Chandra Kumar, R. David

Singh, Raj Kishor

English

Singh Raj Kishor

Chandra Kumar R. David

University of Szeged

Acta Sci. Math., 49 (1985), 339—344 Compact weighted composition operators on L2(X) R. K. SINGH and R. DAVID CHANDRA KUMAR 1. Introduction. Let (X, IF, X) be a sigma-finite measure space and let T: X-~X be a non-singular measurable transformation such that the composit ion transforma-tion CT defined as CTf=foT is bounded linear operator on L2(?.). If 6£L°°(X), then the multiplication operator M0 defined as MBf=Q • f is a bounded linear operator on L\X). The product CTMe is an operator on L\X) and we call it a weighted composit ion operator on L\X). This class of operators includes some o f the well known operators such as multiplication operators, weighted shifts and composit ion operators [1]. In this note we are interested in studying compact weighted composit ion opera-tors on L2(X). By B(H) we denote the C*-algebra o f all bounded linear operators on a Hilbert space H. If CT is a composi t ion operator on L~{)), then / „ denotes the R a d o n — N i k o d y m derivative o f the measure XT-1 with respect to the measure • X. For any complex valued function on X, Zg~{x : 0 ( x ) = O } and Z'g is the complement o f Ze. 2. Some general results. It has been proved in [4] that if CT£B(L?(X)), then CyCT=Mfa, where / „ is the R a d o n — N i k o d y m derivative o f the measure XT-1 with respect to the measure X. A l so it has been proved in [6] that / 0 o 7 V O (a.e ). By using these results w e prove the fo l lowing theorem. T h e o r e m 2.1. Let CT£B(L2(X)). Then CT has dense range if and only if CTC£=Mj-gOT. P r o o f . Suppose CT has dense range and let f£L2(X). Then there exists a sequence { / „ } in L?(X) such that {CT/„} converges to / . N o w ? C r C i / = lim (CTC*TCTf) - l i m ( C r M / o / „ ) = Mfa0T(\\m Crfn) = MUoTf Hence CTC£=M/oOT. Received September 9, 1983. 22» 340 R. K. Singh and R. David Chandra Kumar Conversely, suppose CTC^=M/oOT. Since f0oT^O (a.e.), M/<>oT and hence CTCj is an injection. This implies that ker Cy = ker CTCj = {0} and hence CT has dense range. C o r o l l a r y 2.2. Let CT£B(12). Then CTC^=MfoOT if and only if CT is onto. P r o o f . Since the range o f CT is always closed in I2, the result fo l lows imme-diately. T h e o r e m 2.3. Let CTMe£B(L2(X)). Then (CrMg)*CTM9=MWfo. P r o o f . If CTMg£B(L2(X)), then (CTMg)*CTMe = MgCrCTMe = M|9|s/0. C o r o l l a r y 2.4. Let 0€L°°(A) be such that Z 9 c ( r a n T)', the complement of the range of T. Then (CTMe)(CT M e ) * ! / o ) 0 T if and only if CTMB has dense range. P r o o f . Suppose ( C T M a ) ( C r M e ) * = M ( | 9 | ! / o ) o T . Since Z f l c ( r a n T)', 0 o 7 V O (a.e.) and hence ( | 0 | 2 / o ) o r ^ O (a.e.). This implies that ker ( C T M „ ) * = { 0 } and hence CTMe has dense range. The converse o f this theorem fol lows from Theorem 2.1. T h e o r e m 2.5. Let CT, Me£B(L2(X)). Then (i) CTMe=MeoTCT, (ii) M „ C T = 0 if and only if 0 = 0 (a.e.), (iii) CTMe=0 if and only if 0 o T = O (a.e.), and (iv) CTMe—MeCT if and only if 6=8 oT (a.e.). P r o o f , (i) Let / €L- (A) . Then CTMef=(0oT)(foT)=M6oTCTf H e n c e CTMe=M6oTCT. (ii) Suppose MeCT=0. Then M„CTf= 0 for e v e r y / i n L\X). Since (X, i f . A) is a sigma-finite measure space, there exists an / € L2(X) such that / ^ 0 (a.e.). Hence 6-(foT)=0 (a.e.) implies that 0 = 0 (a.e.). The converse is obvious. (iii) Since CTMe=M$aTCT, the proof fo l lows from (ii). (iv) The sufficiency o f this result is obvious. T o prove the necessary part, sup-pose CTMe=MeCT. Then M0aTCT=MeCT and hence M f l o T _ e C x = 0 . T h u s the result fo l lows from (ii). The fo l lowing examples illustrate that there are CT and MB in B(l2), such that CT commutes wi th MB. Here / 2 denotes the Hilbert space o f square summable sequences o f complex numbers. Compact weighted composition operators 341 E x a m p l e 2.6. Let X=N, the set o f natural numbers and A be the count ing measure on it. Def ine T: X-X by T(n)=1, if n= 1 , 2 and r ( 3 n + m ) = n + 2 , if m = 0 , 1 , 2 and n£N. Then CT€B(l2). Def ine 9: X-C by 0 ( « ) = 2 , if « = 1 , 2 and 0 ( n ) = 3 , if n > 2 . Then Mg£B(l2) and C T commutes wi th M e . E x a m p l e 2.7. If 0 £ L ~ ( A ) is a constant, then Mg commutes with every C r €f l (L 2 (A) ) . E x a m p l e 2.8. Let CTdB{L\Xj) be such that T(E)=E for some EdSf and 0 < A ( £ ) < o ° - Define 6=XE, the characteristic function o f E. Then CT commutes with Me. 3. Compact weighted composition operators. Let (X, ¿7, A) be a sigma-finite measure space. A n element E d i ? is said to be an atom if for every non-null measurable subset F o f E, either A ( F ) = 0 or A ( F ) = A ( £ ) . A measure space (X, y , A) is said to be atomic if every element o f if contains an atom. A measure space (X, y , A) is said to be non-atomic if it does not contain any atom. It has been proved in [4], that n o compos i t ion operator on L2 of a non-atomic measure space is compact. It is interesting to note that the weighted composit ion operator on L2(A) is compact if and only if it is the zero operator. This is evident from the fol lowing theorem. T h e o r e m 3.1. The weighted composition operator CTMe on L2 of a non-atomic measure space is compact if and only if 0 = 0 (a.e.) on Zfj . P r o o f . Suppose CTMe is compact . Then C£CTMe and hence Mgf is com-pact. By a theorem of [5], 0 / o = 0 (a.e.). If 0 ^ 0 (a.e.) on Z'f<¡, then / „ = 0 (a.e.). This implies that CT=0. But n o non-singular measurable transformation induces the zero operator. Hence 0 = 0 (a.e.) on Z'f¡¡. The converse is obvious. C o r o l l a r y 3.2. The weighted composition operator CTMe on L2(X) is compact if and only if it is the zero operator. P r o o f . Suppose CTM9 is compact . Then (CTM0)*CTMe and hence M^lft¡ is the zero operator. Hence CTMe is the zero operator. C o r o l l a r y 3.3. No composition operator on L2(A) is compact. Let 0€ L°° (A). We denote X¡ = {xeX: 0(x)><5} and M¡ = { / € ¿ 2 ( A ) : f ( x ) = 0 on X- X%}. It has been proved in [5] that the multiplication operator Me on L2(A) is compact if and only if M l is finite dimensional. W e shall characterize compact weighted composit ion operators on L2 o f an atomic measure space. Since (X,£f,k) is a 342 R. K. Singh and R. David Chandra Kumar s igma. finite measure space, without loss of generality we write X as a countable union o f a toms and we denote the fth atom by i. • • . . • - : T h e o r e m 3.4. Let CTMe£B(L2(X)). Then CTM9 is compact if and only if either { / 0 ( 0 } or {0(z')} converges to zero. P r o o f . Suppose CrMe is compact. Then Mei is compact and hence is finite dimensional. This shows that X$ f<> contains finite number o f atoms. It fo l lows from this that the sequence {0/iCO} converges to zero. Since 0 and f 0 are essentially bounded functions, either { / 0 ( 0 } or {0( / ) } converges to zero. This completes the necessary part o f the theorem. Conversely, suppose either { / 0 ( 0 } or ( 0 ( / ) } converges to zero. Then either CT or M„ is compact . Hence CTMB is compact . It fo l lows from this theorem that there are plenty of compact weighted composi t ion operators on L 2 o f an atomic measure space as is shown in the fo l lowing example. . E x a m p l e 3:5. Let X=N and ! ( « ) = « " , 0 < a < 1. Then I2 denotes the weighted sequence space. Def ine T: X-+X by T(n) = n +1, i f « is odd and T(n)=n — 1, if n is even. Then CT£B(l2) and / 0 ( « ) = a " _ 1 ( l +a). Hence CT is compact . If Me is any multiplication operator o n I2, then CTMe is always compact . T h e fo l lowing theorem characterizes compact weighted composi t ion operators on I2, the Hilbert space of square summable sequences o f complex numbers on N, the set of natural numbers. . . • T h e o r e m 3.6. Let CTMedB(l2). Then CTMB is compact if and only if {0(n)} converges to zero. P r o o f . Suppose CTMe is compact . Then is finite dimensional and hence N? f° contains finite number o f elements o f N. If N$ contains infinite number o f elements of N, then N{" must contain only finite number of elements o f N. Th i s shows that / „ = 0 for all but finitely many elements o f TV and hence the range o f T contains finitely many elements of N. By taking T(N)=E, we have XT~\E)^ ^ MA(E) for any finite 0. Hence by Theorem 1 o f [3], CT is n o t bounded. This proves that iV® contains finitely many elements o f N. Hence {6 («)} converges to zero. Conversely, if {6{n)} converges to zero, then Me is compact, and hence CTMe is compact . C o r o l l a r y 3.7. No composition operator on I- is compact. P r o o f . The proof fol lows from Theorem 3.6, when 9(x)= I. : Compact weighted composition operators 343 Theorem 3.6 implies that the necessary condit ion for a weighted compos i t ion operator CTMe on I2 to be compact is that 0 is not bounded away from zero. But this condit ion is not sufficient as is shown in the fol lowing example. E x a m p l e 3.8. Let X=N and let C r £ S ( / 2 ) . Define 0 : A T - C by 0 ( 1 ) = O and 0 ( r i ) = 1, if 2. Hence 0 is not bounded away from zero, but CTMg is n o t compact . D e f i n i t i o n : A subalgebra si of B(H) is said to be transitive if si is weakly closed, contains the identity operator and Lat si—{0, H}, where Lat — = Q Lat A. It has been proved in [2] that if si is a transitive algebra o f B(H) AZst containing a compact operator, then si=B(H). Let {»vj be a bounded sequence o f non-zero complex numbers and let {en} be an orthonormal basis of H. The operator W on H defined by the requirements We0=0 and fVe„=w„e„-1 (n=l, 2, . . . ) is called a weighted unilateral (backward) shift wi th the weight sequence {vv„}. C o r o l l a r y 3.9. The weighted shift W on 12 is compact if and only if the sequence of weights {H>„} converges to zero. C o r o l l a r y 3.10. If si is a transitive algebra of B{P) containing a weighted composition operator CTM9 such that 9(n)—0 as n-*-then si=B(l2). The fol lowing result of YADAV and CHATTARJEE [7] fo l lows immediately from Theorem 3.6. C o r o l l a r y 3.11. If si is a transitive algebra of B(l2) containing a weighted shift with weights {»'„} such that <*(«)= J " H- fc+2 . . . iv t + „/u ' 2 iv 3 . . . wn T=O tends to zero as n —°o (for nS2), then si =B(l2). P r o o f . Since the sequence {<5(«)} converges to zero, the corresponding sequence of weights {w„} converges to zero. Hence the weighted shift is compact . T h u s the result fo l lows (cf. [2]). References [1] E. A. NORDGREN, Composition operators on Hilbert space, in: Proc. of Long Beach Con-ference, Lecture Notes in Math. 693, Springer-Verlag (Berlin—Heidelberg—New York, 1978), pp. 37—63. [2] H. RADJAVI and P. ROSENTHAL, Invariant subspaces, Springer-Verlag (Berlin—Heidelberg—New York, 1973). 344 R. K. Singh and R. David Chandra Kumar: Compact weighted composition operators [3] R. K. SINGH, Composition operator induced by rational function, Proc. Amer. Math. Soc., 59 (1976), 329—333. [4] R. K. SINGH, Compact and quasi-normal composition operators, Proc. Amer. Math. Soc., 49 (1974), 82—85. [5] R. K. SINGH and ASHOK KUMAR, Compact composition operators, J. Austral. Math. Soc., 28 (1979), 309—314. [6] R. K. SINGH and T. VELUCHAMY, Non-atomic measure spaces and composition operators, pre-print. [7] B. S. YADAV and S. CHATTARJEE, On a partial solution of the transitive algebra problem, Acta Sci. Math., 4 2 (198Ô), 211—215. DEPARTMENT OF MATHEMATICS UNIVERSITY OF JAMMU NEW CAMPUS JAMMU — 180001, INDIA 

Compact weighted composition operators on L2(λ)

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Compact weighted composition operators on L2(λ)

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