DIFFERENCE OF COMPOSITION OPERATORS OVER THE HALF-PLANE

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1 TRANSACTIONS OF TE AMERICAN MATEMATICAL SOCIETY Volume 00, Number 0, Pages S (XX) DIFFERENCE OF COMPOSITION OPERATORS OVER TE ALF-PLANE BOO RIM COE, YUNGWOON KOO, AND WAYNE SMIT Abstract. We study the differences of composition operators acting on weighted Bergman spaces over the upper half-plane. In this setting not all composition operators are bounded and none are compact. The idea of joint pullback measure is used to give a Carleson measure characterization of when the difference of two composition operators is bounded or compact. Alternate characterizations, not using Carleson measures, are also given for certain large classes of the inducing maps for the operators. The relationship between angular derivatives and compact differences of composition operators is also explored, which, in particular, reveals a new phenomenon due to the upper half-plane not being bounded. Our results produce a variety of examples of distinct composition operators whose difference is compact, including examples when the individual operators are not bounded. The paper closes with a characterization of when the difference of composition operators is ilbert-schmidt.. Introduction Let be the upper half of the complex plane C, i.e., := {z C : Im z > 0} and let S be the class of all holomorphic self-maps of. Each φ S induces a composition operator C φ defined by C φ f = f φ for functions f holomorphic on. It is clear that C φ takes the space of holomorphic functions on into itself. Efforts to understand the topological structure of the space of composition operators have led to the study of the difference C φ C ψ of two composition operators. In the setting of the unit disk D, all composition operators, and hence all differences of two composition operators, are bounded on the ardy spaces and the Bergman spaces. Thus the question of when the difference of composition operators is compact was naturally a focus of attention. Shapiro and Sundberg [23] first raised and studied such a question on the ardy space, motivated by the isolation phenomenon observed by Berkson []. Their work initiated subsequent efforts [0, 8] on the Received by the editors Apr. 26, Mathematics Subject Classification. Primary 47B33; Secondary Key words and phrases. Difference of composition operators; Joint pullback measure; Carleson measure; Bounded operator; Compact operator; ilbert-schmidt operator; Weighted Bergman space; alf-plane. B. R. Choe was supported by NRF(203RAA ) of Korea and. Koo was supported by NRF(204RAA205445) of Korea. c XXXX American Mathematical Society

2 2 B. COE,. KOO, AND W. SMIT ardy space and [5, 7, 9, 20] on the Bergman space; see also [3, 4, 2, 4, 6] for work on the Bergman spaces over the ball or the polydisk. The purpose of the current paper is to study the differences of composition operators acting on the weighted Bergman spaces over. We first recall the spaces we work on. For α >, put da α (z) := c α (Im z) α da(z) where c α = 2α (+α) π is a normalizing constant and A is the area measure on. For 0 < p <, we denote by A p α() the weighted Bergman space consisting of all holomorphic functions f on such that the norm { } /p f A p α := f p da α is finite. As is well known, each space A p α() is a closed subspace of L p (da α ). Thus A p α() is a Banach space for p < and, in particular, A 2 α() is a ilbert space. Also, when 0 < p <, the space A p α() is a complete metric space under the translation-invariant metric (f, g) f g p A. p α Unlike the disk case, not all composition operators are bounded on the spaces A p α(). For example, it is not hard to check that the spaces A p α() are not Möbius invariant. Recently, Elliott and Wynn [9] characterized bounded composition operators on A p α(); see Theorem 2.4 below. Their work also shows that no composition operator on A p α() is compact; see [22] for a more general result in this direction. Nevertheless, we will show that there are many and varied examples of distinct composition operators with compact difference, including examples when the individual composition operators are not bounded. In the next section we collect some basic material that will be needed subsequently. Then in 3 the idea of joint pullback measure is used to give a Carleson measure characterization of when the difference of composition operators is bounded or compact on A p α(); see Theorem 3.3. Alternate characterizations not using Carleson measures are given in 4 for special classes of inducing maps for the composition operators, in particular when they map to relatively compact subsets of ; see Theorems 4. and 4.5. Angular derivatives are used in 5 to investigate compactness of the difference of composition operators. In the settings of classical bounded domains, such as the disk([7]), the polydisk([4]), or the ball([3]), it is known that a necessary condition for compactness is that the inducing maps have exactly the same first-order data. This means that their angular derivatives exist at the same points, and at these points the angular limits of the functions and their derivatives match. In the setting of there is a new phenomenon due to the domain being unbounded. We show that while matching first-order data is usually necessary (see Theorem 5.4), in general it is not (see Example 5.). Then, under the assumption that the individual composition operators are bounded, we use angular derivatives to characterize when the difference is compact; see Theorem 5.5. Some applications of our characterization of boundedness are collected in 6. In particular, we show that in the context of bounded differences of composition operators on A p α(), automorphisms of exhibit a certain rigidity; see Theorem 6.2. Finally, ilbert-schmidt differences of composition operators on A 2 α() are characterized in the last section by means of square-integrability with respect to

3 DIFFERENCES OF COMPOSITION OPERATORS 3 the Möbius invariant measure of a quantity that naturally appears in the characterizations of bounded/compact differences in earlier sections; see Theorem 7.6. Constants. Throughout the paper we use the same letter C to denote various positive constants which may vary at each occurrence but do not depend on the essential parameters. Variables indicating the dependency of constants C will be often specified in parenthesis. For nonnegative quantities X and Y the notation X Y or Y X means X CY for some inessential constant C. Similarly, we write X Y if both X Y and Y X hold. 2. Prerequisites In this section we collect some basic facts and preliminary results to be used throughout the paper. 2.. Reproducing Kernel. Given α >, subharmonicity yields a constant C = C(α) > 0 such that f(z) 2 C (Im z) α+2 f 2 A 2 α, z for f A 2 α(). In particular, this shows that each point evaluation is a continuous linear functional on A 2 α(). Thus, to each z corresponds a unique reproducing kernel K z (α) A 2 α(). As is well known, the explicit formula of K z (α) is given as ( ) α+2 i K z (α) (w) = (i = ). w z Note (2.) ( ) (α+2)/2 K z (α) A 2 α = K z (α) (z) =. 2Im z 2.2. Compact Operator. It seems better to clarify the notion of compact operators, since the spaces under consideration are not Banach spaces when 0 < p <. Suppose X and Y are topological vector spaces whose topologies are induced by complete metrics. A continuous linear operator T : X Y is said to be compact if the image of every bounded sequence in X has a subsequence that converges in Y. We have the following convenient compactness criterion for a linear combination of composition operators acting on the weighted Bergman spaces. Lemma 2.. Let α > and 0 < p <. Let T be a linear combination of composition operators and assume that T is bounded on A p α(). Then T is compact on A p α() if and only if T f n 0 in A p α() for any bounded sequence {f n } in A p α() such that f n 0 uniformly on compact subsets of. A proof can be found in [7, Proposition 3.] for a single composition operator over the unit disk and it can be easily modified for a linear combination over the half-plane.

4 4 B. COE,. KOO, AND W. SMIT 2.3. Pseudo-hyperbolic Distance. The pseudo-hyperbolic distance ρ(z, w) between z, w is given by ρ(z, w) := z w z w. Note that ρ is invariant under dilation and horizontal translation. Also, note So, we have by (2.) (2.2) ρ 2 (z, w) = z w 2 z w 2 z w 2 = = [ ρ 2 (z, w)] (α+2)/2 = 2Re [z(w w)] z w 2 4(Im z)(im w) z w 2. K (α) z K w (α) (z) α K w (α) α for any z, w and α >. For z and 0 < δ <, let E δ (z) denote the pseudohyperbolic disk centered at z with radius δ. A straightforward calculation show that E δ (z) is the Euclidean disk with (2.3) (center) = x + i + δ2 2δ y and (radius) = δ2 δ 2 y where x = Re z and y = Im z. Thus it is easily seen that (2.4) δ + δ < Im z Im w < + δ δ whenever w E δ (z). Also, one may verify that (2.5) δ z a < + δ w a < + δ δ whenever w E δ (z) and a ; see [6, Lemma 3.3]. In particular, (2.6) δ z w < + δ 2Im z < + δ δ whenever w E δ (z). Given 0 < δ < and α >, note that there is a constant C = C(α, δ) > 0 such that (2.7) C (Im z) α+2 A α [E δ (z)] C(Im z) α+2 for all z. This yields the submean value type inequality f(z) p C (2.8) (Im z) α+2 f p da α, z E δ (z) for 0 < p < and functions f holomorphic on ; see [6, Lemma 3.6] or, for details on the disk, [25, Proposition 4.3]. In particular, we have (2.9) f(z) p C (Im z) α+2 f p A p α, z

5 DIFFERENCES OF COMPOSITION OPERATORS 5 and (2.0) lim (Im z) α+2 f(z) p = 0 z Ĥ for functions f A p α(). ere, Ĥ := { } and lim z Ĥ g(z) = 0 means that sup \K g 0 as the compact set K expands to the whole of, or equivalently that g(z) 0 as Im z 0 + and g(z) 0 as z Carleson Measure. Let α > and µ be a locally finite positive Borel measure on. ere, the term locally finite means that µ(k) < for any compact set K. For 0 < δ < and 0 < p <, we have (2.) and (2.2) the embedding A p α() L p (dµ) is bounded µ[e δ (z)] sup z A α [E δ (z)] < the embedding A p α() L p (dµ) is compact lim z Ĥ µ[e δ (z)] A α [E δ (z)] = 0. In fact these are proved in [5, Theorem 4.] and [5, Theorem 5.3] for the unweighted L p -harmonic Bergman space, < p <, over the upper half-space; we have however added the missing hypothesis, namely, the local finiteness. Working on the holomorphic Bergman space, one can check that the restriction < p < can be removed. One may also extend the proofs to the weighted cases. Alternatively, one may modify the proofs in [25, Section 7.2] for the disk setting. We say that µ is an α-carleson measure if (2.) holds. Also, we say that µ is a compact α-carleson measure if (2.2) holds. Note that the notion of (compact) α-carleson measures is independent of the size of p or δ. The connection between composition operators and Carleson measures comes from the standard identity (see [, p. 63]) (2.3) (h φ) da α = h d(a α φ ) valid for holomorphic self-maps φ of and Borel functions h 0. ere, A α φ denotes the pullback measure defined by (A α φ )(E) = A α [φ (E)] for Borel sets E. One can easily see from (2.3) that C φ is bounded (compact, resp.) on A p α() if and only if A α φ is a (compact) α-carleson measure Angular Derivative. We first recall the well-known notion of angular derivatives on D. A holomorphic self-map f of D is said to have finite angular derivative at η D, denoted by f (η) C, if f has nontangential limit f(η) D at η such that f(ζ) f(η) lim = f (η) ζ η ζ η where lim stands for the nontangential limit. As is well known by the Julia- Carathéodory Theorem(see [8, Theorem 2.44]), f (η) exists if and only if lim inf ζ η f(ζ) 2 ζ 2 <.

6 6 B. COE,. KOO, AND W. SMIT In this case, the left-hand side of the above is equal to f (η). In particular, we have f (η) f(0) + f(0) > 0 by the Schwarz-Pick Lemma. Before introducing angular derivatives in the half-plane setting, we need to clarify the notion of nontangential limits at boundary points of Ĥ. Of course, those at a finite boundary point refer to the standard notion. Meanwhile, those at Ĥ refer to those associated with nontangential approach regions Ω γ, γ > 0, consisting of all z C such that Im z > γ Re z. For a function h : and x Ĥ, we write h(x) = L (possibly ) if h has nontangential limit L, i.e., lim z x h(z) = L. We now introduce the notion of angular derivatives on via the Caley transformation κ(ζ) := i + ζ ζ, ζ D, which conformally maps D onto. Note that a region Γ D is contained in a nontangential approach region in D if and only if κ(γ) is contained in a nontangential approach region in. Given φ S, let (2.4) φ κ = κ φ κ. We say that φ has finite angular derivative at x Ĥ if φ κ has finite angular derivative at x := κ (x) D. In this case, note that φ is forced to have nontangential limit φ(x) Ĥ at x Ĥ and we have φ κ( x) = lim z x φ(z) i φ(z)+i φ(x) i φ(x)+i z i z+i x i x+i Depending on whether x or φ(x) is equal to, this can be simplified and rephrased through straightforward calculations as follows: (i) For x, φ(x), [ ] 2 φ(z) φ(x) φ(x) + i (2.5) lim = φ z x z x κ( x) ; x + i (ii) For x and φ(x) =, [ ] 2 (2.6) lim z x (z x)φ(z) = φ κ( x) ; x + i (iii) For x = and φ( ), (2.7) (iv) For x = = φ( ), (2.8) lim z z[φ(z) φ( )] = φ κ() [φ( ) + i] 2 ; z lim z φ(z) = φ κ(). In each case of (2.5)-(2.8) the limit in the left-hand side is naturally denoted by φ (x). Note φ (x) 0. The following Julia-Carathéodory Theorem for the upper half-plane, taking care of the case (iv), is proved in [9, Prop. 2.2]..

7 DIFFERENCES OF COMPOSITION OPERATORS 7 Proposition 2.2. For φ S, the following statements are equivalent: (a) φ( ) = and φ ( ) exists; Im z (b) sup z Im φ(z) < ; Im z (c) lim sup z Im φ(z) <. Moreover, the quantities in (b) and (c) are both equal to φ ( ). While not needed in the current paper, we remark that one may deduce from Proposition 2.2 the Julia-Carathéodory Theorem for the remaining cases (i)-(iii). For example, if φ( ) = 0, then apply Proposition 2.2 to the function φ(z). The following is another form of the half-plane version of the Julia-Carathéodory Theorem. Proposition 2.3. Let φ S and x Ĥ. Then φ (x) exists if and only if 2 lim sup φ(z) + i Im z z x z + i Im φ(z) > 0. Proof. The proposition follows from the Julia-Carathéodory Theorem and the identity ζ 2 φ κ (ζ) 2 = z i z+i 2 2 φ(z) i 2 = φ(z) + i Im z (2.9) z + i Im φ(z) φ(z)+i where ζ = κ (z). Elliott and Wynn [9] showed that bounded composition operators are characterized by one of the conditions in Proposition 2.2. Theorem 2.4 (Elliott-Wynn). Let α > and 0 < p <. Let φ S. Then C φ is bounded on A p α() if and only if φ( ) = and φ ( ) exists. In fact the above theorem is stated in [9] only for p = 2, but remains true for all p by the Carleson measure characterization mentioned in Section Test Functions. For a, let τ a be the function on defined by (2.20) τ a (z) := z a. Powers of these functions will be the source of test functions in conjunction with Lemma 2.. The norms of such kernel-type functions can be computed by means of the next lemma. Lemma 2.5. Given α > and s real, the equality { cα,s z a α+2+s da (Im a) if s > 0 α(z) = s if s 0 holds for a. ere, c α,s is a constant.

8 8 B. COE,. KOO, AND W. SMIT Proof. Fix α > and a real number s. Given a, denoting by J α,s (a) the integral in question and making the change-of-variable w = z Re a Im a, we obtain J α,s (a) = (Im a) s w + i α+2+s da α(w) = 2c α (Im a) s which yields the asserted equality. 0 dx (x 2 + ) (α+2+s)/2 0 y α dy (y + ) +α+s, As a consequence, we have τ s a A p α() if and only if sp > α + 2. Also, when sp > α + 2, we have (2.2) and thus τ s a p A p α = (constant) (Im a) sp 2 α (2.22) as a Ĥ. τ s a τ s a A p α 0 uniformly on compact subsets of 3. Carleson Measure Characterization The notion of joint pullback measures has been introduced by Koo and Wang [6] quite recently in their study of differences of composition operators over the ball. Earlier, such notion was implicitly considered by Saukko [9] in his study of operator (essential) norm estimates for differences of composition operators over the disk. ere, we introduce the half-plane analogue and use it to characterize boundedness (compactness, resp.) of the difference of two composition operators by the property of the joint pullback measure being a (compact) Carleson measure. Before proceeding, we first set some notation to be used for the rest of the paper. Given φ, ψ S, we put σ(z) = σ φ,ψ (z) := ρ(φ(z), ψ(z)) for short. Use of this function σ has become standard in the study of the difference of two composition operators, going back to [3] in the setting of the ardy space 2 over D. Given α > and 0 < p <, we define the joint pullback measure ω α,p = ω α,p;φ,ψ by (3.) ω α,p (E) := σ p da α + σ p da α φ (E) ψ (E) for Borel sets E. So, ω α,p is actually the sum of two pullback measures (σ p da α ) φ and (σ p da α ) ψ. By a standard argument one can verify (3.2) g dω α,p = [g(φ) + g(ψ)]σ p da α for any positive Borel function g on. Given 0 < δ <, we use the notation (3.3) ω α,p [E δ (z)] ω α,p δ := sup z A α [E δ (z)] ; recall that E δ (a) denotes the pseudohyperbolic disk of radius δ centered at a.

9 DIFFERENCES OF COMPOSITION OPERATORS 9 We need a couple of technical lemmas. In what follows, we use the notation for a and N > 0. a N := Re a + in(im a) Lemma 3.. Given s > 0 and 0 < δ <, there are constants N = N(s, δ) > 0 and C = C(s, δ) > 0 such that τ s a(z) τ s a(w) + τ s a N (z) τ s a N (w) Cρ(z, w) τ s a(z) for all a, w and z E δ (a). Proof. Fix s > 0 and 0 < δ <. Let a. By horizontal translation we may assume a = iy, y > 0. Let z E δ (a). Let w. Consider the case w a Ny where N > 4 is a number to be chosen later. In this case we have and thus by (2.6) w a w a a a ( N 2)y z a w a 4 ( δ)( N 2). So, choosing N so large that [ 4 (3.4) we obtain ( δ)( N 2) ] s 2, ( τa(z) s τa(w) s = τa(z) s τ a(w) s ) τa(z) s 2 τ a(z) s, which implies the asserted inequality. Now, consider the case w a < Ny. Note τa s N (z) τa s N (w) τa s τa(z) s = N (z) ( ) s z + iny τa(z) s w + iny τ s = an (z) ( τa(z) s w z ) s (3.5). w + iny For the first factor of the above, since z + iny (N )y z + iy + z + iy < + N 2 where the second inequality holds by (2.6), we have τa s ( ) N (z) s δ τa(z) s >. N + δ δ < N δ To estimate the second factor of (3.5), we first note from (2.6) w z w a + a a + a z < w + iny Ny N So, choosing N so large that (3.6) ( ζ) s s 2 ζ for ζ N ( N δ ( ) N , δ ).

10 0 B. COE,. KOO, AND W. SMIT we have 2 s ( w z ) s w z w + iny w + iny z w = ρ(z, w) w + iny Im z ρ(z, w) w a + a + iny ρ(z, w) δ N + N + 2 Accordingly, choosing N satisfying (3.4) and (3.6), we obtain τ s an (z) τ s a N (w) Cρ(z, w) τ s a (z) by (2.4). for some positive constant C = C(α, δ, N) = C(α, δ). This completes the proof. Lemma 3.2. Let α >, 0 < p < and 0 < δ < δ <. Then there is a constant C = C(α, p, δ, δ ) > 0 such that f(z) f(w) p C ρp (z, w) f p da α A α [E δ (z)] E δ (z) for all z, w with w E δ (z) and functions f holomorphic on E δ (z). Proof. Let z, w and assume w E δ (z). Let f be an arbitrary holomorphic function on E δ (z). Then we have ( ) (3.7) f(z) f(w) z w sup f (ζ) ζ E δ (z) In conjunction with this, we note (by the subharmonicity of f p ) that there is a constant C = C(α) > 0 such that f(a) p C d α+2 f p da α, a E δ (z) (a) E δ (z) where d(a) denotes the distance from a to the boundary of E δ (z). Using this and the Cauchy Estimates, one may check by (2.3) that sup f (ζ) p C (3.8) ζ E δ (z) (Im z) α+2+p f p da α for some constants C = C(α, δ, δ ) > 0. Now, since E δ (z) 4(Im z) (3.9) z w = ρ(z, w) z w ρ(z, w) δ by (2.6), we conclude the lemma by (3.7), (3.8) and (2.7). We are now ready to characterize the boundedness/compactness of the difference of composition operators by means of the joint pullback measures. In the next theorem we use notation as in (3.)-(3.3). Theorem 3.3. Let α >, 0 < p < and 0 < δ <. For φ, ψ S the following statements hold: (a) C φ C ψ is bounded on A p α() if and only if ω α,p is an α-carleson measure. In this case, the operator norm of C φ C ψ is comparable to ω α,p /p δ ;.

11 DIFFERENCES OF COMPOSITION OPERATORS (b) C φ C ψ is compact on A p α() if and only if ω α,p is a compact α-carleson measure. Proof. Put T := C φ C ψ for brevity. First, we consider the boundedness. To prove the sufficiency, assume ω α,p δ <. Let f A p α(). Setting we have T f p da α D = D δ/2 := {z : σ(z) < δ/2}, D T f p da α + ( f(φ) p + f(ψ) p ) da α \D =: I(f) + II(f). The second term of the above is easily handled. Namely, we have (3.0) II(f) δ p ω α,p δ f p A p α by the Carleson measure characterization. We now estimate the first term. We have by Lemma 3.2 and Fubini s Theorem { σ p } (z) I(f) f(w) p da α (w) da α (z) D A α [E δ (φ(z))] E δ (φ(z)) { } σ p (z) (3.) = A α [E δ (φ(z))] da α(z) f(w) p da α (w). Note from (2.7) and (2.4) D φ [E δ (w)] A α [E δ (φ(z))] [Im φ(z)] α+2 (Im w) α+2 A α [E δ (w)] for all z φ [E δ (w)]. So, we see from (3.) that ω α,p [E δ (w)] (3.2) I(f) A α [E δ (w)] f(w) p da α (w) and thus I(f) ω α,p δ f p α,p. So, we see from this and (3.0) that T is bounded on A p α with norm at most ω α,p /p δ times some constant depending only on α, p and δ. For the necessity, assume that T is bounded on A p α. Let s = (α + 3)/p and N = N(s, δ) > 0 be a number provided by Lemma 3.. Let w and put Note from Lemma 3. that w N := Re w + in(im w). T τ s w(z) + T τ s w N (z) σ(z) τ s w(φ(z)), z φ [E δ (w)] for all w. Also, note from (2.6) τ s w(φ(z)) (Im w) s, z φ [E δ (w)].

12 2 B. COE,. KOO, AND W. SMIT Accordingly, we obtain J(w) : = T τ s w(z) p + T τ s w N (z) p da α (z) φ [E δ (w)] (Im w) α+3 σ p (z) τ s w(φ(z)) p da α (z) φ [E δ (w)] σ p (z) da α (z). for all w. The same argument works with ψ in place of φ and hence we have (3.3) for all w. Since we have (2.2) we may use (2.7) to rephrase (3.3) as J(w) ω α,p[e δ (w)] (Im w) α+3 τw s N p A = p α Im w N N(Im w) τ w s p A, p α (3.4) T τ s w p A p α τ s w p A p α + T τ s w N p A p α τ s w N p A p α ω α,p[e δ (w)] A α [E δ (w)] for all w. Consequently, the boundedness of T on A p α yields ω α,p δ T p ; the constant suppressed above depends only on α, p and δ. proof of the boundedness part. This completes the We now turn to the proof of the compactness part. First, assume that T is compact on A p α(). Note from (a) that ω α,p is locally finite. Also, note w N Ĥ as w Ĥ. So, we deduce from (2.22), (3.4) and Lemma 2. that ω α,p is a compact α-carleson measure. Conversely, suppose that ω α,p is a compact α-carleson measure, i.e., that ω α,p is locally finite and (3.5) ω α,p [E δ (w)] lim = 0. w Ĥ A α [E δ (w)] Let {f n } be an arbitrary bounded sequence in A p α() such that f n 0 uniformly on compact subsets of. Since the embedding A p α() L p (dω α,p ) is compact, we can pick a subsequence {f nj } that converges (necessarily to 0) in L p (dω α,p ). So, we obtain by (3.2) ( f nj (φ) p + f nj (ψ) p )σ p da α = f nj p dω α,p 0 and thus II(f nj ) 0. Also, since ω α,p is locally finite, it is not hard to see I(f nj ) 0 by (3.2) and (3.5). Accordingly, we have T f nj 0 in A p α(). This actually implies T f n 0 in A p α(), as one may easily check. This completes the proof.

13 DIFFERENCES OF COMPOSITION OPERATORS 3 4. Other Characterizations: Two Special Cases In this section we obtain alternate characterizations not involving Carleson measures, in the case of two special types of inducing maps. The first case concerns inducing maps whose images are relatively compact. Theorem 4.. Let α > and 0 < p <. Let φ, ψ S and assume that their images are relatively compact in. Then the following statements are equivalent: (a) C φ C ψ is compact on A p α(); (b) C φ C ψ is bounded on A p α(); (c) φ ψ A p α(); (d) σ L p (da α ). One may use Theorem 3.3 to prove Theorem 4.. proof. ere, we provide a direct Proof. Put T := C φ C ψ. Let K be a convex compact set containing φ() ψ(). The implication (a) = (b) is clear. Also, the equivalence (c) (d) is clear, because the pseudohyperbolic distance is equivalent to the Euclidean distance on a compact set. We now prove the implications (b) = (c) = (a). First, we assume (b) and prove (c). Pick s > (α + 2)/p and put f N (z) := τ s Ni(z) = (z + Ni) s Ap α() where N > 0 is a large number to be fixed in a moment. Note z w w+ni 0 uniformly in z, w K as N. Thus, fixing N > 0 sufficiently large, we have f N(w) f N (z) s 2 z + Ni w + Ni = s z w (4.) 2 w + Ni for all z, w K. It follows that T f N = f N (φ) f N(ψ) f N (φ) s 2 f N(φ) φ ψ ψ + Ni C φ ψ on for some constant C = C(K, N, s) > 0. So, we conclude (c). Next, we assume (c) and show (a). Note ( ) T g sup g K φ ψ on for all g A p α(). Thus, given a bounded sequence {f n } converging to 0 uniformly on compact subsets of, we have ( ) T f n p A = f p α n (ϕ) f n (ψ) p da α sup f n p φ ψ p da α. K It follows that T f n p A 0, because {f p α n} also converges to 0 uniformly on compact sets. Thus T is compact on A p α() by Lemma 2.. This completes the proof. The second special case concerns the characterization of compactness when the operator in question is already known to be bounded on a certain smaller space. Such a characterization is motivated by the necessary conditions described in Theorem 4.3 below. We need the following lemma.

14 4 B. COE,. KOO, AND W. SMIT Lemma 4.2. Given s > 0, there is a constant C = C(s) > 0 such that ( ) s z z Cρ(z, w) w z for z, w with Im z Im w. Proof. Let s > 0. Note ρ(z, w) = z z w z for z, w. Also, note that there is δ 0 (0, ) such that ζ s s ζ 2 for ζ C with ζ δ 0. Thus the inequality in question holds for any z, w with ρ(z, w) δ 0. Now, assume ρ(z, w) > δ 0 and Im z Im w. Since (Im z) 2 (Im z)(im w), we have ( ) s z z s z z w z w z [ ] s/2 4(Im z)(im w) w z 2 = [ ρ 2 (z, w) ] s/2 Now, since t s/2 t for 0 t <, we obtain ( ) s z z ρ 2 (z, w) > δ 0 ρ(z, w), w z which completes the proof. by (2.2). Theorem 4.3. Let α > and 0 < p <. For φ, ψ S the following statements hold: (a) If C φ C ψ is bounded on A p α(), then [ Im z Im φ(z) + Im z ] (α+2)/p (4.2) σ(z) < ; Im ψ(z) sup z (b) If C φ C ψ is compact on A p α(), then [ Im z Im φ(z) + Im z ] (α+2)/p (4.3) σ(z) = 0. Im ψ(z) lim z Ĥ Proof. Before proceeding, we first introduce some temporary notation. Pick a number s > (α + 2)/p and put τ s a f a := τa s A p α for a ; recall that τ a denotes the function defined in (2.20). Put T := C φ C ψ again for brevity. To prove (a), it is sufficient to show (4.4) T f φ(z) p A p α + T f ψ(z) p A p α C [ Im z Im φ(z) + Im z ] α+2 σ p (z) Im ψ(z)

15 DIFFERENCES OF COMPOSITION OPERATORS 5 for all z and for some constant C = C(p, α) > 0. To see this let z. Note from (2.8) that T f φ(z) p A f p α φ(z) (φ) f φ(z) (ψ) p da α E δ (z) (Im z) α+2 fφ(z) ( φ(z) ) fφ(z) ( ψ(z) ) p. Meanwhile, we have ( ) ( ) ( ) fφ(z) φ(z) fφ(z) ψ(z) = fφ(z) φ(z) f ( ) φ(z) ψ(z) ( ) f φ(z) φ(z) ( ) s [Im φ(z)] (α+2)/p φ(z) φ(z) ψ(z) φ(z) ; the last estimate comes from (2.2). It follows that [ ] ( ) α+2 T f φ(z) p Im z s A p α Im φ(z) φ(z) φ(z) p (4.5) ; ψ(z) φ(z) the same estimate holds when the roles of φ and ψ are exchanged. Now, assuming Im φ(z) Im ψ(z) by symmetry, we obtain by (4.5) and Lemma 4.2 that [ ] α+2 T f φ(z) p Im z (4.6) A σ p (z); p α Im φ(z) the constant suppressed above is independent of z. This yields (4.4), because Im φ(z) Im ψ(z). We now turn to the proof of (b). Assume T is compact on A p α() but (4.3) fails. Then we can find a sequence {z k } such that [ Im zk z k Ĥ but inf k Im φ(z k ) + Im z ] α+2 k σ p (z k ) > 0. Im ψ(z k ) By passing to a subsequence if necessary, we may assume for all k and thus Im ψ(z k ) Im φ(z k ) [ ] α+2 Im zk inf σ p (z k ) > 0. k Im φ(z k ) It follows from (4.6) and Lemma 2. that φ(z k ) Ĥ. So, by passing to another subsequence if necessary, we may further assume {φ(z k )} converges to some point, say b, in. Assuming so, we have (4.7) Note from (2.0) (4.8) inf k (Im z k) α+2 σ p (z k ) > 0. (Im z k ) α+2 T f b (z k ) p 0 as k. Since inf k (Im z k ) > 0 from (4.7), this implies T f b (z k ) 0, which in turn implies ( )s b b. ψ(z k ) b

16 6 B. COE,. KOO, AND W. SMIT Moreover, this holds for any s > (α + 2)/p, because the choices of s and {z k } are independent. Accordingly, we have ψ(z k ) b. It follows that and therefore φ(z k) b ψ(z k ) b = ψ(z k) ϕ(z k ) 0 ψ(z k ) b ( )s T f b (z k ) = τb s A p ϕ(z α k) b s ϕ(zk ) b ψ(z k ) b (Im b)s (α+2)/p ψ(z k ) ϕ(z k ) (Im b) s ψ(z k ) b σ(z k ) ψ(z k ) ϕ(z k ) = (Im b) (α+2)/p ψ(z k ) b σ(z k ) (Im b) (α+2)/p for all large k. This, together with (4.8), is a contradiction to (4.7). The proof is complete. One may suspect that the necessary conditions in Theorem 4.3 would also be sufficient. The next example shows that it is not the case. Example 4.4. Let α > and 0 < p <. Let φ(z) = 2i and ψ(z) = 2i + (z + i) (α+2)/p [log(z + ei)]. /p Then (4.3) holds but C φ C ψ is bounded/compact on A q α() if and only if q > p. Proof. Note ψ(z) 2i for all z, because z + i and log(z + ei) log z + ei. Thus ψ is a holomorphic self-map of whose image is relatively compact. It follows that [ Im z Im φ(z) + Im z Im ψ(z) ] α+2 σ p (z) ( Im z ) α+2 z + i log(z + ei) which easily yields (4.3). On the other hand, we see by Theorem 4. that C φ C ψ is bounded/compact on A q α() if and only if ( ) q/p z + i α+2 da α (z) <, log(z + ei) which, in turn, is equivalent to q > p. The above example can be generalized a little bit by Theorem 4.. Namely, for any φ whose image is relatively compact, one may take ψ of the form c ψ(z) := φ(z) + (z + i) (α+2)/p [log(z + ei)] /p where c > 0 is a sufficiently small constant. We also remark that there are inducing maps fixing, in addition to having the properties stated in Example 4.4; see the remark after Example 7.9. When C φ C ψ is already known to be bounded on a smaller space, it turns out that the necessary conditions in Theorem 4.3 are also sufficient on a certain

17 DIFFERENCES OF COMPOSITION OPERATORS 7 larger space. In connection with this remark, we note A q β () Ap α() under the parameter conditions given below; this can be verified via (2.9) and the factorization f p = f q f p q. Theorem 4.5. Let α > β > and 0 < q < p < with α+2 q. Let φ, ψ S and assume that C φ C ψ is bounded on A q β (). Then the following statements hold: (a) C φ C ψ is bounded on A p α() if and only if (4.2) holds; (b) C φ C ψ is compact on A p α() if and only if (4.3) holds. p = β+2 Proof. Fix δ (0, ). Note ω β,q δ < by assumption. By Theorem 4.3 we only need to prove the sufficiency. We provide details for the compactness part; the boundedness part is simpler. We now turn to the proof of the sufficiency. So, assume (4.3). Using the notation as in Theorem 3.3, it suffices to show that ω α,p is a compact α-carleson measure. Since C φ C ψ is bounded on A q α+2 β () and p = β+2 q, we see from (a) that C φ C ψ is bounded on A p α() and thus that ω α,p is an α-carleson measure by Theorem 3.3. So, it suffices to show ω α,p [E δ (z)] (4.9) lim = 0. z Ĥ A α [E δ (z)] Let ɛ > 0. By (4.3) there is a compact set K such that [ Im w Im φ(w) + Im w ] (α+2)/p (4.0) σ(w) ɛ Im ψ(w) for w \ K. Put V := w φ(k) E δ (w). Since φ(k) is compact, we see that V is a compact subset of. Note that, for z / V, we have E δ (z) φ(k) = and thus φ [E δ (z)] K =. Accordingly, (4.0) holds when w φ [E δ (z)] and z V. Assume z \ V. For w φ [E δ (z)], note and by (4.0) σ p (w)(im w) α = σ q (w)(im w) β σ p q (w)(im w) α β σ p q (w)(im w) α β ɛ p q ( Im φ(w) Im w = ɛ p q (Im φ(w)) α β ) (α+2)(p q)/p (Im w) α β ɛ p q (Im z) α β by (2.6); the equality above comes from α+2 q. Consequently, we obtain by (2.6) σ p ɛ p q (w) da α (w) σ q (w) da β (w); A α [E δ (z)] A β [E δ (z)] φ [E δ (z)] p = β+2 φ [E δ (z)] the constant suppressed here is independent of z \ V. The same argument also works for ψ instead of φ. Consequently, we obtain ω α,p [E δ (z)] A α [E δ (z)] ɛ p q ω β,q δ, z \ V

18 8 B. COE,. KOO, AND W. SMIT and thus conclude (4.9), as required. The proof is complete. An immediate consequence of Theorem 4.3 (and Theorem 4.5) is the following: Corollary 4.6. Let α > β > and 0 < q < p < with α+2 q. Let φ, ψ S. If C φ C ψ is bounded(compact, resp.) on A q β (), then it is bounded(compact) on A p α(). Im z Im φ(z) and Im z Im ψ(z) p = β+2 When are already bounded, it is elementary to see that condition (4.3) is independent of the parameters α and p. So, as another consequence, we obtain the next corollary. Corollary 4.7. Let α > and 0 < p <. Let φ, ψ S and assume that C φ and C ψ are both bounded on A p α(). Then C φ C ψ is compact on A p α() if and only if (4.) [ Im z lim z Ĥ Im φ(z) + Im z ] σ(z) = 0. Im ψ(z) Of course, condition (4.) is not sufficient without the boundedness of C φ and C ψ ; see Example 4.4 (with α = 0 and p = 2). We now close the section with another example with inducing maps whose images are not relatively compact in. Example 4.8. For s > 2, let φ(z) := 2(z + i) and ψ(z) := 2(z + i) +. (z+i) s Then (4.) holds but C φ C ψ is not even bounded on A p α() for p α+2 s+2. Proof. Note Im φ(z) = Im z + 2 z + i 2 and 2Im z Im (z + i) s 2Im z + Im ψ(z) = 2(z + i) + 2 (z+i) 2(z + i) + 2. s (z+i) s Thus φ and ψ are holomorphic self-maps of. Note ( σ(z) = ρ φ(z), ) = ψ(z) z + i s (z+i) + 4i(Im z + ) ; s the first equality holds by the Möbius invariance of the pseudohyperbolic distance. Thus (4.) is easily verified, because s > 2. Now, given α > and 0 < p <, put f := τ (3+α)/p i A p α(). Since f is one-to-one in a small closed disk D centered at the origin, the function F (z, w) := { f(z) f(w) z w f (z) if z w if z = w is continuous and zero-free on D D. Thus F has a positive minimum, say m, on D D. It follows that (4.2) f(z) f(w) m z w, z, w D.

19 DIFFERENCES OF COMPOSITION OPERATORS 9 Note φ(z) 0 and ψ(z) 0 as z. Thus, we see from (4.2) that (C φ C ψ )f(z) m φ(z) ψ(z) z + i 2+s for all z with z sufficiently large. Thus, for p α+2 2+s, we conclude (C φ C ψ )f / A p α() by Lemma 2.5 and thus that C φ C ψ cannot be bounded on A p α(). 5. Compactness and Angular Derivatives In this section we investigate compactness by means of angular derivatives. We first fix some notation and terminology. Given φ S, we denote its angular derivative set by F (φ) := {x Ĥ : φ (x) exists} and put (5.) D[φ, x] := ( φ(x), φ (x) ) Ĥ C 0, x F (φ) where C 0 = C\{0}; recall φ (x) 0. Naturally, we refer to D[φ, x] as the first-order data of φ at x. In the settings of the classical bounded domains such as the disk, the ball and the polydisk, it is known that a difference of composition operators cannot be compact unless the inducing maps have exactly the same first-order data; such necessary condition is first observed in [5, Theorem 5.2] on the disk and then extended to the polydisk [3, Corollary 5.9] and to the ball [4, Corollary 3.7]. To our surprise, the coincidence of the first-order data is no longer necessary in general on the half-plane, as the next example shows. Example 5.. Given a 0 with a, let φ, ψ S be given by φ(z) := z + i z and ψ(z) := z + i a z. Then C φ C ψ is compact on any A p α(), but φ and ψ do not have the same firstorder data at 0. Proof. Note that φ(0) = and φ (0) =. On the other hand, we have (a) ψ(0) = i and 0 / F (ψ) if a = 0; (b) ψ(0) = and ψ (0) = /a if a > 0 (and a ). Thus, in either case φ and ψ do not have the same first-order data at 0. Note that C φ and C ψ are both bounded on any A p α() by Theorem 2.4. Thus, to prove the compactness of C φ C ψ on any A p α(), we may assume p = 2 and α = 0 by Corollary 4.7. Now, by Theorem 3.3, it suffices to show that the embedding A 2 0() L 2 (dω) is compact where ω := ω 0,2;φ,ψ is the joint pullback measure induced by φ and ψ. So, given a weak null-sequence {f n } in A 2 0(), we need to show f n 0 in L 2 (dω), or more explicitly by (3.2), (5.2) I n := ( f n (φ) 2 + f n (ψ) 2 )σ 2 da 0. In conjunction with this we note by the uniform boundedness principle and the standard normal family argument that any weak null-sequence in A 2 0() is normbounded and converges to 0 uniformly on each compact subset of. Put M := sup n f n A 2 0 () <.

20 20 B. COE,. KOO, AND W. SMIT We now prove (5.2) for the case a > 0; the case a = 0 is a bit simpler. Given R > > ɛ > 0, put U ɛ := {z : z < ɛ}, V R := {z : z > R} and decompose the defining integral of I n into the sum of three pieces I n = ( f n (φ) 2 + f n (ψ) 2 )σ 2 da V R + U ɛ + \(U ɛ V R ) = : I n, + I n,2 + I n,3. First, we have I n,3 0, because φ and ψ map \(U ɛ V R ) onto relatively compact sets in. Next, to estimate the first integral, note φ ψ Im φ + Im ψ 2 and thus σ(z) 2 a z where c = a 2 > 0. It follows that c R, z V R I n, c2 M 2 R ( C φ 2 + C ψ 2 ) for all n where denotes the operator norm on A 2 0(). Finally, for the second integral, setting ψ (w) := ψ( w ) = aw w +i, we have by an elementary change of variables I n,2 ( f n (φ) 2 + f n (ψ) 2 ) da U ɛ = ( f n (φ(w)) 2 + f n (ψ (w)) 2 ) da(w) V /ɛ w 4 ɛ 4 M 2 ( C φ 2 + C ψ 2 ) for all n; note that C ψ is also bounded on A 2 0() by Theorem 2.4. Combining the observations in the preceding paragraph, we obtain ( lim sup f n 2 dω CM 2 ɛ 4 + ) n R for some constant C = C(φ, ψ) > 0. Since this holds for arbitrary 0 < ɛ < and R >, we conclude (5.2), as required. In view of Example 5., we now proceed to the investigation of a subclass of S for which the coincidence of first-order data is necessary for compact difference. Note that one of the inducing maps in Example 5. maps a finite boundary point to and has finite angular derivative at that boundary point. Such a property will turn out to be the main cause responsible for the pathology demonstrated in Example 5.; see Theorem 5.4 below. We first introduce some notation needed for a couple of technical lemmas. Given ɛ > 0 and x, let R ɛ,x be the ray emanating from x with slope ɛ, i.e., For x =, we put R ɛ,x := {z : Im z = ɛre (z x)}. R ɛ, := {z : Im z = ɛre z}. Note that R ɛ,0 is mapped onto R ɛ,, and vice versa, by the automorphism z. Clearly, each R ɛ,x is a nontangential curve having x as one of the end points.

21 DIFFERENCES OF COMPOSITION OPERATORS 2 Lemma 5.2. Let φ, ψ S. Then (5.3) lim lim ɛ 0 + z x z R ɛ,x for x F (φ) F (ψ). φ(z) φ(z) φ(z) ψ(z) = { if D[ϕ, x] = D[ψ, x] 0 otherwise Proof. Let x F (φ) F (ψ). Clearly, the z-limit in (5.3) is 0 when φ(x) ψ(x). So, assume φ(x) = ψ(x) for the rest of the proof. First, assume x φ(x). Since x F (φ), we have by (2.9) and (2.5) (5.4) Im z lim z x Im φ(z) = φ (x) and the same holds with ψ in place of φ. Note Also, for any ɛ > 0, note and φ(z) ψ(z) φ(z) φ(z) = Im z φ(z) ψ(z) Im φ(z) z z = Im z Im φ(z) z x φ(z) ψ(z) Im ψ(z) + z z z x Im φ(z). z x z z = + ɛi 2ɛi for z R ɛ,x φ(z) ψ(z) lim = φ (x) ψ (x). z x z x We see from these and (5.4) that the z-limit in (5.3) is equal to { + ɛi 2ɛi φ (x) ψ (x) φ (x) + ψ (x) φ (x) }. Now, taking the limit ɛ 0 +, we conclude (5.3). Next, assume x = φ(x). In this case we consider the functions φ (z) := /φ(z) and ψ (z) := /ψ(z). Clearly, φ (x) = ψ (x) = 0. Moreover, it is easily verified that x F (φ ) F (ψ ) with φ (x) = φ (x) and ψ (x) = ψ (x). Also, since ψ(z) lim z x φ(z) = lim (z x)ψ(z) z x (z x)φ(z) = φ (x) ψ (x), we note φ (z) φ (z) lim z x z R ɛ,x φ (z) ψ (z) = lim φ(z) φ(z) z x z R ɛ,x φ(z) ψ(z) ψ(z) φ(z) = φ (x) φ(z) φ(z) lim ψ z x (x) z R ɛ,x φ(z) ψ(z) for any ɛ > 0. So, applying the preceding case to φ and ψ, we obtain (5.3). Finally, when x =, one may apply the previous cases to the functions φ 2 (z) := φ( /z) and ψ 2 (z) := ψ( /z). The proof is complete. In the next lemma the restriction φ(x) cannot be removed. For example, consider φ(z) = z, ψ(z) = i and x = 0.

22 22 B. COE,. KOO, AND W. SMIT Lemma 5.3. Let φ, ψ S. Then for x F (φ) \ F (ψ) with φ(x). lim z x φ(z) φ(z) φ(z) ψ(z) = 0 Proof. By a straightforward calculation we have φ κ (ζ) 2 φ κ (ζ)ψ κ (ζ) = ψ(z) i φ(z) φ(z) φ(z) i φ(z) ψ(z) where ζ = κ (z). Thus, since ψ(z) + i, we have φ(z) φ(z) φ(z) ψ(z) φ(z) + i φ κ(ζ) 2 ζ 2 ζ 2 ψ κ (ζ). Now, for x F (φ)\f (ψ) with φ(x), note by the Julia-Carathéodory Theorem that the right-hand side of the above tends to φ(x) + i φ κ( x) 0 = 0 where x = κ (x) as z x nontangentially. This completes the proof. Put S f := {φ S : φ(x) for all x F (φ) }. We are now ready to prove that compactness implies the coincidence of first-order data, when the inducing maps are confined to the above subclass of S. Theorem 5.4. Let φ, ψ S f and assume that C φ C ψ is compact on some A p α(). Then the following statements hold: (a) F (φ) = F (ψ) =: F ; (b) φ and ψ have the same first-order data at every point of F. Proof. In the proof below we continue using the notation introduced in the proof of Theorem 4.3. To prove (a) it is sufficient to show F (φ) F (ψ) by symmetry. Let x F (φ). Since φ S f, we have either φ(x) or x = = φ( ). In the latter case C φ is bounded on A p α() by Theorem 2.4 and so is C ψ. Thus F (ψ) again by Theorem 2.4. So, assume φ(x). Since φ has nontangential limit φ(x), we have φ(z) as z x along any R ɛ,x. So, given ɛ > 0, we have by (2.22) and thus by (4.5) (5.5) lim z x z R ɛ,x lim T fφ(z) A p = 0 z x α z R ɛ,x [ ] ( (α+2)/p Im z Im φ(z) Meanwhile, since φ(x), note from (2.9) that ) s φ(z) φ(z) ψ(z) φ(z) = 0. Im z lim z x Im φ(z) = x + i 2 φ κ( x) φ(x) + i 2 > 0. It follows from (5.5) (because s > (α + 2)/p is arbitrary) that (5.6) φ(z) φ(z) lim z x z R ɛ,x ψ(z) φ(z) =.

23 DIFFERENCES OF COMPOSITION OPERATORS 23 Thus, we conclude x F (ψ) by Lemma 5.3, as required. Note that (b) is a consequence of (a), (5.6) and Lemma 5.2. The proof is complete. Now, under the additional assumption of boundedness, we characterize compact differences of composition operators with inducing maps in S f by means of angular derivative sets. Theorem 5.5. Let α > and 0 < p <. Let φ, ψ S f and assume that C φ and C ψ are bounded on A p α(). Then the following statements are equivalent: (a) C φ C ψ is compact on A p α(); (b) F (φ) = F (ψ) =: F and (5.7) for all x F. lim z x [ Im z Im φ(z) + Im z ] σ(z) = 0 Im ψ(z) Proof. The implication (a) = (b) is immediate from Corollary 4.7 and Theorem 5.4. Conversely, assume (b). Let y / F. Note y by Theorem 2.4, because C φ and C ψ are bounded on A p α(). Also, note from Proposition 2.3 so that 0 = lim z y φ(z) + i z + i 2 Im z Im φ(z) Im z lim sup y + i 2 z y Im φ(z) Im z lim z y Im φ(z) = 0; the same is true with ψ in place of φ. It follows that lim z y [ Im z Im φ(z) + Im z Im ψ(z) ] σ(z) = 0 for each y F. This, together with (5.7), implies (4.). So, we conclude (a) by Corollary Some Consequences In this section we notice some consequences of our results on boundedness. We begin with a lemma. Lemma 6.. Let φ, ψ S, x and γ : [0, ] be a continuous curve such that Im γ(t) > 0 if t [0, ) and γ() = x. Assume that (6.) lim sup σ(γ(t)) < and lim Im φ(γ(t)) = 0. t t Then φ(γ(t)) ψ(γ(t)) 0 as t. Proof. By the first assumption in (6.) there is some 0 < t 0 < such that and thus we obtain by (2.6) sup σ((γ(t)) < t 0<t< φ(γ(t)) ψ(γ(t)) sup. t 0<t< Im φ(γ(t))

24 24 B. COE,. KOO, AND W. SMIT Now, by this and the second assumption in (6.) we conclude that completing the proof. lim φ(γ(t)) ψ(γ(t)) = 0, t Recall that Aut(), the set of all Möbius transformations from onto, consists of functions φ of the form (6.2) φ(z) = az + b cz + d where a, b, c, d are real and ad bc > 0. Clearly, automorphisms fixing must be linear polynomials and thus induce bounded composition operators. On the other hand, other automorphisms turn out to behave quite rigidly in the sense of the next theorem. Theorem 6.2. Let φ Aut() with φ( ) and ψ S. If C φ C ψ is bounded on some A p α(), then φ = ψ. Proof. Let φ be a function as in (6.2). Note c 0, because φ( ). Note Im φ(z) = (ad bc) Im z cz+d. So, assuming that C 2 φ C ψ is bounded on A p α(), we have by Theorem 4.3(a) sup z + d 2(α+2)/p c σ(z) <. z In particular, for any x > 0 sufficiently large, we have sup σ(z) <. Re z=x Note lim y 0 + Im φ(x + iy) = 0 for any x d c. It follows from Lemma 6. that φ ψ has vanishing boundary values (= radial limit) on a set of positive measure. Thus we conclude ψ = φ on. This completes the proof. The rigidity phenomenon as in Theorem 6.2 does not extend to univalent symbols, as the next proposition shows. Proposition 6.3. There exist distinct univalent φ, ψ S such that C φ and C ψ are both bounded on A p α(), and C φ C ψ is compact on A p α() for any α > and 0 < p <. Proof. Recall that D denotes the unit disk of the complex plane. Let Q D be a simply connected domain with smooth boundary such that Q D = {} and makes order β-contact with D, where < β < 2. Let η be a Riemann map from D onto Q with (radial limit) η() =. It follows from < β that η has (positive) finite angular derivative at ; see [2, p. 72]. Now, choose φ S so that φ κ = η where φ κ is as in (2.4). Then φ is a univalent mapping of onto κ(q) =: Ω, with the properties (i) φ( ) = and φ has finite angular derivative at ; (ii) lim Im w = ; w w Ω (iii) inf w Ω Im w > 0.

25 DIFFERENCES OF COMPOSITION OPERATORS 25 Indeed, property (i) is immediate from the relationship of φ to η; see Section 2.5. To see (ii), let be a disk contained in D with intersecting D at. Since Q makes order β < 2 contact with D, there is a small neighborhood U of such that U Q. Mapping by κ to, we see that there is a neighborhood V := κ(u) of such that V Ω κ( ). By varying, we can get κ( ) to be any half-plane {Im w > t}, 0 < t <, and this establishes (ii). Last, (iii) follows from (ii) and the property Q D = {}. Now define ψ(z) := φ(z) + i, so that ψ also satisfies (i). Thus C φ and C ψ are both bounded on any A p α() by Theorem 2.4. To see that C φ C ψ is compact on A p α(), by Corollary 4.7 it suffices to show that (4.) holds. From (iii) we see that lim Im z 0 + [ Im z Im φ(z) + Im z Im ψ(z) ] = 0. ence (4.) holds as z, since σ(z). Next, as z in, φ(z) and by (ii) Im φ(z). ence φ(z) (i + φ(z)) lim σ(z) = lim = lim z z φ(z) ψ(z) z + 2Im φ(z) = 0. Also, from (i) and Proposition 2.2(b), we have [ Im z Im φ(z) + sup z Im z Im ψ(z) ] <, and hence [ Im z lim z Im φ(z) + Im z ] σ(z) = 0. Im ψ(z) This completes the proof that (4.) holds as z C φ C ψ is compact on A p α(). Ĥ, and thus the proof that In general we observe the following. Proposition 6.4. Let φ, ψ S and assume that C φ C ψ is bounded on some A p α(). Assume φ( ). Then there is a sequence {z n } such that z n nontangentially and φ(z n ) ψ(z n ) 0 as n. Proof. Since φ( ), there is a sequence {z n } such that z n nontangentially and {φ(z n )} stays bounded. So, passing to a subsequence if needed, we may assume φ(z n ) a for some a. Note that Im z n, because z n nontangentially. It follows that Im z n Im φ(z n). So, we have (6.3) σ(z n ) 0 by Theorem 4.3(). This clearly implies ψ(z n ) a if a. On the other hand, if a, then Im φ(z n ) Im a = 0. So, we conclude ψ(z n ) a by Lemma 6.. We remark that φ( ) is essential in the above proposition. For example, consider the bounded difference C z C 2z.

26 26 B. COE,. KOO, AND W. SMIT 7. ilbert-schmidt Differences In this section we characterize ilbert-schmidt differences of composition operators on A 2 α() by means of square-integrability (with respect to the Möbiusinvariant measure) of the quantity [ Im z Im φ(z) + Im z ] (α+2)/2 σ(z), Im ψ(z) which already appeared in the previous sections. Recall that a bounded linear operator T on a separable ilbert space X with an orthonormal basis {e β } is called ilbert-schmidt if the norm T S(X) := T e β 2 is finite. As is well known, the value (possibly ) of the sum on the right-hand side of the above does not depend on the choice of orthonormal basis. It is well known that every ilbert-schmidt operator is compact; see [24, Section 6.2] for details. Throughout the section the parameters α > and p = 2 are fixed. So, we simplify notation by writing We have (7.) β /2 K z := K (α) z and K z α := K z A 2 α. C φ C ψ 2 S(A 2 α ) = K φ(z) K ψ(z) 2 α da α (z), for φ, ψ S; this is proved in [2, Proposition 3.] (in fact for general linear combinations) over the disk and the proof remains precisely the same over the half-plane. Motivated by (7.), we will establish the optimal norm estimate K z K w α K z α + K w α ρ(z, w) for z, w. Based on the identity (2.2), this estimate was first observed over the disk in [2] and then extended to the polydisk in [3]. One may modify the earlier proofs to fit to the setting of the half-plane. Details are included below for readers convenience. We need Lemma 7.3 below, whose disk and polydisk analogues were the key tools in [2] and [3]. We need a couple of preliminary lemmas. Lemma 7.. Given s > 0, the inequality z s w s 8s ρ(z, w) ρ(z, w) holds for z, w. Proof. Fix s > 0 and let z, w. Note z s = s(z w) ws 0 ( Im z + ) s Im w dt [tz + ( t)w] s+.

27 Also, note Im ξ Im z + Im w DIFFERENCES OF COMPOSITION OPERATORS 27 for any ξ on the line segment connecting z and w. Thus we have z s ( w s s z w Im z + ) s+ Im w ( = sρ(z, w) z w Im z + Im w ρ(z, w) 4s (Im z) ρ(z, w) ) s+ ( Im z + Im w ) s+ ; the last inequality comes from (2.6). By symmetry we obtain z s ( w s 4s ρ(z, w) min(im z, Im w) ρ(z, w) Im z + ) s+, Im w which implies the asserted inequality. Lemma 7.2. Given s real and 0 < δ <, there is a constant C = C(s, δ) > 0 such that ( ) s z a w a Cρ(z, w) for a, z, w with ρ(z, w) < δ. Proof. By symmetry we may assume s > 0. Note that ξ s / ξ stays bounded as ξ. Thus we have { ξ s M := sup ξ : δ + δ < ξ < + δ }, ξ / R < δ where R denotes the set of all negative real numbers. Now, given a, z, w with ρ(z, w) < δ, we have by (2.5) ( ) s z a w a M z w w w a Mρ(z, w) z Im w. So, we conclude the lemma by (2.6). As a consequence of Lemma 7.2 we see that [( ) s ] z a (7.2) Re > 0 w a for a, z, w with ρ(z, w) < δ when δ = δ(s) is sufficiently small. Using this and Lemma 7., we now prove the next lemma. Lemma 7.3. Given s > 0, there is a constant C = C(s) > 0 such that ( ) s i z w s Re ρ 2 (z, w) C z w (Im z) s/2 (Im w) s/2 for z, w. Proof. Fix s > 0 and let z, w. Since z w Im z + Im w, the inequality in question is elementary when ρ(z, w) stays away from 0. So, we only need to consider the case when ρ(z, w) is small. Fix a sufficiently small δ = δ(s) > 0 for which (7.2) holds.