Compact differences of composition operators
aa r X i v : . [ m a t h . F A ] J un COMPACT DIFFERENCES OF COMPOSITION OPERATORS
KATHERINE HELLER, BARBARA D. MACCLUER, AND RACHEL J. WEIR
Abstract.
When ϕ and ψ are linear-fractional self-maps of the unit ball B N in C N , N ≥
1, we show that the difference C ϕ − C ψ cannot be non-triviallycompact on either the Hardy space H ( B N ) or any weighted Bergman space A α ( B N ). Our arguments emphasize geometrical properties of the inducingmaps ϕ and ψ . Introduction
For a domain Ω in C N , where N ≥
1, and an analytic map ϕ : Ω → Ω, we definethe composition operator C ϕ by C ϕ ( f ) = f ◦ ϕ , where f is analytic in Ω. In thecase that Ω = D , the unit disk in C , every composition operator acts boundedly onthe Hardy space H ( D ) = (cid:26) f analytic in D : k f k H ≡ lim r → − Z π | f ( re iθ ) | dθ π < ∞ (cid:27) and weighted Bergman spaces A α ( D ) = (cid:26) f analytic in D : k f k α ≡ ( α + 1) Z D | f ( z ) | (1 − | z | ) α dA ( z ) < ∞ (cid:27) , where α > − dA is normalized area measure. If D is replaced by the unitball B N in C N , N >
1, it is no longer the case that every composition operatoris bounded on the Hardy or weighted Bergman space of the ball (these spaces aredefined in Section 3). However for a large class of maps ϕ , including the rich classof linear-fractional maps, boundedness does continue to hold.Many properties of composition operators have been studied over the past fourdecades; the monographs [4] and [14] give an overview of the work before themid-1990’s. Recently there has been considerable interest in studying algebrasof composition operators, often modulo the ideal of compact operators (see, forexample, [5], [6], [8], [9], [10]). In this direction, the question of when a difference C ϕ − C ψ is compact naturally arises.The main result of this paper shows that if ϕ and ψ are linear-fractional self-maps of B N , then C ϕ − C ψ cannot be non-trivially compact; i.e. if the difference iscompact, either C ϕ and C ψ are individually compact (this happens precisely when k ϕ k ∞ < k ψ k ∞ < ϕ = ψ . While our focus is on the several variablecase, we begin with a simplified proof of this result in one variable. The factthat a difference of linear-fractional composition operators cannot be non-triviallycompact on H ( D ) or A α ( D ) was first obtained by P. Bourdon [2] and J. Moorhouse[12] as a consequence of results on the compactness of a difference of more general Date : June 8, 2010.The third named author would like to thank the Allegheny College Academic Support Com-mittee for funding provided during the development of this paper. composition operators in one variable. Our approach here is self-contained, andtakes a geometric perspective, which will allow us to generalize our arguments toseveral variables. The analogy between the one and several variable arguments isnot perfect, owing to a number of phenomena that are present when
N > N = 1. Nevertheless, our geometric approach when N = 1 leadsus to a tractable way to proceed when N >
1, and highlights the new phenomenawhich must be addressed. Since our arguments are essentially the same for eitherthe Hardy or weighted Bergman spaces, in what follows we will let H denote any ofthese spaces. Our starting point, in either the disk or the ball, will be the followingnecessary condition for compactness of C ϕ − C ψ . Theorem 1. [ [11] , [12] ] Suppose ϕ, ψ are holomorphic self-maps of D (respectively, B N ) and suppose that there exists a sequence of points z n tending to the boundaryof D ( B N ) along which (1) ρ ( ϕ ( z n ) , ψ ( z n )) (cid:18) − | z n | − | ϕ ( z n ) | + 1 − | z n | − | ψ ( z n ) | (cid:19) does not converge to zero, where ρ ( ϕ ( z n ) , ψ ( z n )) is defined by (2) 1 − ( ρ ( ϕ ( z n ) , ψ ( z n ))) = (1 − | ϕ ( z n ) | )(1 − | ψ ( z n ) | ) | − h ϕ ( z n ) , ψ ( z n ) i| . Then C ϕ − C ψ is not compact on H . For z and w in D or B N , the quantity ρ ( z, w ) will be referred to as the pseudo-hyperbolic distance between z and w , so that the first factor in (1) is the pseudo-hyperbolic distance between ϕ ( z n ) and ψ ( z n ). In the disk, the pseudohyperbolicdistance has the simpler expression(3) ρ D ( z, w ) = (cid:12)(cid:12)(cid:12)(cid:12) z − w − zw (cid:12)(cid:12)(cid:12)(cid:12) . Results in one variable
Throughout this section H denotes either the Hardy space H ( D ) or a weightedBergman space A α ( D ), as defined in the previous section. Theorem 2.
Suppose that ϕ and ψ are linear-fractional self-maps of D . If thedifference C ϕ − C ψ is compact on H , then either ϕ = ψ , or both k ϕ k ∞ and k ψ k ∞ are strictly less than , so that C ϕ and C ψ are individually compact. The key step in our proof of Theorem 2 is contained in the following result.
Theorem 3.
Suppose ϕ and ψ are non-automorphism linear-fractional self-mapsof D with ϕ ( ζ ) = ψ ( ζ ) ∈ ∂ D and ϕ ′ ( ζ ) = ψ ′ ( ζ ) for some ζ ∈ ∂ D . If ϕ = ψ then C ϕ − C ψ is not compact on H .Proof. By pre- and post- composing with rotations, we may assume without loss ofgenerality that ζ = 1 and ϕ ( ζ ) = ψ ( ζ ) = 1. Since ϕ and ψ are linear-fractional, wemay also assume without loss of generality that ϕ ( D ) ⊆ ψ ( D ), so that τ ≡ ψ − ◦ ϕ is a well-defined linear-fractional self-map of D . Note that τ (1) = 1 and τ ′ (1) = 1.Since ϕ = ψ , τ is not the identity. Thus τ is conjugate via the Cayley transform C ( z ) = i z − z OMPACT DIFFERENCES OF COMPOSITION OPERATORS 3 to a translation w → w + b of the upper half-plane H = { w : Im w > } for some b = 0 with Im b ≥
0. Moreover, it is easy to see that ψ ◦ τ = τ ◦ ψ for some linear-fractional τ which is also conjugate to a translation in the upperhalf-plane. Specifically, if τ is conjugate to translation by b , then τ is conjugateto translation by c = b/ | ψ ′ (1) | ; see Lemma 5 in [9]. Since b = 0, so also c = 0.For any positive number k , the line { Im w = k } corresponds under the Cayleytransform to E k ≡ { z : | − z | = k (1 − | z | ) } , which is an internally tangent circlein D passing through 1. The radius of this circle is equal to ( k +1) − . By choosing k sufficiently large, this circle will be contained in ψ ( D ) ∪{ } . Fix such a k and choosepoints w n on { Im w = k } with w n → ∞ . The corresponding points v n = C − ( w n )in the disk tend to 1 along the circle E k , and each v n is the image under ψ of some z n belonging to the internally tangent circle ψ − ( E k ) = E k ′ . Notice that z n → n → ∞ .Next we compute the pseudohyperbolic distance between ϕ ( z n ) and ψ ( z n ). Tosimplify the computations, we define the pseudohyperbolic distance in the upperhalf-plane H by ρ H ( u, v ) = ρ D ( C − u, C − v )for u and v in H . Using this definition and (3) it is straightforward to see that ρ H ( u, v ) = (cid:12)(cid:12)(cid:12)(cid:12) u − vu − v (cid:12)(cid:12)(cid:12)(cid:12) . Since ϕ = ψ ◦ τ = τ ◦ ψ , ρ D ( ϕ ( z n ) , ψ ( z n )) = ρ D ( τ ( ψ ( z n )) , ψ ( z n )) = ρ D ( τ ( v n ) , v n )= ρ H ( C ( τ ( v n )) , C ( v n )) = ρ H ( C ( τ ( C − ( w n )) , CC − ( w n ))= ρ H ( w n + c, w n )= (cid:12)(cid:12)(cid:12)(cid:12) c i Im w n + c (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) c ik + c (cid:12)(cid:12)(cid:12)(cid:12) . Thus for all n , the pseudohyperbolic distance between ϕ ( z n ) and ψ ( z n ) is a positiveconstant.Turning to the second factor in Equation (1), we have1 − | z n | − | ψ ( z n ) | = k ′ ( | − z n | ) k ( | − ψ ( z n ) | )by the geometry of the sequence { z n } already noted. Thus since ψ is differentiableat 1 with ψ ′ (1) = 0 and ψ (1) = 1,lim n →∞ − | z n | − | ψ ( z n ) | = k ′ k | ψ ′ (1) | = 0 . Thus we have shown that ρ ( ϕ ( z n ) , ψ ( z n )) (cid:18) − | z n | − | ψ ( z n ) | (cid:19) has a positive limit as n → z n → C ϕ − C ψ is not compact on H . (cid:3) KATHERINE HELLER, BARBARA D. MACCLUER, AND RACHEL J. WEIR
Proof of Theorem 2.
If either k ϕ k ∞ < k ψ k ∞ <
1, then the compactness of thedifference C ϕ − C ψ implies the compactness of each operator individually. Thus wemay assume k ϕ k ∞ = k ψ k ∞ = 1. Suppose ϕ ( ζ ) = η where ζ, η are in ∂ D . Since ϕ and ψ are linear fractional, both ϕ ′ ( ζ ) and ψ ′ ( ζ ) exist and are non-zero. If either ϕ ( ζ ) = ψ ( ζ ) or ϕ ′ ( ζ ) = ψ ′ ( ζ ), then by Theorem 9.16 of [4] the essential norm of C ϕ − C ψ satisfies k C ϕ − C ψ k e ≥ | ϕ ′ ( ζ ) | β for some positive number β depending on the particular choice of H in question;when H = H ( D ) we may take β = 1, and when H = A α ( D ), β = α + 2. Thisgives a positive lower bound on the essential norm of the difference, so that if thedifference is compact we must have ϕ ( ζ ) = ψ ( ζ ) and ϕ ′ ( ζ ) = ψ ′ ( ζ ). Note that thisargument also shows that if C ϕ − C ψ is compact but non-zero, neither ϕ nor ψ canbe an automorphism. An appeal to Theorem 3 finishes the proof. (cid:3) Results in several variables
In this section H will denote either the Hardy space H ( B N ) or a weightedBergman space A α ( B N ), where B N is the ball { ( z , z , · · · , z N ) ∈ C N : N X j =1 | z j | < } in C N , N >
1. These Hilbert spaces are defined by H ( B N ) = { f analytic on B N : k f k ≡ sup
0) = (1 , ′ ). The (linear-fractional) Cayley transform C ( z ) = i e + z − z is a biholomorphic map of the ball B N onto H N ; its inverse is C − ( w ) = (cid:18) w − iw + i , w ′ w + i (cid:19) . If b = ( b , b ′ ) ∈ C N , an H -translation is a map of the form(7) h b ( w , w ′ ) = ( w + 2 i h w ′ , b ′ i + b , w ′ + b ′ ) . If Im b ≥ | b ′ | it maps H N into itself. It is an automorphism of H N if Im b = | b ′ | .The following two facts, which generalize results we used in the previous section,are easily checked: • An H -translation h b maps the set Γ k ≡ { ( w , w ′ ) : Im w − | w ′ | = k } intothe corresponding set { ( w , w ′ ) : Im w − | w ′ | = ˜ k } where ˜ k = k + Im b −| b ′ | . • For any k > C − (Γ k ) = E ( k, e ) ≡ { z ∈ B N : | − z | = 1 k (1 − | z | ) } . The set E ( k, e ) is an internally tangent ellipsoid at e = (1 , ′ ); a computationshows that E ( k, e ) consists of the points ( z , z ′ ) satisfying (cid:12)(cid:12)(cid:12)(cid:12) z − k k (cid:12)(cid:12)(cid:12)(cid:12) + 11 + k | z ′ | = (cid:18)
11 + k (cid:19) . In particular, for t real, points of the form ( t + i ( k + | w ′ | ) , w ′ ) in Γ k pull backunder C − to points on the ellipsoid E ( k, e ). For fixed w ′ , these pull-back pointstend to e as t → ∞ , and for fixed t , they tend to e as | w ′ | → ∞ . KATHERINE HELLER, BARBARA D. MACCLUER, AND RACHEL J. WEIR
Recall that the pseudohyperbolic metric ρ B N ( · , · ) on B N is defined by1 − ρ B N ( z, w ) = (1 − | z | )(1 − | w | ) | − h z, w i| . For points v, u in H N , write ρ H ( v, u ) for ρ B N ( C − v, C − u ); we will call this thepseudohyperbolic metric on H N . Since the pseudohyperbolic metric ρ B N is easilyseen to be automorphism invariant, it follows that for any automorphism Λ of H N , ρ H (Λ v, Λ u ) = ρ H ( v, u ) . In the next theorem, we will use this observation with Λ an automorphic H -translation.By a parabolic linear-fractional map in B N fixing e we mean a linear-fractionalmap τ of B N into B N with τ ( e ) = e and D τ ( e ) = 1, but fixing no other pointin B N . By [1] any parabolic linear-fractional self-map ϕ of B N that fixes e isconjugate to a self-map of H N of the formΦ( w , w ′ ) = ( w + 2 i h w ′ , δ i + b, Aw ′ + γ )(where δ and γ are in C N − , b ∈ C , and certain conditions hold, including | A | ≤ H -translations of Equation (7) are a special case of this family ofmaps. Conjugating Φ by the Cayley transform C we see that the first coordinatefunction of C − Φ C is(8) (2 i − b ) z − h z ′ , δ i + b − bz − h z ′ , δ i + 2 i + b . We will need this explicit formula in the proof of the next result.
Theorem 5.
Suppose ϕ and ψ are parabolic linear-fractional self-maps of the ballfixing e , so that D ϕ ( e ) = 1 = D ψ ( e ) . If ϕ = ψ , then C ϕ − C ψ is not compact on H .Proof. We will show that for distinct maps ϕ and ψ as in the hypothesis, thereexists a sequence of points { z ( n ) } in B N such that(a) z ( n ) → e as n → ∞ .(b) For all n , ρ B N ( ϕ ( z ( n ) ) , ψ ( z ( n ) )) has a strictly positive constant value.(c) (1 − | ϕ ( z ( n ) ) | ) / (1 − | z ( n ) | ) has finite positive limit as n → ∞ .An appeal to Theorem 1 will then complete the proof.We will use the corresponding upper case letters for a self-map of B N conjugatedto H N , so that Φ = CϕC − and Ψ = CψC − . These maps have the formsΦ( w , w ′ ) = ( w + 2 i h w ′ , δ i + b , A w ′ + γ )and Ψ( w , w ′ ) = ( w + 2 i h w ′ , δ i + b , A w ′ + γ )for some δ i and γ i in C N − , scalars b i and ( N − × ( N −
1) matrices A i , i = 1 , { w ( n ) } = { ( w ( n )1 , c ) } in H N , where c is a constant in C N − to be determined, satisfying(i) Im w ( n )1 − | c | = k (ii) C − ( w ( n )1 , c ) → e where k > C − ( w ( n ) ) lie on the ellipsoid E ( k, e ). We can ensure that condition (ii) holds by requiring that Re w ( n )1 → ∞ .Let P ( n )1 = Φ( w ( n ) ) = ( w ( n )1 + 2 i h c, δ i + b , A c + γ )and P ( n )2 = Ψ( w ( n ) ) = ( w ( n )1 + 2 i h c, δ i + b , A c + γ ) . Since the pseudohyperbolic metric is automorphism invariant, we have ρ H ( P ( n )1 , P ( n )2 ) = ρ H ( h ( P ( n )1 ) , h ( P ( n )2 ))where h is the automorphic H -translation given by h ( v , v ′ ) = ( v − Re w ( n )1 , v ′ ) . Thus for any positive integer n , the distance ρ H ( P ( n )1 , P ( n )2 ) is equal to ρ H (( i ( k + | c | ) + 2 i h c, δ i + b , A c + γ ) , ( i ( k + | c | ) + 2 i h c, δ i + b , A c + γ )) . Note that this quantity is independent of the particular point w ( n ) in our sequencechosen to satisfy of (i) and (ii), and that if ϕ = ψ (so that not all of b = b , δ = δ , A = A and γ = γ hold) we may certainly choose c so that this quantity isnot 0. Thus for such a choice, condition (ii) gives the existence of a sequence ofpoints { z ( n ) } in B N tending to e along E ( k, e ) for which ρ B N ( ϕ ( z ( n ) ) , ψ ( z ( n ) ))is a positive constant value; the z ( n ) ’s being just the inverse images under theCayley transform C of our chosen points w ( n ) in H N . Hence conditions (a) and (b)hold.For property (c), first note that the images under Φ of points of the form ( w , c )with Im w − | c | = k look like( w + 2 i h c, δ i + b , A c + γ ) , and for these points we see thatIm ( w + 2 i h c, δ i + b ) − | A c + γ | = k + | c | + Im(2 i h c, δ i + b ) − | A c + γ | , which is constant, say k ′ . In other words, the image under ϕ of our points z ( n ) lieon some ellipsoid E ( k ′ , e ) and | − ϕ ( z ( n ) ) | = 1 k ′ (1 − | ϕ ( z ( n ) ) | ) . Moreover, since the points z ( n ) lie on E ( k, e ), we have | − z ( n )1 | = 1 k (1 − | z ( n ) | ) . Thus(9) 1 − | ϕ ( z ( n ) ) | − | z ( n ) | = k ′ k | − ϕ ( z ( n ) ) | | − z ( n )1 | . KATHERINE HELLER, BARBARA D. MACCLUER, AND RACHEL J. WEIR
Since ϕ is differentiable at e , we have a Taylor series expansion of ϕ in aneighborhood of e : ϕ ( z ) = ϕ ( e ) + D ϕ ( e )( z −
1) + N X j =2 D j ϕ ( e ) z j + 12! D ϕ ( e )( z − + N X j =2 D j ϕ ( e )( z − z j + N X k,j =2; k = j D kj ϕ ( e ) z k z j + 12! N X j =2 D jj ϕ ( e ) z j + · · · . Recall that by hypothesis D ϕ ( e ) = 1. Direct computation using Equation (8)shows that D j ϕ ( e ) = 0 , for j = 2 , ..., N (this also follows more generally fromthe fact that e is a fixed point of ϕ ; see Lemma 6.6 in [4]) and D kj ϕ ( e ) = 0 for k, j = 2 , ..., N . Thus ϕ ( z ) − z −
1) + 12! D ϕ ( e )( z − + 2 N X j =2 D j ϕ ( e )( z − z j + · · · , where the + · · · indicates higher order terms of the form D ν ϕ ( e )( z − e ) ν ν ! , and ν is a multi-index of order at least 3. Since for z ∈ B N we have | z | + · · · + | z N | | − z | ≤ − | z | | − z | ≤ − | z || − z | ≤ − ϕ ( z )1 − z → z → e in B N . By (9) this implies that 1 − | ϕ ( z ( n ) ) | − | z ( n ) | has a positive finite limit as n → ∞ , and property (c) holds as desired. (cid:3) To prove Theorem 4 we will use the preceding result and the following qualitativegeneralization of Theorem 9.16 in [4], specialized to linear-fractional maps. In thestatement we use the notation ψ ζ for the coordinate of ψ in the ζ − direction, thatis ψ ζ ( z ) = h ψ ( z ) , ζ i . Moreover the derivative of ψ ζ in the η direction, denoted D η ψ ζ , is defined by D η ψ ζ ( z ) ≡ h ψ ′ ( z ) η, ζ i . Note that when ζ = η = e this is just D ψ ( z ).For η ∈ ∂B N , write [ η ] for the complex line containing η and 0; that is, theone-dimensional subspace of B N consisting of all points of the form { αη : α ∈ C } .In particular, the complex line [ e ] intersected with B N consists of all points in theball whose last N − Theorem 6.
Suppose ϕ and ψ are linear-fractional self-maps of B N , and suppose ϕ ( ζ ) = ζ for some ζ ∈ ∂B N . If either ψ ( ζ ) = ζ , or ψ ( ζ ) = ζ and D ζ ϕ ζ ( ζ ) = D ζ ψ ζ ( ζ ) , then C ϕ − C ψ is not compact. OMPACT DIFFERENCES OF COMPOSITION OPERATORS 9
Proof.
The argument follows that of Theorem 9.16 in [4]. Without loss of generalitywe may assume ζ = e .First suppose ψ ( e ) = e . If we can find a sequence of points z ( n ) tending to ∂B N , so that (cid:13)(cid:13)(cid:13)(cid:13) ( C ϕ − C ψ ) ∗ (cid:18) K z ( n ) k K z ( n ) k (cid:19)(cid:13)(cid:13)(cid:13)(cid:13) is bounded away from 0, where K z ( n ) denotes the kernel function in H at z ( n ) (seeEquations (4) and (5)), then C ϕ − C ψ is not compact, since the normalized kernelfunctions K z ( n ) / k K z ( n ) k tend weakly to 0 as z ( n ) → ∂B N . Using the fact that forany bounded composition operator C τ we have C ∗ τ ( K z ) = K τ ( z ) , we see that k ( C ϕ − C ψ ) ∗ ( K z ) k = k K ϕ ( z ) k + k K ψ ( z ) k − K ϕ ( z ) ( ψ ( z )) , and thus(10) k ( C ϕ − C ψ ) ∗ ( K z / k K z k ) k ≥ (cid:18) − | z | − | ϕ ( z ) | (cid:19) β − K ϕ ( z ) ( ψ ( z )) k K z k where β = N in H ( B N ) and β = N + 1 + α in A α ( B N ). With our assumptionthat ψ ( e ) = e , it is easy to see that as z → e radially, the second term on theright hand side of Equation (10) tends to 0. By Julia-Caratheodory theory (see forexample, [13], Section 8.5), the first term tends to the positive value ( D ϕ ( e )) − β .This shows that C ϕ − C ψ is not compact if ψ ( e ) = ϕ ( e ).Now suppose ψ ( e ) = ϕ ( e ) = e but D ψ ( e ) = D ϕ ( e ). As before, if wecan find a sequence of points z ( n ) in B N tending to e along which k ( C ϕ − C ψ ) ∗ ( K z ( n ) / k K z ( n ) k ) k is bounded away from 0, then we can conclude that C ϕ − C ψ is not compact. Thesequence { z ( n ) } will chosen so that z ( n ) = ( z ( n )1 , ′ ) where | − z ( n )1 | − | z ( n ) | = | − z ( n )1 | − | z ( n )1 | = M for a fixed and suitably large value of M ; that is, the points z ( n ) will approach e along the boundary of a non-tangential approach region, of large aperture, inthe complex line [ e ]. To analyze the second term on the right hand side of Equa-tion (10), we first consider(11)1 − h ϕ ( z ) , ψ ( z ) i − | z | = 1 − | ϕ ( z ) | − | z | + h ϕ ( z ) − e , ϕ ( z ) − ψ ( z ) i − | z | + h e , ϕ ( z ) − ψ ( z ) i − | z | . The third term on the right hand side of Equation (11) has modulus | ϕ ( z ) − ψ ( z ) | − | z | = (cid:12)(cid:12)(cid:12)(cid:12) − ψ ( z )1 − z − − ϕ ( z )1 − z (cid:12)(cid:12)(cid:12)(cid:12) | − z | − | z | , and if z is chosen to approach e in [ e ] along the curve | − z | / (1 − | z | ) = M thiswill tend to | D ψ ( e ) − D ϕ ( e ) | M . Since D ϕ ( e ) = D ψ ( e ) by assumption,this can be made as large as desired by choosing M large.The first term on the right hand side of Equation (11) tends to | D ϕ ( e ) | alongany sequence of points approaching e non-tangentially in [ e ]. We claim that the second term in (11) tends to 0 along any such sequence. To see this, it’s enough toshow that | ϕ ( z ) − e || ϕ ( z ) − ψ ( z ) | − | z | tends to 0 as z approaches e non-tangentially in [ e ]. Since | ϕ ( z ) − ψ ( z ) | ≤ | ϕ ( z ) − e | + | e − ψ ( z ) | it suffices to show that | ϕ ( z ) − e | − | z | and | ϕ ( z ) − e || ψ ( z ) − e | − | z | both tend to 0. We have | ϕ ( z ) − e | − | z | = | ϕ ( z ) − | + | ϕ ′ ( z ) | | − z | | − z | − | z | where ϕ ′ ( z ) denotes the ( N − ϕ ( z ) , . . . , ϕ N ( z )). We’re considering points z ( n ) = ( z ( n )1 , ′ ) tending to e for which | − z ( n )1 | / (1 −| z ( n )1 | ) is some constant value M . Along such a sequence, | − ϕ ( z ) | / | − z | tends to | D ϕ ( e ) | , so that | ϕ ( z ) − | | − z | | − z | − | z | → . By the Julia-Caratheodory theorem ([13], Theorem 8.5.6), we also have for 2 ≤ k ≤ N , | ϕ k ( z ) | | − z | → e in [ e ]. Since a similar analysisapplies to show that | ψ ( z ) − e | − | z | → | ϕ ( z ) − e || ψ ( z ) − e | − | z | → ϕ ( e ) = ψ ( e ) = e but D ϕ ( e ) = D ψ ( e ), then given ǫ > M < ∞ so that if z ( n ) = ( z ( n )1 , ′ ) ap-proaches e as n → ∞ , where | − z ( n )1 | − | z ( n )1 | = M, then lim sup n →∞ | K ϕ ( z ( n ) ) ( ψ ( z ( n ) )) |k K z ( n ) k < ǫ. By Equation (10) this says C ϕ − C ψ is not compact. (cid:3) OMPACT DIFFERENCES OF COMPOSITION OPERATORS 11
Remark: It’s clear that a version of Theorem 6 holds, with essentially the sameproof, when ϕ and ψ are more general analytic self-maps of B N , if in the statementof the theorem the values of ϕ, ψ, D ζ ϕ ζ and D ζ ψ ζ at ζ are replaced by their radiallimits there. Also a version of the result, with the hypothesis ϕ ( ζ ) = ζ replacedby ϕ ( ζ ) = η for ζ, η ∈ ∂B N can be formulated. Since we do not need these moregeneral versions, we leave the precise statements to the interested reader. Proposition 1.
Suppose that τ is a linear-fractional self-map of B N such that therestriction of τ to the the complex line [ e ] in B N is the identity on [ e ] ∩ ∂B N .Then τ ( z , z , · · · , z N ) = ( z , A ′ z ′ ) , where z ′ denotes ( z , · · · , z N ) and A ′ is an ( N − × ( N − matrix.Proof. Since ϕ is linear-fractional we have ϕ ( z ) = Az + Bc z + c z + · · · c N z N + 1for some N × N matrix A = ( a jk ), N × B , and constants c k . By hypothesiswe must have a k λ + b k = 0for 2 ≤ k ≤ N and all λ ∈ C with | λ | = 1. Thus a k = b k = 0 for 2 ≤ k ≤ N . From ϕ ( e ) = e and ϕ ( − e ) = − e we see that b = c and a = 1. Using this, and therequirement that a λ + b = λ ( c λ + 1) for | λ | = 1, we must have b = 0 and thusalso c = 0.Moreover, since ϕ fixes e , we have by Lemma 6.6 of [4] that(12) D k ϕ ( e ) = 0 for k = 2 , , · · · , N. Since ϕ ( − e ) = − e , we may apply the same lemma to − ϕ ( − z ) to see that(13) D k ϕ ( − e ) = 0 for k = 2 , , · · · , N. Equations (12) and (13) together say a k = c k = 0 for k = 2 , · · · , N. This completes the proof. (cid:3)
To move from Theorem 5, which deals with parabolic maps, to the full resultof Theorem 4, we will need the notion of the Krein adjoint of the linear-fractionalmap ϕ . If ϕ is as given in Equation (6), its Krein adjoint is defined to be the linearfractional map(14) σ ϕ ( z ) = A ∗ z − C h z, − B i + d . This will be a self-map of B N whenever ϕ is, and when ϕ is an automorphism, itsKrein adjoint is equal to ϕ − . Moreover, ϕ and σ ϕ have the same fixed points on ∂B N ; for these and other basic facts, see [3] and [11]. Properties of the map ϕ ◦ σ ϕ appear in the next result. Theorem 7.
Suppose ϕ and ψ are linear fractional maps with k ϕ k ∞ = k ψ k ∞ = 1 .Assume further that at least one of the maps ϕ, ψ is univalent. If C ϕ − C ψ iscompact on H , then ϕ = ψ . Proof.
By the symmetric roles of ϕ and ψ we may assume that ϕ is univalent. Thehypothesis k ϕ k ∞ = 1 implies that there exists ζ in ∂B N with | ϕ ( ζ ) | = 1. Composingon the left and right by unitaries, there is no loss of generality in assuming ζ = e and ϕ ( e ) = e . By Theorem 6, we must have ψ ( e ) = e as well.Let σ ϕ be the Krein adjoint of ϕ as defined in Equation (14). Since C ϕ − C ψ iscompact and C σ ϕ is bounded, C σ ϕ ( C ϕ − C ψ ) = C ϕ ◦ σ ϕ − C ψ ◦ σ ϕ is also compact. Set τ = ϕ ◦ σ ϕ and ξ = ψ ◦ σ ϕ . We have τ ( e ) = ϕ ◦ σ ϕ ( e ) = e and thus by Theorem 6, ξ ( e ) = e and D τ ( e ) = D ξ ( e ). A computationshows that D τ ( e ) = 1, (details of this computation can be found in the proof ofTheorem 2 in [11]) so that D ξ ( e ) = 1 . We claim that τ = ξ . To see this, note that it is immediate by Theorem 5 ifneither τ nor ξ have any fixed point in the open ball B N . Suppose next that τ hasa fixed point in the open ball and lying in the complex line [ e ]. Restricting τ tothe intersection of [ e ] and the ball, we see that τ must be the identity on [ e ] ∩ B N ,since D τ ( e ) = 1 (see, for example, Problem 2.38 in [4], p. 60). By Proposition 1we see that τ has the form τ ( z , z ′ ) = ( z , Az ′ ) for some ( N − × ( N −
1) matrix A .We label the entries of A as a jk for j, k = 2 , · · · , N . Since C τ − C ξ is compact, weappeal to Theorem 6 to see that, since τ is the identity at each point of [ e ] ∩ ∂B N ,so is ξ . Applying Proposition 1 again, we see that ξ ( z , z ′ ) = ( z , M z ′ ) for an( N − × ( N −
1) matrix M = ( m jk ), j, k = 2 , · · · , n . Our goal is to show that A = M .Fix j with 2 ≤ j ≤ N , and consider the pseudohyperbolic distance ρ ( τ ( ω t ) , ξ ( ω t ))at points of the form(15) ω t = ( t, , · · · , √ − t, · · · ,
0) = ( t, ′ , √ − t, ′′ ) , for 0 < t <
1, where the √ − t appears in the j th component. These points lie inthe ball B N . A computation shows that1 − ρ ( τ ( ω t ) , ξ ( ω t ))is equal to [1 − t − (1 − t ) P Nk =2 | a kj | ][1 − t − (1 − t ) P Nk =2 | m kj | ] | − t − (1 − t ) P Nk =2 a kj m kj | , and further computation shows that as t ↑ − λ )(2 − γ )(2 − P k a kj m kj )(2 − P k a kj m kj )where λ = N X k =2 | a kj | ! / and γ = N X k =2 | m kj | ! / . OMPACT DIFFERENCES OF COMPOSITION OPERATORS 13
Observe that λ and γ are at most 1, since τ and ξ map the ball into itself. Write Z =( a j , a j · · · , a Nj ) and W = ( m j , m j , · · · , m Nj ) , so that λ = k Z k and γ = k W k .Moreover, the denominator in (16) is | − h Z, W i| , and by the Cauchy-Schwarzinequality, |h Z, W i| ≤ λγ with equality only if either Z = cW for some c ∈ C orone of Z, W is 0.We investigate the condition under which the expression in (16) is equal to 1.We have (2 − λ )(2 − γ ) | − h Z, W i| ≤ (2 − λ )(2 − γ )(2 − |h Z, W i| ) ≤ (2 − λ )(2 − γ )(2 − λγ ) ≤ , with the last inequality following from its equivalence to ( λ − γ ) ≥
0. Thus if theexpression in (16) is equal to 1, we must have λ = γ and | − h Z, W i| = 2 − |h Z, W i| = 2 − k Z kk W k . Together these force Z = W , which says that the ( j − st column of A is the sameas the ( j − st column of M . Thus if A = M , ρ ( τ, ξ ) has a strictly positive limitalong some path as in Equation (15). The ratio1 − | z | − | τ ( z ) | has the positive limit (2 − λ ) − along the same path. Applying Theorem 1 we havea contradiction to the hypothesis that C τ − C ξ is compact. Thus A = M , verifyingthe claim under the assumption that τ has a fixed point in [ e ] ∩ B N .Finally, suppose τ has a fixed point in the intersection of the open ball andthe complex line through η and e for some η ∈ ∂B N , but not in [ e ]. Since theautomorphisms act doubly transitively on ∂B N , we may find an automorphism Φof the ball, fixing e so that e τ ≡ Φ − τ Φ fixes e and a point of [ e ] ∩ B N . Acomputation shows that D e τ ( e ) = 1; this computation is aided by the fact that D e τ ( e ) = h e τ ′ ( e ) e , e i and the observation that since τ, Φ and Φ − all fix e , we have D k τ ( e ) = 0 , D k Φ ( e ) = 0 , D k Φ − ( e ) = 0 for all k = 2 , , . . . , N ([4], Lemma 6.6). Conjugating ξ by Φ as well to get e ξ ≡ Φ − ξ Φ, we apply theprevious argument to see that e τ = e ξ , and hence τ = ξ , in this case as well.Thus compactness of C ϕ − C ψ implies that τ = ξ, or equivalently(17) ϕ ◦ σ ϕ = ψ ◦ σ ϕ on B N , where σ ϕ is the Krein adjoint of ϕ . From this we see that ϕ and ψ agreeon the range of σ . Since we are assuming that ϕ is univalent, so is σ ϕ ([3]) and itfollows that ϕ = ψ , since the range of σ ϕ is an open set in B N . (cid:3) The final step is to remove the hypothesis of univalence in the last result toobtain the full proof of Theorem 4, which we turn to next. It will be helpful torecast our Hilbert space H as weighted Hardy spaces, defined below, and considerrestriction and extension operators on these weighted Hardy spaces.If f is analytic in B N , then f has a homogeneous expansion f = X s f s , where, for each z ∈ B N , we have(18) f s ( z ) = X | α | = s c α z α . Here, α = ( α , . . . , α N ) and | α | = α + · · · + α N . The Hardy space H ( B N ) isequivalently defined as(19) { f analytic in B N : ∞ X s =0 k f s k < ∞} , where k · k is the norm in L ( σ ). The sum in (19) is k f k H ( B N ) . More generally,given a suitable sequence of positive numbers { β ( s ) } , the weighted Hardy space H ( β, B N ) is the set of functions f which are analytic in B N and which satisfy k f k H ( β,B N ) ≡ ∞ X s =0 k f s k β ( s ) < ∞ . Note that, since the monomials z α are orthogonal on L ( σ ) ([13], Section 1.4), k f s k = X | α | = s | c α | k z α k = X | α | = s | c α | ( N − α !( N − s )! . The next result realizes the weighted Bergman spaces A γ ( B N ) as weighted Hardyspaces. Lemma 1. (a)
Let γ > − and set β ( s ) = ( s +1) − ( γ +1) . We have H ( β, B N ) = A γ ( B N ) , with equivalent norms. (b) Let K be an integer with ≤ K < N , and let β ( s ) = ( N − K − s )!( K − N − s )! ( s + 1) − ( γ +1) where γ ≥ − . Then H ( β, B K ) = A N − K + γ ( B K ) , with equivalent norms.Proof. We prove (a) first. Let f be analytic in B K with homogeneous expansion f = P f s , where f s is as in Equation (18). If β ( s ) = ( s + 1) − ( γ +1) , we have k f k H ( β,B N ) = ∞ X s =0 X | α | = s | c α | α !( N − N + s ) · s + 1) γ +1 and k f k A γ ( B N ) = ∞ X s =0 X | α | = s | c α | α !Γ( N + γ + 1)Γ( N + s + γ + 1) . This last formula follows from the fact that k z α k A γ ( B K ) = α !Γ( N + γ + 1)Γ( N + | α | + γ + 1)(see Lemma 1.11 in [15].)The result in (a) will follow if we can show that (cid:18) α !( N − N + s ) · s + 1) γ +1 (cid:19) · (cid:18) Γ( N + s + γ + 1) α !Γ( N + γ + 1) (cid:19) OMPACT DIFFERENCES OF COMPOSITION OPERATORS 15 is bounded above and below by positive constants, depending only on N and γ , forall s ≥
0. This follows easily from the fact that, by Stirling’s formula,lim s →∞ Γ( N + s + γ + 1)( s + 1) γ +1 Γ( N + s ) = 1 . From (a) we know that A N − K + γ ( B K ) = H ( e β, B K ) where e β ( s ) = ( s +1) − ( N − K + γ +1) .Thus it suffices to show that (cid:20) ( N − K − s )!( K − N − s )! ( s + 1) − ( γ +1) (cid:21) · h ( s + 1) ( N − K + γ +1) i is bounded above and below by positive constants (depending on N and K ) for all s ≥
0. Straightforward estimates show that( N − K − ≥ ( N − K − s )!( K − N − s )! · ( s + 1) N − K ≥ ( N − K − (cid:18) N + 1 (cid:19) N − K for all s ≥
0, and the desired result follows. (cid:3)
Since the proof of Theorem 4 ultimately relies on an inductive argument, we willwork with certain restriction and extension operators on weighted Hardy spaces.These are defined next.Let
K, N ∈ N with 1 ≤ K < N . Given a sequence { β ( s ) } of positive numbers,we define the associated sequence { e β ( s ) } by e β ( s ) = ( N − K − s )!( K − N − s )! β ( s ) . We can then define the extension operator E : H ( e β, B K ) → H ( β, B N ) by( Ef )( z , . . . , z N ) = f ( z , . . . , z K ) , and the restriction operator R : H ( β, B N ) → H ( e β, B K ) by( Rf )( z , . . . , z K ) = f ( z , . . . , z K , ′ ) . The next result establishes properties of these operators; it is an extension of Propo-sition 2.21 in [3] which applies to the case K = 1. Lemma 2. (a)
The extension operator E is an isometry of H ( e β, B K ) into H ( β, B N ) . (b) The restriction operator R is a norm-decreasing map of H ( β, B N ) onto H ( e β, B K ) .Proof. For (a), let f ∈ H ( e β, B K ) with homogeneous expansion f = P f s with f s as in Equation (18) for z ∈ C K . Then Ef = P f s also, and writing k f s k ,K for the norm of f s in L ( ∂B K , σ ) we have k f k H ( e β,B K ) = X s k f s k ,K e β ( s ) = X s X | α | = s | c α | ( K − α !( K − s )! e β ( s ) = X s X | α | = s | c α | ( K − α !( K − s )! · ( N − K − s )!( K − N − s )! β ( s ) = X s X | α | = s | c α | ( N − α !( N − s )! β ( s ) = X s k f s k ,N β ( s ) = k Ef k H ( β,B N ) . Therefore, E is an isometry.For (b) let f ∈ H ( β, B N ) have homogeneous expansion f = P f s with f s asin Equation (18) for z ∈ B N . For each nonnegative integer s , let A s = { α : α =( α ′ , ′ ) } , where α ′ is a multi-index with K entries and 0 ′ denotes the zero vectorin C N − K , and let B s consist of all other multi-indices α with N entries satisfying | α | = s . Writing f s ( z ) = X A s c α z α + X B s c α z α , it follows that ( Rf )( z , . . . , z K ) = X s X A s c α z α · · · z α K K , and k Rf k H ( e β,B K ) = X s X A s | c α | ( K − α !( K − s )! e β ( s ) = X s X A s | c α | ( N − α !( N − s )! β ( s ) ≤ X s X | α | = s | c α | ( N − α !( N − s )! β ( s ) = k f k H ( β,B N ) . Therefore, R is norm-decreasing. To see that R is surjective, let f ∈ H ( e β, B K )with f ( z , . . . , z K ) = X s X | α ′ | = s c α ′ z α ′ · · · z α ′ K K , where α ′ is a multi-index with K entries. Then f = RF , where F ( z ) = X s X A s c α z α . OMPACT DIFFERENCES OF COMPOSITION OPERATORS 17 and c α ≡ c α ′ for α = ( α ′ , ′ ) as before. Also, k F k H ( β,B N ) = X s X A s | c α | ( N − α !( N − s )! β ( s ) = X s X A s | c α | ( K − α !( K − s )! e β ( s ) = X s X | α ′ | = s | c α ′ | k z α ′ · · · z α ′ K K k ,K = k f k H ( e β,B K ) < ∞ , so F ∈ H ( β, B N ). (cid:3) Recall that by an affine subset of dimension k in B N , we mean the intersectionof B N with a translate of a k -dimensional subspace of C N . Proof of Theorem 4.
Suppose that ϕ and ψ are linear-fractional maps with k ϕ k ∞ = k ψ k ∞ = 1. We will show that if C ϕ − C ψ is compact on H , then ϕ = ψ . Thehypothesis k ϕ k ∞ = 1 implies that there exists a point ζ in ∂B N with | ϕ ( ζ ) | = 1.Composing on the left and right by unitaries, there is no loss of generality inassuming ζ = e and ϕ ( e ) = e . By Theorem 6, we must have ψ ( e ) = e as well.The argument proceeds exactly as in the proof of Theorem 7 up to the pointwhere the relationship(20) ϕ ◦ σ ϕ = ψ ◦ σ ϕ , is obtained. Since Theorem 7 covers the case that at least one of ϕ and ψ is one-to-one, we now only consider the case that neither is univalent. This implies thatthere is a smallest k with 1 ≤ k < N so that ϕ ( B N ) is contained in an affineset of dimension k , and there is a smallest k with 1 ≤ k < N so that ψ ( B N )is contained in an affine set of dimension k . Since the roles of ϕ and ψ can bereversed, there is no loss of generality in assuming k ≥ k .Our first goal is to show that equality k = k holds as a consequence of Equa-tion (20). By Proposition 13 in [3], σ ϕ ( B N ) is also contained in an k -dimensionalaffine set. Set σ ϕ (0) = p , and let φ p be an automorphism of B N sending p to 0and satisfying φ p = φ − p . Since φ p ◦ σ ϕ fixes the origin, and maps the ball into a k -dimensional affine set, we may write, as in [3], φ p ◦ σ ϕ = L ◦ τ where L is linear of rank k and τ is a one-to-one linear fractional map. Specifically,if φ p ◦ σ ϕ = Az h z, C i + 1we can choose L ( z ) = Az and τ ( z ) = z h z, C i + 1 . (Note that L and τ need not be self-maps of B N , though their composition is,and both are defined and analytic on a neighborhood of the closed ball). From this it follows that σ ϕ = φ p ◦ L ◦ τ . Taking Krein adjoints on both sides we have ϕ = σ τ ◦ L ∗ ◦ σ φ p , where σ τ ( z ) = z − C . We have σ φ p = φ − p = φ p , so that(21) ϕ ◦ σ ϕ = σ τ ◦ L ∗ ◦ σ φ p ◦ φ p ◦ L ◦ τ = σ τ ◦ L ∗ L ◦ τ, where L ∗ L is linear with rank k , τ is univalent, and σ τ is a translation. Thus theimage of the ball B N under ϕ ◦ σ ϕ cannot be contained in a k dimensional affineset for any k < k , and the relation ϕ ( σ ϕ ( B N )) = ψ ( σ ϕ ( B N )) ⊆ ψ ( B N ) says thatstrict inequality k > k is impossible, and therefore k = k as desired. We denotethe common value of k and k by K .Thus ϕ ( B N ) is contained in a K -dimensional affine set A and is not containedin any affine set of smaller dimension, and ψ ( B N ) is contained in a K -dimensionalaffine set A , and is not contained in any affine set of smaller dimension. We have A ⊇ ψ ( B N ) ⊇ ψ ( σ ϕ ( B N )) = ϕ ( σ ϕ ( B N ))where A ⊇ ϕ ( σ ϕ ( B N )) = σ τ L ∗ Lτ ( B N ) for linear L ∗ L of rank K and univalentlinear fractional τ and σ τ . This forces A = A ; that is, the range of ϕ and therange of ψ are contained in the same K -dimensional affine set, which we will simplydenote A . Note that e ∈ A .Our goal is to show that ϕ = ψ . Let ζ ∈ ∂B N with ζ = e . Let Λ ζ be a K -dimensional affine subset of B N , containing e and ζ in its boundary, whose intersec-tion with B N is a K -dimensional ball. We will write g B K for { ( z , z , . . . , z K , ′′ ) ≡ ( z ′ , ′′ ) ∈ B N } , where 0 ′′ denotes the 0 in C N − K . Let ρ be an automorphism of B N fixing e with ρ ( g B K ) = Λ ζ and let ρ be an automorphism of B N fixing e with ρ ( A ) = g B K ; such automorphisms exist because of the two-fold transitivity ofthe automorphisms on ∂B N . Note that ρ ◦ ϕ ◦ ρ and ρ ◦ ψ ◦ ρ are linear-fractionalself-maps of B N with ρ ◦ ϕ ◦ ρ ( g B K ) ⊆ g B K and ρ ◦ ψ ◦ ρ ( g B K ) ⊆ g B K . Let π bethe projection of C N onto C K defined by π ( z ′ , z ′′ ) = z ′ and define maps µ and ν on B K by µ ( z ′ ) = π ◦ ρ ◦ ϕ ◦ ρ ( z ′ , ′′ )and ν ( z ′ ) = π ◦ ρ ◦ ψ ◦ ρ ( z ′ , ′′ ) . These are linear-fractional self-maps of B K fixing (1 , . . . , ∈ ∂B K .Write H as H ( β, B N ) with β ( s ) = ( s +1) − ( γ +1) , where γ = − H = H ( B N )and γ = α if H = A α ( B N ), up to an equivalent norm. We claim that C µ − C ν iscompact on the weighted Hardy space H ( e β, B K ) where e β ( s ) = ( N − K − s )!( K − N − s )! β ( s ) . Since, by Lemma 1, H ( e β, B K ) = A N − K + γ ( B K ) with equivalent norms, it willfollow that C µ − C ν is compact on A N − K + γ ( B K ).To prove the claim it suffices to show that if { f n } is a bounded sequence on H ( e β, B K ) and f n → B K , then k ( C µ − C ν ) f n k H ( e β,B K ) → . Let f n be such a sequence in H ( e β, B K ). Define functions F n on B N by F n = Ef n , OMPACT DIFFERENCES OF COMPOSITION OPERATORS 19 where E : H ( e β, B K ) → H ( β, B N ) is the extension operator defined by( Ef )( z , . . . , z N ) = f ( z , . . . , z K ) . By Lemma 2, E is an isometry, so the functions F n form a bounded sequence in H and F n → B N . Since C ρ ◦ ϕ ◦ ρ − C ρ ◦ ψ ◦ ρ = C ρ ( C ϕ − C ψ ) C ρ is compact on H , we have k F n ◦ ρ ◦ ϕ ◦ ρ − F n ◦ ρ ◦ ψ ◦ ρ k H ( β,B N ) → R : H ( β, B N ) → H ( e β, B K ) by( Rf )( z , . . . , z K ) = f ( z , . . . , z K , ′′ ) . By Lemma 2, R is a norm-decreasing map of H ( β, B N ) onto H ( e β, B K ) and so k R ( F n ◦ ρ ◦ ϕ ◦ ρ ) − R ( F n ◦ ρ ◦ ψ ◦ ρ ) k H ( e β,B K ) → R ( F n ◦ ρ ◦ ϕ ◦ ρ ) = f n ◦ µ on B K and R ( F n ◦ ρ ◦ ψ ◦ ρ ) = f n ◦ ν on B K , sothe claim is verified, and C µ − C ν is compact on A N − K + γ ( B K ). Since K < N and µ and ν are linear-fractional self-maps of B K with k µ k ∞ = k ν k ∞ = 1, by inductionthis forces µ = ν , which in turn says that ϕ = ψ on the affine set Λ ζ containing ζ and e . Since ζ is an arbitrary point in ∂B N , this says ϕ = ψ in B N . (cid:3) References [1] F. Bracci, M. Contreras, and S. Diaz-Madrigal, Classification of semigroups of linear fractionalmaps in the unit ball, Adv. Math., 208 (2007), 318–350.[2] P. Bourdon, Components of linear-fractional composition operators, J. Math. Anal. Appl.279 (2003), 228–245.[3] C. Cowen and B. MacCluer, Linear fractional maps of the ball and their composition opera-tors, Acta. Sci. Math. (Szeged), 66 (2000), 351–376.[4] C. Cowen and B. MacCluer,
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Katherine Heller, Department of Mathematics, P. O. Box 400137, University of Vir-ginia, Charlottesville, VA 22904
E-mail address : [email protected] Barbara D. MacCluer, Department of Mathematics, P.O. Box 400137, University ofVirginia, Charlottesville, VA 22904
E-mail address : [email protected] Rachel J. Weir, Department of Mathematics, Allegheny College, Meadville, PA16335
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