→X = sup 

kT (u)k ^X : u ∈ L ^Φ , % Φ (u) < ∞

= sup 

kT (u)k ^X : u ∈ L ^Φ , p Φ (u) ¬ 1 .

Proof Since T ∈ L ^s (L ^Φ , X), there exists an ideal M in L ^Φ that supp M = Ω and T (u) = 0 for all u ∈ M. There exists a sequence (Ω ⁿ ) ⊂ Σ such that Ω ⁿ ↑, S ∞

n=1 = Ω and 1 Ω

∈ M (see [S, Corollary 1.7.4]). Let % ^Φ (u) < ∞ and let for

n ∈ ℕ,

u n (ω) =

( u(ω) if |u(ω)| ¬ n and ω ∈ Ω ⁿ , 0 elsewhere .

Then |u ⁿ (ω)| ¬ n 1 ^Ω

(ω) for ω ∈ Ω, so u ⁿ ∈ M. Note that |(u ⁿ − u)(ω)| ¬

|u(ω)| and u ⁿ (ω) → u(ω) for ω ∈ Ω. Hence using the Lebesgue dominated conver- gence theorem, we get % Φ (u − u ⁿ ) → 0. Thus % ^Φ (u − u ⁿ

) ¬ 1 for some n ⁰ ∈ ℕ, and hence

kT (u)k ^X = kT (u − u ⁿ

) + T (u n

)k ^X = kT (u − u ⁿ

)k ^X ¬ kT k L

→X . It follows that

sup 

kT (u)k ^X : u ∈ L ^Φ , % Φ (u) < ∞

¬ kT k L

→X .

Since kT k L

→X ¬ sup {kT (u)k ^X : u ∈ L ^Φ , % Φ (u) < ∞}, we have kT k ^L

→X = sup 

kT (u)k ^X : u ∈ L ^Φ , % Φ (u) < ∞ .

Now let u ∈ L ^Φ with p Φ (u) ¬ 1. Let ε > 0 be given. Since p

(u)+ε ^u ∈ L ^Φ 0 , we get

kT k L

→X ­ T  u p Φ (u) + ε 

_X = 1

p Φ (u) + ε k T (u) k ^X ­ 1

1 + ε k T (u) k ^X .

It follows that kT k L

→X ­ kT (u)k ^X . Hence kT k L

→X ­ sup 

kT (u)k ^X : u ∈ L ^Φ , p Φ (u) ¬ 1 . On the other hand, we have

kT k ^L

→X = sup 

kT (u)k ^X : u ∈ L ^Φ , % Φ (u) ¬ 1

¬ sup 

kT (u)k ^X : u ∈ L ^Φ , p Φ (u) ¬ 1 . Thus

kT k L

→X = sup 

kT (u)k ^X : u ∈ L ^Φ , p Φ (u) ¬ 1 .

Corollary 3.2 We have

L ^s (L ^Φ , X) = {T ∈ L(L ^Φ , X) : T is (p Φ , k · k ^X ) − continuous}

= {T ∈ L(L ^Φ , X) : T (u) = 0 f or all u ∈ E ^Φ }.

Proof Assume that T ∈ L ^s (L ^Φ , X). Let u ∈ L ^Φ and ε > 0 be given. Since

u

p

(u)+ε ∈ L ^Φ 0 , in view of Proposition 3.1, we get kT k L

→X ­

T

 u

p Φ (u) + ε

 _X = 1

p Φ (u) + ε k T (u) k ^X .

Hence we have that kT (u)k ^X ¬ p ^Φ (u)·kT k L

→X , so T is (p Φ , k·k ^X )-continuous.

Now assume that T is (p Φ , k · k ^X )-continuous. Thus for some L > 0, kT (u)k ^X ¬ L · p ^Φ (u) for u ∈ L ^Φ . Since ker p Φ = E ^Φ , T (u) = 0 for all u ∈ E ^Φ . Note that if T ∈ L(L ^Φ , X) and T (u) = 0 for all u ∈ E ^Φ , then T ∈ L ^s (L ^Φ , X).

Thus the proof is complete. _■

References

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[RR] M.M. Rao and Z.D. Ren, Theory of Orlicz spaces, Marcel Dekker, New York, Basel, Hong Kong, 1991.

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ipzig, 1984.

Agnieszka Oelke

Faculty of Mathematics, Computer Science and Econometrics, University of Zielona Góra ul. Szafrana 4A, 65–516 Zielona Góra, Poland

E-mail: A.Oelke@wmie.uz.zgora.pl

(Received: 21.12.2009)