Hoover, Thomas B.

Lambert, Alan

English

Hoover Thomas B.

Lambert Alan

University of Szeged

Acta Sci. Math.. 55 (1991), 403—402. Essentially normal composition operators on L2 THOMAS HOOVER and ALAN LAMBERT 1. Preliminaries. Let (X, I , m) be a complete, sigma-finite measure space and let T be a Z-measurable mapping (T*1! c i ) of X into X. The composition operator C induced by T on the set of complex valued, measurable functions on X is defined by Cf=foT. Throughout this article L2=L2(X, I , m). For S£l, L2(S) is the L2 space of functions on S, with the appropriate restrictions of I and m. We will regard this space as the subspace of L2 consisting of those functions with support in S. In general the support of the function / will be denoted S f . F o r / i n LT, Mf will denote the operator of multiplication by / on L2. We will be concerned with those composition operators C which are bounded linear operators on L2. A detailed description of the general properties of such operators is given in [3]. In particular, it is shown that C is a bounded operator on L2 if and only if (i) m o f - 1 is absolutely continuous with respect to m, and dmoT'1 (ii) — dm Conditions (i) and (ii) are assumed to hold throughout. We set dmoT~x h = dm We will make use of the following notation. For / in L2 or measurable and non-negative, E ( f ) is the conditional expectation E{f\T~lI). For f£L2, E ( f ) is the orthogonal projection of / onto L2(X,T~iI,m). Verifications of the following properties are found in [1], [2], and [5]. (iii) | | C f = | | A | L . (iv) For each / there is a function F such that E(f)=FoT. If E(f)=GoT as well, then F=G on Sh. In particular the function h • [ F ( / ) ] o T - 1 is well defined even if F i s not invertible. In fact, C*f=h-[E(f)]oT-\ C*Cf=hf, and CC*f= =hoTE(f). Received February 10, 1989. 404 Thomas Hoover and Alan Lambert (v) For measurable / and g, E((foT) g)=(f°T)Eg. For f£L°° this equa-tion has the operator theoretic form MfoTE=EMfoT. 2 . Essential normality. In [ 5 ] R . W H I T L E Y proved that C is normal if and only if T is invertible and bi-measurable, and h=hoT. Recall that an operator A is essentially normal if its image in the Calkin algebra is a normal element. Equiva-lent^ A is essentially normal if and only if A* A —A A* is compact. R. K. S I N G H and T . V E L U C H A M Y ( [ 4 ] ) have examined the question of essential normality for certain •composition operators. Their result in this regard is stated below. T h e o r e m . If (X, I , m) is completely nonatomic, and if C is essentially normal with dense range, then C is normal. In this article we will develop characterizations of essentially normal composi-tion operators. It will be shown that the dense range hypothesis in the above result is unnecessary. We first note that in the atomic case it is possible to have a non-normal, •essentially normal composition operator. 2.1. E x a m p l e . Let ^ = N = { 1 , 2, ...} and let m be the counting measure. Set T ( l ) = l and T(n +1)=«. Then C is a rank one perturbation of the unilateral •shift. In particular, it is an essentially normal operator with index — 1, and so is not normal. For convenience, let D=C*C—CC*—Mh—MhoTE. We will examine D with Tespect to the orthogonal decomposition of L2 as EL2®(i—E)L2. We note that EL2 consists of those L2 functions which are T_1I measurable. The range of C is dense in EL2 ([1]). Also, ( I — E ) L 2 consists of those L2 functions / for which J' f dm=0 for every I-set A. 2.2. L e m m a . D is compact if and only if both Mh(l—E) and Mh_hoTE are compact. P r o o f . D is compact if and only if both DE and D(I—E) are compact. But D=Mh-MhoTE, so DE = (Mh-Mh0TE)E = Mh.hoTE, and D(I—E) = Mh(I—E). 2.3. C o r o l l a r y . Suppose that D is compact. Then Mh,^h_haT) is compact. P r o o f . Mh(l—E) and Mh_hoTE are compact. But Mh(Mh-h»TEr + (Mh(I-E))Mh-hoT = MhEMh..hoT + Mh(I-E)Mh-koT = = MhMh-hoT — MH.(H-HOT)-Essentially normal composition operators on L- 405 Write X— Xc U {a,: / £ / ) where m is completely nonatomic on Xc and {«¡1 i£J) consists of the atoms for m. Let A = T~lXc and Ai=T~1ai, i f j . These sets are pairwise disjoint, so that the corresponding subspaces of L2 are orthogonal. Note that for any measurable set S, L2(T~1S) is a reducing subspace for D, because if S f C T ^ S , then hf-hoTEf = hf-hoTE(fxT-'s) = hf-hoT(Ef)XT-*s = 0 off T~lS. We have established the following result. 2.4. T h e o r e m . C is essentially normal if and only if D\L^iA) and D\Li(A^(i^J) are compact, and lim ||Z>k*Ui)|| = 0. J — C O This result is strengthened somewhat by Lemma 2.6 below. Its proof depends on. the following fact. 2.5. L e m m a . If S is a subset of Xc with 0 < m ( S ) < then there is a subset A of S with jm(S) - m(A) < jm(S). P r o o f . Suppose no such set A exists. Then for every measurable subset E of S, either m(E)<^m(S) or m(E)>^m(S). Let m(E)>^m(S)|. If E and F are in S, then m(Ef]F) = m(E) + m(F)-m(EUF) > ~m(S). Thus £TIF£<?. Let a = i n f {m(E): E££), and let {En} be a decreasing sequence 3 of sets in 8 whose measures converge to a. Let G= f)En. Then m(G)=a^—m(S). 4 Now, there is a measurable subset B of G with 0 <m(B)<m(G). But then neither B nor G—B are in S. It then follows that both B and G—B must have measures 3 1 less than —m(S), which implies that the measure of G is less than —m(S). This contradicts the location of G in S. 2.6. L e m m a . If D\l*(A) compact then it is 0. P r o o f . Assume D0=D\L,iA) is compact. Since D is selfadjoint and reduced by L2(A), D0 is selfadjoint. In particular, if Z>0 is not 0 then it has a nonzero eigen-value r. Let Sr be the corresponding finite dimensional eigenspace, and let cp be any L°° function with S9czXe. Then S-9oT=T~lS9c:A. Now, MipoTL2(A)(zL2(A) 12 406 Thomas Hoover and Alan Lambert and for any / i n L?(X), M9.TDf = (<poT)(hf-hoTEf) = (h) • (<poT) •/— (hoT) • E{(<poT) •/) = DMvoTf. It follows that M<poT leaves <?, invariant. But Sr is finite dimensional and so there is a function /£<?, other than 0, and a scalar X such that (<poT)f—Xf a.e. dm. In parti-cular, q>oT=X on a set of positive measure. This shows that every L°°(XC) func-tion is constant on a set of positive measure. But by definition Xc is completely non-atomic. Let S1 be a set of finite, positive measure in Xc. Via Lemma 2.5 we partition S into two measurable sets, each of measure no more than 3/4 that of S. Define the function fx to take the values 1/2, 1 respectively on the sets. Repeat this procedure by replacing S by each of the sets of constancy of f and defining / 2 to take the value of / i on one part of each of the original two subsets and to be 1/4, 3/4 respectively on the remaining two sets. Continuation of this procedure gives rise to a mono-tonically decreasing sequence of functions whose pointwise limit is bounded and not constant on any set of positive measure in Xc. Indeed, we have for each x, 0 = § / „ ( - Y ) - / „ + 1 ( X ) SI ^ ¿ R , so that Thus, for any 0 and any positive integer n, {x: f ( x ) = r}c {x: r S/„(x) ^ r + But this latter set contains at most two sets of constancy for fn, so m |x: r s / ^ g r + i ) ^ . It then follows that f ^ r a.e. dm. This contradiction completes the proof of the lemma. Note that the result of Singh and Veluchamy as stated in Section 1 of this paper follows as a special case of Lemma 2.6, for in the completely nonatomic case A=X. But then D=0, i.e. C is normal. It is interesting to see that one basic prop-erty from Whitley's characterization of normality carries over to the general essen-tially normal setting. 2.7. C o r o l l a r y . If C is essentially normal then h=hoT a.e. on T~1XC. Essentially normal composition operators on L- 407 P r o o f . Assume that C is essentially normal. Then } = 0 . Let Y be a subset of Xc of finite measure. Since h£L°° we have m(T~1Y)= J hdm^oo y and in particular ^ - l y f Z.2(T - 1 A^). But then we see that 0 = D / j - j y = h • Xr - ' y — (H°T) • E(XT-'Y) - (h-hoT)xT->R-It then follows that h=hoT ale. on Xc. We will conclude this paper with an example establishing the existence of an essentially normal composition operator for which /i > 0 a.e. and for which there is an atom a with T_1a infinite. First we investigate the structure of the sets T^a,, when C is essentially normal. Let a be an atom for m and let B=T~1a. Then D\L2(B) is compact. Let f£L?(B). Since m is sigma-finite and h is essentially bound-ed, B is a set of finite measure. Noting that Ef is constant on B=T~1a, we see that Jfdm — f fdm = f h - (EfioT^dm = m(a)h(a)(Ef)oT~\a) = T-'A miT^a) = m { a ) m{d) { E f ) ° T ~ y { a ) = It then follows that Ef=—-— ( f d m on B. J m(B) B m(B) Also, for x in B, hoT(x)=h(a)= - . In particular (MhoTE)\,i(B) is the m(a) rank one operator /— (fdm. But then the compactness of D\L2(m implies m{a) g Mh\LHB) is compact. This in turn shows that BDSb = {bk\ kiK} where each bk is an atom. 2.8. Example . Let O be the origin in the plane and let X={0}U(NXN). Define m by m(0)= 1; m(i,j)=l/2iJ. Finally, define T on X by T(0) = O ; T(i, 1) = O ; T(i,j) = (i,j-1) for j > 1. Then T-\0) = {0} U (Nx{l}), and i(0) while ur - w(r"1('">/)) _ m(i,j+1) _ 2 - " + 1 ) _ nKhJ} m(ij) m(i,j) 2-'J — I 12* 408 Thomas Hoover and AÎan Lambert : Essentially normal composition operators on L1 F o r / suppor ted on T~lO, Df=hf^2 f f dm. Since l im h(n, 1 ) = 0 , D\LHt^0) T-'O is compac t . On the o ther hand , (Df)(i,j+1) = 2~ ' / ( / , y + 1 ) — 2 _ ' / ( / , j + l ) = 0. T h u s C is essentially n o r m a l . References [1] J . CAMPBELL and J . JAMISON, On some classes of weighted composition operators, Glasgow Math. J., to appear. [2]' A. LAMBERT, Hyponormal composition operators, Bull. London Math. Soc., 18 ( 1986) , 3 9 5 — 400 . [3] E. NORDGREN, Composition operators on Hilbert space, in: Hilbert Space Operators, Lecture Notes in Math., vol. 693 , Springer-Verlag (Berlin and New York, 1978) , pp. 3 7 — 6 3 . [4 ]R . SINGH and T . VELUCHAMY, Non atomic measure spaces and Fredholm composition operators, Acta Sci. Math., 51 (1987) , 4 6 1 — 4 6 5 . [5] R. WHITLEY, Normal and quasinormal composition operators, Proc. Amer. Math. Soc., 76 (1978) , 114—118 . <T.H.) DEPARTMENT OF MATHEMATICS UNIVERSITY O F HAW AH HONOLULU, HAW AU 96822 U.SA. (A.L.) DEPARTMENT O F MATHEMATICS UNIVERSITY OF NORTH CAROLINA AT CHARLOTTE CHARLOTTE, NORTH CAROLINA 28223 U.S.A. 

Essentially normal composition operators on L2

http://acta.bibl.u-szeged.hu/15321/1/math_055_fasc_003_004_403-408.pdf

Essentially normal composition operators on L2

Abstract

Similar works

Full text

Available Versions

University of Szeged