Gupta, Dharmendra K.

Komal, B. S.

English

Komal B. S.

Gupta Dharmendra K.

University of Szeged

Acta Sei. Math. 47 (1984), 3^1—403 Normal composition operators B. S. KOMAL and D! K. GUPTA 1. Preliminaries Let (X', Of, X) be a (7-finite measure space and let T be a measurable non-singular (XT~1(E)=0 whenever X(E) = 0) transformation from X into itself. Then the equation CTf=foTl for every fZLp(X) defines a composition transformation CT from LP(X) into the space of all complex valued functions on X. If the range of CT is contained in LP(X) and CT turns out to be a bounded operator on LP(X), then we call it a composition operator induced by T. Let X=N, the set of all (non-zero) positive integers and Sf=P(N), the power set of N. Suppose w= {w„} is a sequence of (non-zero) positive real numbers. Define a measure X on P(N) by ' 2 w n for every ^ ^ ( i V ) . Then for p=2, L"(X) is a Hilbert space with the inner product defined on it by </ .*>= l v v „ / ( n ) ^ j fo-r every / , g€Xp(A). n = l This space is called a weighted sequence space with weights {w„: n£N} and is de-noted by The symbol B(PW) denotes the Banach algebra of all bounded linear operators on Pw and the symbol / 0 denotes the Radon—Nikodym derivative of the measure XT'1 with respect to the measure X. Received November 30, 1982. 446 B. S. Komal and D . K . Gupta 2. Normal composition operators A bounded linear operator A on a Hilbert space is called normal if it commutes with A*. The operator A is called quasinormal if it commutes with A*A. In [7] SINGH, KUMAR and GUPTA made the study of these operators on a weighted se-oo • quence space Pw when 2 WHITLY [8] has studied the seoperators on L2(X), n = 1 when the underlying space is a finite measure space. He has proved that the class of unitary composition operators coincides with the class of normal composition operators. In our present note we have generalised this result to £2(A), when the underlying measure space is a special type of c-finite measure space. Some results on quasinormal, isometric and co-isometric composition operators are also reported. We shall first give an example to show that a normal composition operator may not be unitary. Example 2.1. Let T: N—N be the mapping defined by 2 if 71 = 1, i •• T(n) = h + 2 if n is an even integer, n — 2 if n is an odd integer >1. Let the sequence {w„} of weights be defined by w„ = 1 if n = l, 1/2" if n is an ever integer, 2"_1 if n is an odd integer >1. Then / 0 (n)=4 for every n£N. Hence in view of Theorem 1 of [5] CT is a bounded operator. Clearly fo(n)=fo{T(n)) for every n£N. Since Tis an injection, T~1{£f) = =Sf. Hence by Lemma 2 of [8] CT is normal. Since CjCT=Mfo=4i, it is clear that CT is not unitary. If the sequence {vv„} is a convergent sequence of positive real numbers, then every normal operator becomes unitary. It is given in the following theorem. We shall first give a definition. Def in i t ion . Let T: N-»N be a mapping. Then two integers m and n are said to be in the same orbit of T if each can be reached from the other by composing T and T~x (T - 1 means a multivalued function) sufficiently many times. Theorem 2.2. Let CT^B(PW) and let w = {w„} be a convergent sequence of positive real numbers. Then the following are equivalent: (i) CT is unitary, (ii) CT is normal. \ Normal composition operators 447 Proof . The implication (i)=>(ii) is true for any bounded operator A. We show that (ii)=»(i). Assume that CT is normal. Then by Lemma 2 of WHITLEY [8], /<,=/<> o T and T-r{¥)=Sf. From Lemma 1 of WHITLEY [8], CT has dense range and hence CT is onto in view of the normality of CT. Thus by Corollary 2.3 and .Corollary 2.5 of SINGH and KUMAR [6] T is invertible. Let n£T~\{n}). Then' Mnt)=fQ(T(n,))= =fo(n). Similarly, we can show that / 0 is constant on the orbit of.«. Further let n0£N. Then O(n0), the orbit of n0 is either a finite set or it is an infinite set. We first, suppose that £?(«„) is a finite set. Then there is an integer m in N such that r > 0 ) = « 0 . Now fo(T(n))=f0(T2(n0j)= ...=fQ(Tm(n0)). Equivalently, . . where Tk(n0) = nk for k^m, and «„,=«<,.• From this we get w„k=wn/pk for ksm and hence fim=l which implies that p=l. Next, if O(n0) is an infinite" subset of N, then Tk(n0)^n0 for every k£N. Let (Tk)~\n0)=n^k. Then,/„ is constant on the set {nk: k£Z}, where Z is the set of all integers; Thus as shown in the first case w „ k = w J P k (i) and w„_k=Pkw„0 (")• If P ^ h then at least one of the subsequences (i) and (ii) is divergent. This contradicts the fact that every sub-sequence of a convergent sequence is convergent. Hence p = 1. Thus /„(«)= 1 for every n£N. This implies that CT is an isometry by [3]. Since CT has dense range, we can conclude that CT is unitary. T h e o r e m 2.3. Let CT^B(PW). Then C*T is an isometry if and only if w=woT and T is an injection. Proof . Suppose CT is a co-isometry. Then CTC%e'n=e'n, where e'n=X{n)/w„, !"{„} being the sequence each terms of which is 0, except for the «th one which equals 1. Using the definition of [5], we get CTe'T{n)=e'n. This implies that ^T-H{T(n)})/wT(n)=X{n)/wn. Hence we can conclude that T is an injection and w=woT. Conversely, if w = woTi.e . wn — vvT(n) for every nf_N and T is an injection then CrC*e'n=e'n. Let f£Pw. Then (CTC}fj(.n) = (CTC}f,ed = (f,CTC}e'n) = (f,e'„)=f(n) for every n£N. Hence CTC*Tf=f for every This completes the proof of the theorem. Theo rem 2.4. Let T: N-+N be an injection such that CT£B(PW). Then the following are equivalent: (i) C*T is an isometry, (ii) CT is a partial isometry, (iii) w=woT. 448 B. S. Komal and D. K. Gupta: Normal composition operators Proof , (i) implies (ii): Suppose Cj is an isometry. Then C*T is a partial iso-metry. Hence CT is a partial isometry [1, p. 96]. (ii) implies (iii): If CT is a partial isometry, then from a corollary to Problem 98 of [2] CTC^CT=CT. Since C £ C r = =Mf , this implies that MfaOTCT=CT. Thus f0oT=l on the range of CT. Now T is an injection, therefore by Corollary 2.4 of [6] CT has dense range. Hence ( / 0 oJ ) (« )= l for every n^N. Thus T~1({T(n)})/T(n)=l for every n£N which implies that wn=wTW for every n£N. Hence w=woT. (iii) implies (i): This proof is given in Theorem 2.3. WHITLEY [8] has given an example to show that not every quasinormal compo-sition operator is normal. We show that if T is an injection, then every quasinormal composition operator becomes normal. It is given in the following theorem. Theo rem 2.5. Let T: N—N be an injection such: that CT£B(Pw). Then the following are equivalent: (i) CT is normal, (ii) Cf is quasinormal, (iii) CT is an isometry. Proof . (i)=Kii) is trivial; (ii)=>(iii) follows from Theorem 2 of [8]. (iii)=>(i): By the assumption of the theorem, Corollary 2.4 of [6] guarantees that CT has dense range. References [1] R. G . DOUGLAS, Banach Algebra Techniques in Operator Theory, Academic Press Inc. (New York—London, 1972). [2] P. R. HALMOS, A Hilbert space problem book, Van Nostrand (Princeton, N. J., 1967). [3] R. K. SINGH, Normal and Hermitian composition operators, Proc. Amer. Math. Soc., 47 (1975), 348—350. [4] R. K. SINGH, Composition operators induced by rational functions, Proc. Amer. Math. Soc., 59 (1976), 329—333. [5] R. K. SINGH and D. K. GUPTA, Normal and invertible composition operators, Bull. Austral. Math. Soc., 18 (1978), 45—54. . " [6] R. K . SINGH and A. KUMAR, Characterization of invertible, unitary and normal-composition operators, Bull. Austral. Math. Soc., 19 (1978), 81—95. [7] R. K . SINGH, A. KUMAR and D. K . GUPTA, Quasinormal composition operators, Indian J. Pure Appl. Math., 11 (1980), 904—907. [8] R. WHITLEY, Normal and quasinormal composition operators, Proc. Amer. Math. Soc., 70 (1978), 114—118. DEPARTMENT OF MATHEMATICS UNIVERSITY OF JAMMU JAMMU-180001, INDIA 

Normal composition operators

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Normal composition operators

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