MATEMATIQKI VESNIK
UDK 517.984
originalniresearch
nauqnipaper
rad
57 (2005), 109–112
COMPACT COMPOSITION OPERATORS ON LORENTZ SPACES
Rajeev Kumar and Romesh Kumar
Abstract. We give a necessary and sufficient condition for the compactness of composition
operators on the Lorentz spaces.
Let X = (X, Σ, µ) be a σ-finite measure space. By L(µ), we denote the linear
space of all equivalence classes of Σ-measurable functions on X. Let T : X →
X be a non-singular measurable transformation. Then T induces a composition
transformation CT from L(µ) into itself defined by
CT f (x) = f (T (x)),
x ∈ X,
f ∈ L(µ).
Here, the non-singularity of T guarantees that the operator CT is well defined as a
mapping of equivalence classes of functions into itself. If CT maps a Lorentz space
L(pq, µ) into itself, then we call CT a composition operator on L(pq, µ) induced
by T .
Composition operators are simple operators, but have wide range of applications in ergodic theory, dynamical systems etc., see [7]. Composition operators
have been studied mostly on H 2 -spaces, Lp -spaces and Orlicz spaces (cf. [2–7]).
So, it is natural to extend the study of composition operators to other measurable
function spaces. In this paper, we have initiated the study of composition operators
on Lorentz spaces, which generalize Lp spaces. See [1] for details on Lorentz spaces.
For f ∈ L(µ), we define the distribution function of |f | on 0 < λ < ∞ by
µf (λ) = µ{x ∈ X : |f (x)| > λ},
and the non-increasing rearrangement of f on (0, ∞) by
f ∗ (t) = inf{λ > 0 : µf (λ) ≤ t} = sup{λ > 0 : µf (λ) > t}.
AMS Subject Classification: Primary 47B33, 46E30; Secondary 47B07, 46B70.
Keywords and phrases: Compact operator, composition operators, Lorentz spaces, measurable transformation.
First named author is supported by CSIR–grant (sanction no. 9/100 (96)/2002-EMR-I,
dated–13-5-2002).
110
R. Kumar, R. Kumar
The Lorentz spaces L(pq, µ) are defined by
L(pq, µ) = {f ∈ L(µ) : kf kL(pq) < ∞},
where
kf kL(pq) =
( R∞
[ 0 (t1/p f ∗ (t))q dt/t]1/q , if 1 < p < ∞, 1 ≤ q < ∞,
sup0<t<∞ t1/p f ∗ (t),
if 1 < p < ∞, q = ∞.
The Lorentz spaces (L(pq, µ, k.kL(pq) ) are Banach spaces for 1 ≤ q ≤ p < ∞, or
p = q = ∞.
The following theorem is easy to prove.
Theorem 1. Let T : X → X be a non-singular measurable transformation.
Then T induces a composition operator CT on L(pq, µ), 1 ≤ q ≤ p < ∞ if and only
if there exists some M > 0 such that
µ◦T −1 (A) ≤ M µ(A),
Moreover,
for each A ∈ Σ.
µ
kCT k =
sup
A∈Σ,0<µ(A)<∞
µ◦T −1 (A)
µ(A)
¶1/p
.
Compact composition operators on Lp -spaces were studied in [4], [5], [6] and [8].
For the compactness of these operators on Orlicz spaces, see [2]. Now we give a
necessary and sufficient condition for the compactness of composition operators on
Lpq (µ), 1 ≤ q ≤ p < ∞.
Theorem 2. Let (X, Σ, µ) be a σ-finite measure space and T : X → X be
a non-singular measurable transformation. Let {An } be all the atoms of X and
assume that µ(An ) = an > 0, for each n. Then CT is a compact composition
operator on Lorentz spaces L(pq, µ), 1 ≤ q ≤ p < ∞ if and only if the measure
space (X, Σ, µ) is purely atomic and
bn =
µT −1 (An )
→ 0.
µ(An )
Proof. Suppose CT is compact. Then, using the similar techniques as in [6], it
is easy to show that µ is purely atomic.
Next, we claim that bn → 0. Suppose the contrary. Then there exists some
² > 0 and a subsequence {bnk }k≥1 of the sequence {bn }n≥1 such that bnk ≥ ², for
all k ∈ N.
S∞
Let X = n=1 An , where An ’s are atoms. For each n ∈ N, define
fn =
χ An
kχT −1 (An ) kL(pq)
.
Compact composition operators on Lorentz spaces
111
So, for 1 ≤ q ≤ p < ∞, we have
kfnk kL(pq) =
kχAnk kL(pq)
kχT −1 (Ank ) kL(pq)
µ
=
µ(Ank )
µ(T −1 (Ank ))
¶1/p
p
= 1/b1/p
nk ≤ 1/² ,
for each k ∈ N.
Let gnm = CT fkn −CT fkm , where {fkn }n≥1 is the subsequence of the sequence
{fk }k≥1 . For n 6= m, three cases arise: either µ(T −1 (Akn )) < µ(T −1 (Akm )) or
µ(T −1 (Akn )) = µ(T −1 (Akm )); the third case is similar to the first one, with m and
n interchanging places.
In the first case, we have kχT −1 (Akn ) kL(pq) < kχT −1 (Akm ) kL(pq) and
¯
¯
¾
½
¯ χT −1 (Akn )(x)
χT −1 (Akm )(x) ¯
¯>λ
¯
−
µgnm (λ) = µ x ∈ X : ¯
kχT −1 (Akn ) kL(pq)
kχT −1 (Akm ) kL(pq) ¯

µT −1 (Akm ) + µT −1 (Akn ), if 0 < λ < kχ −1 1 kL(pq) ,

T
(Ak )

m


1
−1
µT
(A
),
if
≤
λ < kχ −1 1 kL(pq) ,
k
n
=
kχT −1 (A
kL(pq)
T
(Ak )
km )
n



1

0,
if λ ≥ kχ −1
.
kL(pq)
T
(Ak )
n
So, we have
∗
gnm
(t) = inf{λ > 0 : µgnm (λ) ≤ t}

1
, if 0 ≤ t < µT −1 (Akn ),

kχ −1
k

 T (Akn ) L(pq)
1
−1
=
(Akn ) ≤ t < µT −1 (Akn ) + µT −1 (Akm ),
kχT −1 (A
kL(pq) , if µT

km )


0,
if t ≥ µT −1 (Akm ) + µT −1 (Akn ).
Thus,
kgnm kqL(pq)
Ã
=1+
¢q/p ¡ −1
¢q/p !
(µT −1 (Akm ) + µT −1 (Akn )
− µT (Akn )
> 1,
(µT −1 (Akm ))q/p
which contradicts the compactness of CT .
In the second case, 1/c = kχT −1 (Akn ) kL(pq) = kχT −1 (Akm ) kL(pq) , we have
kgnm kL(pq) = c( p/q )1/q ( 2 µT −1 (Akn ) )1/p = 21/p > 0
which contradicts the compactness of CT . Hence bn → 0.
Conversely, let (X, Σ, µ) be atomic with atoms An and bn → 0. Note that
P
(N )
(N )
fPand
f (An )χAn are equal µ-a.e. For each N ∈ N, define CT by CT f =
n≤N f (An )χT −1 (An ) . Then for each λ > 0, we have
X
µ(CT −C (N ) )f (λ) ≤
µ(T −1 (An ))
T
n>N,|f (An )|>λ
≤ ( sup bn )
n>N
X
|f (An )|>λ
µ(An ) = ( sup bn )µf (λ).
n>N
112
Therefore
R. Kumar, R. Kumar
(N )
kCT − CT kL(pq)7→L(pq) ≤ ( sup bn )1/p → 0
n>N
(N )
as N → ∞. Since CT is the limit of finite rank operators CT , it is compact.
Acknowledgements. The authors are grateful to the referee for the valuable
suggestions and comments.
REFERENCES
[1] C. Bennett and R. Sharpley, Interpolation of Operators, Pure and Applied Mathematics 129,
Academic Press, London 1988.
[2] Y. Cui, H. Hudzik, R. Kumar and L. Maligranda, Composition operators in Orlicz spaces, J.
Austral. Math. Soc. 76, no.2, (2004), 189–206.
[3] J. Lindenstrauss and L. Tzafriri, Classical Banach spaces II. Function Spaces, Springer Verlag,
Berlin-New York 1979.
[4] S. Petrović, A note on composition operators, Mat. Vestnik, 40 (1988), 147–151.
[5] R. K. Singh, Compact and quasinormal composition operators, Proc. Amer. Math. Soc., 45
(1974), 80–82.
[6] R. K. Singh and A. Kumar, Compact composition operators, J. Austral. Math. Soc., 28 (1979),
309–314.
[7] R. K. Singh and J. S. Manhas, Composition Operators on Function Spaces, North Holland
Math. Studies 179, Amsterdam 1993.
[8] X. M. Xu, Compact composition operators on Lp (X, Σ, µ), Adv. in Math. (China), 20 (1991),
221–225 (in Chinese).
(received 06.08.2004, in revised form 17.08.2005)
Rajeev Kumar, Department of Mathematics, University of Jammu, Jammu-180 006, INDIA.
E-mail: raj1k2@yahoo.co.in
Romesh Kumar, Department of Mathematics, University of Jammu, Jammu-180 006, INDIA.
E-mail: romesh jammu@yahoo.com