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).
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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 < ∞, sup0λ ¯ − 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)
à =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. Petrovi´ c, 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:
[email protected] Romesh Kumar, Department of Mathematics, University of Jammu, Jammu-180 006, INDIA. E-mail: romesh
[email protected]
