Cent. Eur. J. Math. • 12(5) • 2014 • 761-777 DOI: 10.2478/s11533-013-0377-7
Central European Journal of Mathematics
Boundary vs. interior conditions associated with weighted composition operators Research Article Kei Ji Izuchi1∗ , Yuko Izuchi2† , Shûichi Ohno3‡
1 Department of Mathematics, Niigata University, Niigata 950-2181, Japan 2 Aoyama-shinmachi 18-6-301, Niigata 950-2006, Japan 3 Nippon Institute of Technology, Miyashiro, Minami-Saitama 345-8501, Japan
Received 16 January 2013; accepted 20 August 2013 Abstract: Associated with some properties of weighted composition operators on the spaces of bounded harmonic and analytic functions on the open unit disk D, we obtain conditions in terms of behavior of weight functions and analytic self-maps on the interior D and on the boundary ∂D respectively. We give direct proofs of the equivalence of these interior and boundary conditions. Furthermore we give another proof of the estimate for the essential norm of the difference of weighted composition operators. MSC:
47B38, 30H10
Keywords: Weighted composition operator • The space of bounded harmonic functions • The space of bounded analytic functions • Essential norm
© Versita Sp. z o.o.
1.
Introduction
Let h∞ and H ∞ be the spaces of bounded harmonic and analytic functions on the open unit disk D, respectively. For f ∈ h∞ , we write kfk∞ = supz∈D |f(z)|. We may define f ∗ (eiθ ) = limr→1 f(reiθ ) a.e. on the boundary ∂D of D. It is known ∗ † ‡
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Boundary vs. interior conditions associated with weighted composition operators
that {f ∗ : f ∈ h∞ } = L∞ (∂D). We denote by kf ∗ k∂D the essential supremum norm of f ∗ on ∂D, then kf ∗ k∂D = kfk∞ . For b be the harmonic extension of u defined by u ∈ L∞ (∂D), let u Z b(z) = u
∂D
u(eiθ )Pz (eiθ ) dm(eiθ ),
z ∈ D,
b ∈ h∞ . where Pz is the Poisson kernel for the point z ∈ D and m is the normalized Lebesgue measure on ∂D. Then u Besides, for f ∈ h∞ , fb∗ = f on D. Let S(D) be the set of analytic self-maps of D. For each φ ∈ S(D), we may define the composition operator Cφ by Cφ f = f ◦ φ, where f is analytic on D. Composition operators on various analytic function spaces have been studied over the past four decades. See [4, 18] for an overview of these results. One of the recent main subjects is the study of differences of two (weighted) composition operators, which was originally posed by Shapiro and Sundberg [19] to investigate the topological structure of the set of composition operators on the classical Hilbert–Hardy space. To determine whether two composition operators would lie in the same component in the operator norm topology, one must estimate the operator norm of the difference of two composition operators. For a bounded linear operator T on a Banach space X , let kT kX denote the norm of T and kT kX ,e denote the essential norm of T , that is, kT kX ,e = inf kT − K kX : K is compact operator on X . For z, w ∈ D, let
z−w . ρ(z, w) = 1 − wz
On H ∞ , in [14] MacCluer, Ohno and Zhao showed that kCφ − Cψ kH ∞ = sup z∈D
2 1−
p 1 − ρ2 (φ(z), ψ(z)) , ρ(φ(z), ψ(z))
for φ, ψ ∈ S(D) and obtained a characterization of compactness of Cφ − Cψ in terms of behavior of φ and ψ on the open unit disk D. In [8], Hosokawa and Izuchi gave lower and upper estimates of the essential norm kCφ − Cψ kH ∞ ,e . Let u ∈ H ∞ and φ ∈ S(D). In [9], Hosokawa, Izuchi and Ohno considered operators Mu Cφ on H ∞ defined as Mu Cφ f = u(f ◦ φ), f ∈ H ∞ . They gave a characterization of compactness of Mu Cφ − Mv Cψ in terms of certain conditions on functions u, v, φ, ψ on the open unit disk D. We refer to [1, 5, 13, 15] for results on other analytic function spaces. For f ∈ h∞ it is known that Cφ f = f ◦ φ is also harmonic on D. Choa, Izuchi and Ohno introduced in [2] weighted composition operators Mu Cφ on h∞ defined by Z (Mu Cφ f)(z) =
∂D
u(eiθ )(f ◦ φ)∗ (eiθ )Pz (eiθ ) dm(eiθ )
for u ∈ L∞ (∂D), φ ∈ S(D) and z ∈ D. They characterized compactness of Cφ − Cψ on h∞ in terms of certain conditions on φ and ψ on the open unit disk D (see also [16]). Moreover, they gave a characterization of compactness of Mu Cφ and determined the value of the essential norm kCφ − Cψ kh∞ ,e in terms of behavior of φ and ψ on the maximal ideal space of H ∞ . Let u, v ∈ L∞ (∂D) and φ, ψ ∈ S(D). In [10], the authors characterized compactness of Mu Cφ and differences Mu Cφ −Mv Cψ , and provided formulas for operator and essential norms of differences Mu Cφ − Mv Cψ in terms of certain conditions on functions u, v, φ, ψ on the boundary ∂D. Let C(h∞ ) and C(H ∞ ) be the sets of bounded composition operators on h∞ and H ∞ respectively. Let Cw (h∞ ) and Cw (H ∞ ) be the sets of nonzero bounded weighted composition operators on h∞ and H ∞ respectively, i.e. Cw (h∞ ) = Mu Cφ : u ∈ L∞ (∂D), u 6≡ 0, φ ∈ S(D) Cw (H ∞ ) = Mu Cφ : u ∈ H ∞ , u 6≡ 0, φ ∈ S(D) .
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K.J. Izuchi, Y. Izuchi, S. Ohno
The subject of this paper is the following. Let u, v ∈ H ∞ and φ, ψ ∈ S(D). For compactness of Mu Cφ − Mv Cψ on H ∞ , the interior condition on functions u, v, φ, ψ is given in [9] and the boundary condition on them is given in [10]. So these interior and boundary conditions are equivalent. In the next section we recall properties of L∞ (∂D) functions on ∂D investigated in [10] and give a direct proof of the mentioned equivalence in Section 3. Similar situations occur for other properties of weighted composition operators on h∞ and H ∞ . We give direct proofs for the corresponding equivalences in Section 4. In Section 5, we give another proof of the estimate for the essential norm kCφ − Cψ kh∞ ,e obtained in [2] applying the results given in [10]. That is, we prove the main result (Theorem 5.2) using the behavior of φ and ψ on the open unit disk instead of on the maximal ideal space of H ∞ . We expect that the technique discussed here would be valuable in other problems concerning composition operators; e.g. using the argument developed here, the authors have determined path connected components in the spaces of noncompact weighted composition operators on h∞ and H ∞ with the operator and the essential operator norms in [11, 12], respectively.
2.
L∞ (∂D) functions
In order to study L∞ (∂D) functions, we recall some elementary facts and notations used in [10]. For f ∈ h∞ , let Rf be the set of eiθ ∈ ∂D at which f has a radial limit. We define a function f ∗ on Rf by f ∗ (eiθ ) = lim f(reiθ ), r→1
eiθ ∈ Rf .
We have m(Rf ) = 1 and kfk∞ = kf ∗ k∂D = supeiθ ∈Rf |f ∗ (eiθ )|. For u ∈ L∞ (∂D), let Lu be the Lebesgue set of u, that is, Lu is the set of points eit ∈ ∂D satisfying Z 1 m(I)→0 m(I) lim
I
|u(eiθ ) − u(eit )| dm(eiθ ) = 0,
b(reiθ ) where I are subarcs of ∂D centered at eit . By [17, p. 138], m(Lu ) = 1 and by [17, Theorem 11.23], u(eiθ ) = limr→1 u iθ for e ∈ Lu , that is, Lu ⊂ Rub . By the definition of Lu , we have the following result.
Lemma 2.1. For each eit ∈ Lu there are a sequence of open subarcs {In }n of ∂D centered at eit and a sequence of measurable subsets {En }n such that En ⊂ In ∩ Lu , m(In ) → 0, m(En )/m(In ) → 1, and lim sup |u(eiθ ) − u(eit )| = 0.
n→∞ iθ e ∈En
We note that the value of the function u is well defined on Lu and kuk∂D = sup |u(eiθ )|. eiθ ∈Lu
Lemma 2.2. For u, v ∈ L∞ (∂D), we have the following: (i) Lu ∩ Lv ⊂ Luv and kuvk∂D = supeiθ ∈Lu ∩Lv |(uv)(eiθ )|. (ii) Lu ∩ Lv ⊂ Lu−v and ku − vk∂D = supeiθ ∈Lu ∩Lv |u(eiθ ) − v(eiθ )|. (iii) Lu ∩ Lv ⊂ L|u|+|v| and k|u| + |v|k∂D = supeiθ ∈Lu ∩Lv (|u| + |v|)(eiθ ). For f ∈ h∞ , we define a function ef on D ∪ Lf ∗ by ef(z) = f(z) for z ∈ D and ef(eiθ ) = f ∗ (eiθ ) for eiθ ∈ Lf ∗ . For φ ∈ S(D), we have f ◦ φ ∈ h∞ and (f ◦ φ)∗ = ef ◦ φ∗ a.e. on ∂D (see [10]).
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Boundary vs. interior conditions associated with weighted composition operators
3.
Boundary vs. interior conditions
Let u, v ∈ L∞ (∂D) and φ, ψ ∈ S(D). Moreover we assume that φ 6= ψ. Let Ω = Lu ∩ Lv ∩ Lφ ∗ ∩ Lψ ∗ . For each constant 0 < r < 1, let {r < |φ∗ | < 1} = eiθ ∈ Lφ∗ : r < |φ∗ (eiθ )| < 1 , {|φ∗ | > r} = eiθ ∈ Lφ∗ : |φ∗ (eiθ )| > r , {|φ∗ | = 1} = eiθ ∈ Lφ∗ : |φ∗ (eiθ )| = 1 . Similarly we use {|φ∗ | ≤ r} and {|φ∗ | < 1}, etc. The result below follows from Lemma 2.1.
Lemma 3.1. If {r < |φ∗ | < 1} ∩ {r < |ψ ∗ | < 1} = 6 ∅, then m {r < |φ∗ | < 1} ∩ {r < |ψ ∗ | < 1} > 0.
Lemma 3.2 ([2, Theorem 5.1]). b(z) = 0. Let Mu Cφ ∈ Cw (h∞ ) and kφk∞ = 1. Then Mu Cφ is compact on h∞ if and only if lim|φ(z)|→1 u
Lemma 3.3 ([10, Theorem 3.2]). Let Mu Cφ ∈ Cw (h∞ ) and kφk∞ = 1. Then Mu Cφ is compact on h∞ if and only if limr→1 kuχ{|φ∗ |>r} k∂D = 0. b(z) = 0 if and only if limr→1 kuχ{|φ∗ |>r} k∂D = 0. The first one is a condition By the above lemmas, we have that lim|φ(z)|→1 u on functions u, φ on the open unit disk D and the second one is a condition on functions u, φ on the boundary ∂D. There are no direct proofs that both conditions are equivalent. We shall give a direct proof. The arguments of this type are standing viewpoints of this paper. At first we show the equivalence of conditions on functions u, φ on D and ∂D for compactness of weighted composition operators on h∞ .
Lemma 3.4. Let R φ ∈ S(D) and kφk∞ = 1. Let {zn }n be a sequence in D satisfying |φ(zn )| → 1 as n → ∞. Then the integral P (eiθ ) dm(eiθ ) → 1 as n → ∞ for each 0 < r < 1. {|φ∗ |>r} zn
Proof.
We have
Z Z |φ(zn )| = φ∗ (eiθ )Pzn (eiθ ) dm(eiθ ) ≤
{|φ∗ |>r}
∂D
Z ≤ {|φ∗ |>r}
Pzn (eiθ ) dm(eiθ ) + r
Z {|φ∗ |≤r}
Z φ∗ (eiθ )Pzn (eiθ ) dm(eiθ ) +
{|φ∗ |≤r}
φ∗ (eiθ )Pzn (eiθ ) dm(eiθ )
Pzn (eiθ ) dm(eiθ ) ≤ 1.
Since |φ(zn )| → 1 and so {zn }n has no accumulating point in {|φ∗ | ≤ r}, we have R n → ∞. So {|φ∗ |>r} Pzn (eiθ ) dm(eiθ ) → 1 as n → ∞ for each r.
R
{|φ∗ (eiθ )|≤r}
Pzn (eiθ ) dm(eiθ ) → 0 as
Theorem 3.5. b(z) = 0 if and only if limr→1 kuχ{|φ∗ |>r} k∂D = 0. For Mu Cφ ∈ Cw (h∞ ) and kφk∞ = 1, it holds that lim|φ(z)|→1 u
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K.J. Izuchi, Y. Izuchi, S. Ohno
Proof.
b(z) = 0. Suppose limr→1 kuχ{|φ∗ |>r} k∂D 6= 0. Since kuχ{|φ∗ |>r} k∂D is decreasing in r, Suppose that lim|φ(z)|→1 u there is a positive number δ such that kuχ{|φ∗ |>r} k∂D > δ for every r. Take a point eiθr ∈ Lu ∩ {|φ∗ | > r} such that |u(eiθr )| > δ. Since eiθr ∈ {|φ∗ | > r}, we have |φ∗ (eiθr )| > r. Then there exists a constant tr , 0 < tr < 1, such that |b u(tr eiθr )| > δ and |φ(tr eiθr )| > r. This contradicts the starting assumption. b(z) = 0, let {zn }n be a sequence in D satisfying Suppose that limr→1 kuχ{|φ∗ |>r} k∂D = 0. To prove that lim|φ(z)|→1 u R |φ(zn )| → 1. By Lemma 3.4, {|φ∗ |>r} Pzn (eiθ ) dm(eiθ ) → 1 as n → ∞ for each r. Take ε > 0 arbitrarily. Since R limr→1 kuχ{|φ∗ |>r} k∂D = 0, there is a constant r0 , 0 < r0 < 1, such that kuχ{|φ∗ |>r0 } k∂D < ε and {|φ∗ |≤r0 } Pzn (eiθ ) dm(eiθ ) < ε. We have Z |b u(zn )| ≤
Z u(eiθ )Pzn (eiθ ) dm(eiθ ) +
{|φ∗ |>r0 }
Z <ε
Pzn (eiθ ) dm(eiθ ) + kuk∂D
{|φ∗ |>r0 }
u(eiθ )Pzn (eiθ ) dm(eiθ )
{|φ∗ |≤r0 }
Z
Pzn (eiθ ) dm(eiθ ).
{|φ∗ |≤r0 }
b(z) = 0. Hence lim supn→∞ |b u(zn )| ≤ ε. Since ε is arbitrary, we get lim|φ(z)|→1 u Next we see the equivalence between conditions on D and on ∂D for compactness of differences Cφ −Cψ . For a function F in L1 (∂D), we denote by kF k1 the norm of F in L1 (∂D). By [7, p. 42], we have the following.
Lemma 3.6. We have kPz − Pw k1 = 2 −
4 arccos ρ(z, w), π
z, w ∈ D.
Hence kPz − Pw k1 ≤ 2ρ(z, w).
In [2, p. 172], Choa, Izuchi and Ohno pointed out that kCφ − Cψ kh∞ = sup kPφ(z) − Pψ(z) k1 . z∈D
This is the interior description of kCφ − Cψ kh∞ . As an application of [10, Theorem 4.1], we have the following boundary description; n
o kCφ − Cψ kh∞ = max 2 χ{|φ∗ |=1}∪{|ψ ∗ |=1} ∂D , sup Pφ∗ (eiθ ) − Pψ ∗ (eiθ ) 1 , where sup is taken over eiθ ∈ {|φ∗ | < 1} ∩ {|ψ ∗ | < 1}. When m {|φ∗ | = 1} ∪ {|ψ ∗ | = 1} > 0, by the boundary description, we have kCφ − Cψ kh∞ = 2. We shall show that supz∈D kPφ(z) − Pψ(z) k1 = 2. Since φ 6= ψ, φ∗ 6= ψ ∗ a.e. on {|φ∗ | = 1} ∪ {|ψ ∗ | = 1}. Take a point eiθ0 ∈ {|φ∗ | = 1} ∪ {|ψ ∗ | = 1} ∩ Lφ∗ ∩ Lψ ∗ ∗ iθ0 ∗ iθ0 ∗ iθ0 ∗ iθ0 iθ0 iθ0 such that φ∗ (eiθ0 ) 6=
ψ (e ). We may assume that |φ (e )| = 1. Since φ(te ) → φ (e ) and ψ(te ) → ψ (e ),
P iθ0 − P iθ0 → 2 as t → 1. φ(te ) ψ(te ) 1 Suppose that m {|φ∗ | = 1} ∪ {|ψ ∗ | = 1} = 0. Then
kCφ − Cψ kh∞ = sup Pφ∗ (eiθ eiθ ) − Pψ ∗ (eiθ eiθ ) 1 is the boundary description, where sup is taken over eiθ ∈ {|φ∗ | < 1} ∩ {|ψ ∗ | < 1}. We shall give a direct proof that the boundary and interior conditions on analytic self-maps φ, ψ are equivalent. We denote by ball(h∞ ) the closed unit ball of h∞ .
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Boundary vs. interior conditions associated with weighted composition operators
Proposition 3.7. Let φ, ψ ∈ S(D) and φ 6= ψ. Suppose that m {|φ∗ | = 1} ∪ {|ψ ∗ | = 1} = 0. Then we have
sup kPφ(z) − Pψ(z) k1 = sup Pφ∗ (eiθ ) − Pψ ∗ (eiθ ) 1 , z∈D
where sup is taken over eiθ ∈ {|φ∗ | < 1} ∩ {|ψ ∗ | < 1}.
Proof.
Since m {|φ∗ | = 1} ∪ {|ψ ∗ | = 1} = 0, for each z ∈ D we have kPφ(z) − Pψ(z) k1 =
sup
f∈ball(h∞ )
|f(φ(z)) − f(ψ(z))| = sup
f∈ball(h∞ )
Z ≤
sup
f∈ball(h∞ )
{|φ∗ |<1}∩{|ψ ∗ |<1}
Z ≤ {|φ∗ |<1}∩{|ψ ∗ |<1}
Thus we get
Z
∂D
(f ◦ φ)∗ (eiθ ) − (f ◦ ψ)∗ (eiθ ) Pz (eiθ ) dm(eiθ )
∗ f(φ (eiθ )) − f(ψ ∗ (eiθ )) Pz (eiθ ) dm(eiθ )
Pφ∗ (eiθ ) − Pψ ∗ (eiθ ) Pz (eiθ ) dm(eiθ ). 1
sup kPφ(z) − Pψ(z) k1 ≤ sup Pφ∗ (eiθ ) − Pψ ∗ (eiθ ) 1 , z∈D
∗
where sup is taken over eiθ ∈ {|φ | < 1} ∩ {|ψ ∗ | < 1}. On the other hand, for each θ such that eiθ ∈ {|φ∗ | < 1} ∩ {|ψ ∗ | < 1} we have that φ(teiθ ) → φ∗ (eiθ ) and ψ(teiθ ) → ψ ∗ (eiθ ) as t → 1. Hence
Pφ(teiθ ) − Pψ(teiθ ) → Pφ∗ (eiθ ) − Pψ ∗ (eiθ ) 1 1 This shows
as
t → 1.
sup kPφ(z) − Pψ(z) k1 ≥ sup Pφ∗ (eiθ ) − Pψ ∗ (eiθ ) 1 , z∈D
where sup is taken over eiθ ∈ {|φ∗ | < 1} ∩ {|ψ ∗ | < 1}. Lastly, we consider the case of the differences of two weighted composition operators. For u, v ∈ L∞ (∂D) and φ, ψ ∈ S(D), let Ω = Lu ∩ Lv ∩ Lφ∗ ∩ Lψ ∗ . Then m(Ω) = 1. In [10, Theorem 4.4], the authors proved
Lemma 3.8. Let Mu Cφ , Mv Cψ ∈ Cw (h∞ ) and φ 6= ψ. Then Mu Cφ − Mv Cψ is compact on h∞ if and only if the following seven conditions hold:
(a1 ) uχ{|φ∗ |=1} ∂D = 0,
(a2 ) lim uχ({r<|φ∗ |<1}∩{|ψ ∗ |=1}) = 0, ∂D
r→1
(a3 ) lim sup |u(eiθ )|ρ φ∗ (eiθ ), ψ ∗ (eiθ ) = 0, where sup is taken over eiθ ∈ {r < |φ∗ | < 1} ∩ {|ψ ∗ | < 1} ∩ Ω, r→1
(a4 ) vχ{|ψ ∗ |=1} ∂D = 0,
(a5 ) lim vχ({|φ∗ |=1}∩{r<|ψ ∗ |<1}) = 0, ∂D
r→1
r→1
766
φ
∗
(eiθ ), ψ ∗ (eiθ )
= 0, where sup is take over eiθ ∈ {|φ∗ | < 1} ∩ {r < |ψ ∗ | < 1} ∩ Ω,
(a7 ) lim (u − v)χ({r<|φ∗ |<1}∩{r<|ψ ∗ |<1}) ∂D = 0. (a6 )
lim sup |v(eiθ )|ρ r→1
K.J. Izuchi, Y. Izuchi, S. Ohno
We note that if φ = ψ, then by Lemma 3.3, Mu Cφ − Mv Cψ is compact on h∞ if and only if
lim (u − v)χ{|φ∗ |>r} ∂D = 0.
r→1
We note that this condition is not equivalent to (a1 )–(a7 ). Conditions (a1 )–(a7 ) are the boundary conditions on functions u, v, φ, ψ for compactness of Mu Cφ − Mv Cψ on h∞ for φ 6= ψ. But we do not know the equivalent conditions for it on the open unit disk.
Theorem 3.9. Let Mu Cφ , Mv Cψ ∈ Cw (h∞ ) and φ 6= ψ. Suppose that the following three conditions hold: b(zn ) → 0. (b1 ) If {zn }n ⊂ D, |φ(zn )| → 1 and lim inf ρ(φ(zn ), ψ(zn )) > 0, then u n→∞
v (zn ) → 0. (b2 ) If {zn }n ⊂ D, |ψ(zn )| → 1 and lim inf ρ(φ(zn ), ψ(zn )) > 0, then b n→∞
(b3 ) If {zn }n ⊂ D, |φ(zn )| → 1 and |ψ(zn )| → 1, then (b u−b v )(zn ) → 0. Then conditions (a1 )–(a7 ) hold.
Proof.
b(z) = 0. Also conditions Suppose that kψk∞ < 1. Then conditions (b1 )–(b3 ) are equivalent to lim|φ(z)|→1 u (a1 )–(a7 ) are equivalent to limr→1 kuχ{|φ∗ |>r} k∂D = 0. In this case, by Theorem 3.5 we get the assertion. So we assume that kφk∞ = kψk∞ = 1. Since Mu Cφ , Mv Cψ ∈ Cw (h∞ ), we have u 6≡ 0 and v 6≡ 0. To prove (a1 ), suppose that m({|φ∗ | = 1}) > 0. Since φ 6= ψ, by the F. & M. Riesz theorem φ∗ 6= ψ ∗ a.e. on {|φ∗ | = 1} ∩ Lψ ∗ . Let eiθ ∈ {|φ∗ | = 1} ∩ Lψ ∗ and φ∗ (eiθ ) 6= ψ ∗ (eiθ ). For a sequence {tn }n such that tn → 1 with 0 < tn < 1, we iθ iθ have φ(tn eiθ ) → φ∗ (eiθ ) and ψ(tn eiθ ) → ψ ∗ (eiθ ). Since lim inf n→∞ ρ φ(t n e ), ψ(tn e ) > 0, by condition (b1 ) we have
b(tn eiθ ) → 0. This shows that u = 0 a.e. on {|φ∗ | = 1}, so uχ{|φ∗ |=1} ∂D = 0. Similarly, by (b2 ), we may prove (a4 ). u
Suppose (a2 ) does not hold. Then there is a positive number δ such that uχ({r<|φ∗ |<1}∩{|ψ ∗ |=1}) ∂D > δ for every 0 < r < 1. Take a point eiθr ∈ {r < |φ∗ | < 1} ∩ {|ψ ∗ | = 1} ∩ Ω satisfying |u(eiθr )| > δ for each r. Since m {r < |φ∗ | < 1} ∩ {|ψ ∗ | = 1} > 0, we have φ∗ 6= ψ ∗ a.e. on {r < |φ∗ | < 1} ∩ {|ψ ∗ | = 1} ∩ Ω, so we may assume that φ∗ (eiθr ) 6= ψ ∗ (eiθr ). We note that r < |φ∗ (eiθr )| < 1 and |ψ ∗ (eiθr )| = 1. Then there is a number tr , 0 < tr < 1, such that r < |φ(tr eiθr )| < 1, ρ φ(tr eiθr ), ψ(tr eiθr ) > 1/2 and |b u(tr eiθr )| > δ. These facts contradict condition (b1 ). Similarly we may prove (a5 ). Suppose (a3 ) does not hold. Then there is a positive number δ such that sup |u(eiθ )|ρ φ∗ (eiθ ), ψ ∗ (eiθ ) > δ for every r, where sup is taken over eiθ ∈ {r < |φ∗ | < 1} ∩ {|ψ ∗ | < 1} ∩ Ω. We have {r < |φ∗ | < 1} ∩ {|ψ ∗ | < 1} ∩ Ω 6= ∅. By Lemma 3.1, m {r < |φ∗ | < 1} ∩ {ψ ∗ | < 1} ∩ Ω > 0. Since φ∗ 6= ψ ∗ a.e. on ∂D, there is a point eiθr ∈ {r < |φ∗ | < 1} ∩ {|ψ ∗ | < 1} ∩ Ω such that |u(eiθr )|ρ φ∗ (eiθr ), ψ ∗ (eiθr ) > δ. Hence |u(eiθr )| > δ and ρ φ∗ (eiθr ), ψ ∗ (eiθr ) > δ/kuk∂D . Since |φ∗ (eiθr )| → 1 as r → 1, there is a number tr , 0 < tr < 1, such that |b u(tr eiθr )| > δ, ρ φ(tr eiθr ), ψ(tr eiθr ) > δ/kuk∂D and |φ(tr eiθr )| → 1 as r → 1. These facts contradict condition (b1 ). Similarly we may prove (a6 ).
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Boundary vs. interior conditions associated with weighted composition operators
Suppose (a7 ) does not hold. Then there is a number tr , 0 < tr < 1, such that
(u − v)χ({r<|φ∗ |<1}∩{r<|ψ ∗ |<1}) > δ ∂D for every r. Since m {r < |φ∗ | < 1} ∩ {r < |ψ ∗ | < 1} > 0, there is a point eiθr ∈ {r < |φ∗ | < 1} ∩ {r < |ψ ∗ | < 1} ∩ Ω such that |(u − v)(eiθr )| > δ. Since |φ∗ (eiθr )| → 1 and |ψ ∗ (eiθr )| → 1 as r → 1, there is a number tr , 0 < tr < 1, such that |(b u−b v )(tr eiθr )| > δ, |φ(tr eiθr )| → 1 and |ψ(tr eiθr )| → 1 as r → 1. These facts contradict condition (b3 ). Thus we get the assertion. We use the following theorem in Section 5.
Theorem 3.10. Let φ, ψ ∈ S(D), φ 6= ψ and kφk∞ = kψk∞ = 1. Suppose that kφψk∞ = 1, m {|φ∗ | = 1} ∩ {|ψ ∗ | > r} = 0 and m {|φ∗ | > r} ∩ {|ψ ∗ | = 1} = 0 for 0 < r < 1 sufficiently close to 1. Then
lim sup Pφ∗ (eiθ ) − Pψ ∗ (eiθ ) 1 = lim sup kPφ(z) − Pψ(z) k1 ,
r→1
|(φψ)(z)|→1
where sup is taken over eiθ ∈ {r < |φ∗ | < 1} ∩ {r < |ψ ∗ | < 1}.
Proof.
Let
A = lim sup Pφ∗ (eiθ ) − Pψ ∗ (eiθ ) 1 , r→1
B = lim sup kPφ(z) − Pψ(z) k1 , |(φψ)(z)|→1
where sup is taken over eiθ ∈ {r < |φ∗ | < 1} ∩ {r < |ψ ∗ | < 1}. We shall show that A = B. For each z ∈ D, we have kPφ(z) − Pψ(z) k1 =
sup
|f(φ(z)) − f(ψ(z))|
f∈ball(h∞ )
Z ∗ iθ ∗ iθ iθ iθ = sup (f ◦ φ) (e ) − (f ◦ ψ) (e ) Pz (e ) dm(e ) ∞ f∈ball(h ) ∂D Z ∗ f(φ (eiθ )) − f(ψ ∗ (eiθ )) Pz (eiθ ) dm(eiθ ) ≤ sup f∈ball(h∞ ) {|φ∗ |<1}∩{|ψ ∗ |<1} Z ∗ iθ ∗ iθ iθ iθ + (f ◦ φ) (e ) − (f ◦ ψ) (e ) Pz (e ) dm(e ) {|φ∗ |=1}∪{|ψ ∗ |=1}
Z ≤ {|φ∗ |<1}∩{|ψ ∗ |<1}
Pφ∗ (eiθ ) − Pψ ∗ (eiθ ) Pz (eiθ ) dm(eiθ ) + 2 1
Z {|φ∗ |=1}∪{|ψ ∗ |=1}
Pz (eiθ ) dm(eiθ ).
Let {zn }n be a sequence in D such that |(φψ)(zn )| → 1 and kPφ(zn ) − Pψ(zn ) k1 → B as n → ∞. Then |φ(zn )| → 1 and |ψ(zn )| → 1. Hence by Lemma 3.4, Z {|φ∗ |>r}
Pzn (eiθ ) dm(eiθ ) → 1
Z and
{|ψ ∗ |>r}
Pzn (eiθ ) dm(eiθ ) → 1
as n → ∞ for every 0 < r < 1. Therefore Z {|φ∗ |>r}∩{|ψ ∗ |>r}
768
Pzn (eiθ ) dm(eiθ ) → 1
K.J. Izuchi, Y. Izuchi, S. Ohno
as n → ∞, so by the assumption we have Z {r<|φ∗ |<1}∩{r<|ψ ∗ |<1}
Thus we get
Z {|φ∗ |=1}∪{|ψ ∗ |=1}
Pzn (eiθ ) dm(eiθ ) → 1
Pzn (eiθ ) dm(eiθ ) → 0
as n → ∞.
as
n → ∞.
As a consequence, we have Z B = lim kPφ(zn ) − Pψ(zn ) k1 ≤ lim inf n→∞
n→∞
Z = lim inf n→∞
Pφ∗ (eiθ ) − Pψ ∗ (eiθ ) Pz (eiθ ) dm(eiθ ) n 1
{|φ∗ |<1}∩{|ψ ∗ |<1}
Pφ∗ (eiθ ) − Pψ ∗ (eiθ ) Pzn (eiθ ) dm(eiθ ) ≤ sup Pφ∗ (eiθ ) − Pψ ∗ (eiθ ) . 1 1
{r<|φ∗ |<1}∩{r<|ψ ∗ |<1}
So we get B ≤ A.
Take ε > 0 arbitrarily. Since A ≤ sup Pφ∗ (eiθ ) − Pψ ∗ (eiθ ) 1 , for every 0 < r < 1, where sup is taken over eiθ ∈ {r < |φ∗ | < 1} ∩ {r < |ψ ∗ | < 1}, there is a point eiθr ∈ {r < |φ∗ | < 1} ∩ {r < |ψ ∗ | < 1}
such that A − ε < Pφ∗ (eiθr ) − Pψ ∗ (eiθr ) 1 . Then there is a number tr , 0 < tr < 1, such that
A − ε < Pφ(tr eiθr ) − Pψ(tr eiθr ) 1 ,
|φ(tr eiθr )| > |φ∗ (eiθr )| + r − 1 > 2r − 1,
|ψ(tr eiθr )| > |ψ ∗ (eiθr )| + r − 1 > 2r − 1. Hence |(φψ)(tr eiθr )| → 1 as r → 1. Therefore A − ε ≤ B, so we have A ≤ B. As a result we get A = B.
4.
h∞ vs. H ∞
In this section, we consider the equivalences between the compactness of weighted composition operators and their differences on h∞ and H ∞ . We put d∞ (z, w) = sup
|f(z) − f(w)|,
f∈ball(H ∞ )
z, w ∈ D.
For φ, ψ ∈ S(D) with φ 6= ψ, it is known that d∞ (z, w) =
2 1−
p 1 − ρ2 (z, w) ρ(z, w)
and the norm of Cφ − Cψ on H ∞ is given by kCφ − Cψ kH ∞ = sup d∞ (φ(z), ψ(z)) = sup z∈D
z∈D
2 1−
p 1 − ρ2 (φ(z), ψ(z)) ρ(φ(z), ψ(z))
(see [14]). Also we denote by ball(H ∞ ) the closed unit ball of H ∞ . We have that ρ(z, w) ≤ d∞ (z, w) ≤ 2ρ(z, w), so ρ(φ(z), ψ(z)) ≤
sup
f∈ball(H ∞ )
|f(φ(z)) − f(ψ(z))| ≤ 2ρ(φ(z), ψ(z)).
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Boundary vs. interior conditions associated with weighted composition operators
For a, b ∈ C and z, w ∈ D, put Λ(a, b; z, w) =
sup
|af(z) − bf(w)|.
f∈ball(H ∞ )
When a = b = 1, we have Λ(1, 1; z, w) = d∞ (z, w). For u, v ∈ H ∞ and φ, ψ ∈ S(D) with φ 6= ψ, the interior description of the norm of Mu Cφ − Mv Cψ on H ∞ is given by kMu Cφ − Mv Cψ kH ∞ =
sup
f∈ball(H ∞ )
= sup
ku(f ◦ φ) − v(f ◦ ψ)k∞
sup
u(z)f(φ(z)) − v(z)f(ψ(z)) = sup Λ u(z), v(z); φ(z), ψ(z) .
z∈D f∈ball(H ∞ )
z∈D
The following is the boundary description of kMu Cφ − Mv Cψ kH ∞ . Let Ω = Lu∗ ∩ Lv ∗ ∩ Lφ∗ ∩ Lψ ∗ . Then m(Ω) = 1. Let A(D) be the disk algebra on the closure D of D, that is, A(D) is the space of continuous functions on D which are analytic in D.
Theorem 4.1. Let Mu Cφ , Mv Cψ ∈ Cw (H ∞ ) and φ 6= ψ. Then n
o kMu Cφ − Mv Cψ kH ∞ = max (|u∗ | + |v ∗ |)χ({|φ∗ |=1}∪{|ψ ∗ |=1}) ∂D , sup Λ u∗ (eiθ ), v ∗ (eiθ ); φ∗ (eiθ ), ψ ∗ (eiθ ) , where sup is taken over eiθ ∈ {|φ∗ | < 1} ∩ {|ψ ∗ | < 1} ∩ Ω.
Proof.
We have
kMu Cφ − Mv Cψ kH ∞ = sup
f∈ball(H ∞ )
f∈ball(H ∞ ) z∈D
Z ≤ sup
sup
z∈D f∈ball(H ∞ )
∂D
∗ u (eiθ )(f ◦ φ)∗ (eiθ ) − v ∗ (eiθ )(f ◦ ψ)∗ (eiθ ) Pz (eiθ ) dm(eiθ )
Z ≤ sup
sup u(z)f(φ(z)) − v(z)f(ψ(z))
ku(f ◦ φ) − v(f ◦ ψ)k∞ = sup
sup
z∈D f∈ball(H ∞ )
{|φ∗ |=1}∪{|ψ ∗ |=1}
Z +
{|φ∗ |<1}∩{|ψ ∗ |<1}∩Ω
|u∗ (eiθ )| + |v ∗ (eiθ )| Pz (eiθ ) dm(eiθ )
∗ u (eiθ )(f ◦ φ∗ )(eiθ ) − v ∗ (eiθ )(f ◦ ψ ∗ )(eiθ ) Pz (eiθ ) dm(eiθ )
≤ sup (|u∗ | + |v ∗ |)χ({|φ∗ |=1}∪{|ψ ∗ |=1}) z∈D
Z ∂D
Pz (eiθ ) dm(eiθ )
{|φ∗ |=1}∪{|ψ ∗ |=1}
+ sup Λ u∗ (eiθ ), v ∗ (eiθ ); φ∗ (eiθ ), ψ ∗ (eiθ )
Z
Pz (eiθ ) dm(eiθ )
{|φ∗ |<1}∩{|ψ ∗ |<1}∩Ω
n
o ≤ max (|u∗ | + |v ∗ |)χ({|φ∗ |=1}∪{|ψ ∗ |=1}) ∂D , sup Λ u∗ (eiθ ), v ∗ (eiθ ); φ∗ (eiθ ), ψ ∗ (eiθ ) , where sup is taken over eiθ ∈ {|φ∗ | < 1} ∩ {|ψ ∗ | < 1} ∩ Ω in both cases. Next, let A = sup Λ u∗ (eiθ ), v ∗ (eiθ ); φ∗ (eiθ ), ψ ∗ (eiθ ) , where sup is taken over eiθ ∈ {|φ∗ | < 1} ∩ {|ψ ∗ | < 1} ∩ Ω. Take a sequence {eiθn }n in {|φ∗ | < 1} ∩ {|ψ ∗ | < 1} ∩ Ω such that An = Λ u∗ (eiθn ), v ∗ (eiθn ); φ∗ (eiθn ), ψ ∗ (eiθn ) → A as n → ∞. For each n, there is a function fn ∈ ball(H ∞ ) such that An −
770
1 < u∗ (eiθn )fn (φ∗ (eiθn )) − v ∗ (eiθn )fn (ψ ∗ (eiθn )) . n
K.J. Izuchi, Y. Izuchi, S. Ohno
Take 0 < tn < 1 satisfying An −
1 < u(tn eiθn )fn (φ(tn eiθn )) − v(tn eiθn )fn (ψ(tn eiθn )) . n
Then An −
1 < u(fn ◦ φ) − v(fn ◦ ψ) (tn eiθn ) n ≤ ku(fn ◦ φ) − v(fn ◦ ψ)k∞ = k(Mu Cφ − Mv Cψ )fn k∞ ≤ kMu Cφ − Mv Cψ kH ∞ .
Therefore we get A ≤ kMu Cφ − Mv Cψ kH ∞ . Finally, suppose that m {|φ∗ | = 1} ∪ {|ψ ∗ | = 1} > 0. Since φ 6= ψ, φ∗ 6= ψ ∗ a.e. on ∂D. We put Ω0 = {eiθ ∈ Ω : φ∗ (eiθ ) 6= ψ ∗ (eiθ )}. Then m(Ω0 ) = 1, m ({|φ∗ | = 1} ∪ {|ψ ∗ | = 1}) ∩ Ω0 = m {|φ∗ | = 1} ∪ {|ψ ∗ | = 1} ,
∗
(|u | + |v ∗ |)χ({|φ∗ |=1}∪{|ψ ∗ |=1}) = (|u∗ | + |v ∗ |)χ(({|φ∗ |=1}∪{|ψ ∗ |=1})∩Ω ) . 0 ∂D ∂D Take
eiθ ∈ {|φ∗ | = 1} ∪ {|ψ ∗ | = 1} ∩ Ω0 .
Then either |φ∗ (eiθ )| = |ψ ∗ (eiθ )| = 1 or |φ∗ (eiθ )| = 1 > |ψ ∗ (eiθ )| or |φ∗ (eiθ )| < 1 = |ψ ∗ (eiθ )|. Suppose that |φ∗ (eiθ )| = |ψ ∗ (eiθ )| = 1. Since φ∗ (eiθ ) 6= ψ ∗ (eiθ ), there is a function g in A(D) such that kgk∞ = 1, g(φ∗ (eiθ )) = u∗ (eiθ )/|u∗ (eiθ )| and g(ψ ∗ (eiθ )) = −v ∗ (eiθ )/|v ∗ (eiθ )|. Then we have kMu Cφ − Mv Cψ kH ∞ ≥ ku(g ◦ φ) − v(g ◦ ψ)k∞ ≥ lim u(teiθ )g(φ(teiθ )) − v(teiθ )g(ψ(teiθ )) t→1 = u∗ (eiθ )g(φ∗ (eiθ )) − v ∗ (eiθ )g(ψ ∗ (eiθ )) = |u∗ (eiθ )| + |v ∗ (eiθ )|. Suppose that |φ∗ (eiθ )| = 1 > |ψ ∗ (eiθ )|. There is a sequence {gn }n in A(D) such that kgn k∞ = 1, gn (φ∗ (eiθ )) → u∗ (eiθ )/|u∗ (eiθ )| and gn (ψ ∗ (eiθ )) → −v ∗ (eiθ )/|v ∗ (eiθ )|, where a/|a| = 0 if a = 0. Claim.
We write α = φ∗ (eiθ ), β = ψ ∗ (eiθ ), A = u∗ (eiθ )/|u∗ (eiθ )| and B = −v ∗ (eiθ )/|v ∗ (eiθ )|. Then |α| = 1 > |β|, either |A| = 0 or 1, and either |B| = 0 or 1. Since {α, −α} is a peak interpolation set for A(D), there exists a function g ∈ A(D) such that kgk∞ = 1, g(α) = A and g(−α) = B (see [6]). Let {rn }n be a sequence in (0, 1) satisfying rn → 1 and Proof.
z − rn α gn (z) = g , 1 − rn αz
z ∈ D.
Then gn ∈ A(D), kgn k∞ = 1, gn (α) = g(α) for every n ≥ 1, and gn (β) = g
β − rn α 1 − rn αβ
αβ − rn =g α → g(−α) = B 1 − rn αβ
as n → ∞.
Due to the claim we have kMu Cφ − Mv Cψ kH ∞ ≥ u∗ (eiθ )gn (φ∗ (eiθ )) − v ∗ (eiθ )gn (ψ ∗ (eiθ )) → |u∗ (eiθ )| + |v ∗ (eiθ )|
as n → ∞.
Hence |u∗ (eiθ )| + |v ∗ (eiθ )| ≤ kMu Cφ − Mv Cψ kH ∞ . When |φ∗ (eiθ )| < 1 = |ψ ∗ (eiθ )|, we have the same estimate. Thus we get
∗
(|u | + |v ∗ |)χ({|φ∗ |=1}∪{|ψ ∗ |=1}) ≤ kMu Cφ − Mv Cψ kH ∞ . ∂D This completes the proof.
771
Boundary vs. interior conditions associated with weighted composition operators
When u = v = 1, we have the following.
Corollary 4.2. Let φ, ψ ∈ S(D) and φ 6= ψ. Then the norm of Cφ − Cψ on H ∞ is n
o kCφ − Cψ kH ∞ = max 2 χ({|φ∗ |=1}∪{|ψ ∗ |=1}) ∂D , sup d∞ φ∗ (eiθ ), ψ ∗ (eiθ ) , where sup is taken over eiθ ∈ {|φ∗ | < 1} ∩ {|ψ ∗ | < 1}.
The next result follows from Lemma 3.8.
Lemma 4.3. Let Mu Cφ , Mv Cψ ∈ Cw (h∞ ) and φ 6= ψ. Suppose that u, v ∈ H ∞ . Then Mu Cφ − Mv Cψ is compact on h∞ if and only if the following four conditions hold: (c1 ) m({|φ∗ | = 1}) = m({|ψ ∗ | = 1}) = 0. (c2 ) lim sup |u∗ (eiθ )|ρ φ∗ (eiθ ), ψ ∗ (eiθ ) = 0, where sup is taken over eiθ ∈ {r < |φ∗ | < 1} ∩ {|ψ ∗ | < 1} ∩ Ω. r→1 (c3 ) lim sup |v ∗ (eiθ )|ρ φ∗ (eiθ ), ψ ∗ (eiθ ) = 0, where sup is taken over eiθ ∈ {|φ∗ | < 1} ∩ {r < |ψ ∗ | < 1} ∩ Ω. r→1
(c4 ) lim (u∗ − v ∗ )χ({r<|φ∗ |<1}∩{r<|ψ ∗ |<1}) ∂D = 0. r→1
Proof.
Since u ∈ H ∞ and u 6= 0, u∗ χ{|φ∗ |=1} ∂D = 0 if and only if m({|φ∗ | = 1}) = 0. By Lemma 3.8, we get the
assertion. In [9, Theorem 2.2], Hosokawa, Izuchi and Ohno proved
Lemma 4.4. Let Mu Cφ , Mv Cψ ∈ Cw (H ∞ ). Then the following conditions are equivalent: (i) Mu Cφ − Mv Cψ is compact on H ∞ . (ii) There hold: (d1 ) If {zn }n ⊂ D, |φ(zn )| → 1 and lim inf ρ(φ(zn ), ψ(zn )) > 0, then u(zn ) → 0. n→∞
(d2 ) If {zn }n ⊂ D, |ψ(zn )| → 1 and lim inf ρ(φ(zn ), ψ(zn )) > 0, then v(zn ) → 0. n→∞
(d3 ) If {zn }n ⊂ D, |φ(zn )| → 1 and |ψ(zn )| → 1, then (u − v)(zn ) → 0. When u, v ∈ H ∞ , we directly show the equivalence of conditions for the compactness of the differences of two weighted composition operators on H ∞ and h∞ .
Theorem 4.5. Let u, v ∈ H ∞ and φ, ψ ∈ S(D) satisfy u 6≡ 0, v 6≡ 0 and φ 6= ψ. Then the sets of conditions (c1 )–(c4 ) and (d1 )–(d3 ) are equivalent. If kψk∞ < 1, then {|ψ ∗ | > r} = ∅ for 0 < r < 1 sufficiently close to 1. Since φ 6= ψ, φ∗ 6= ψ ∗ a.e.
∗
on ∂D. Hence conditions (c1 )–(c4 ) are equivalent to limr→1 u χ{|φ∗ |>r} ∂D = 0. Also conditions (d1 )–(d3 ) are equivalent to lim|φ(z)|→1 u(z) = 0. Moreover if kφk∞ < 1, then there is nothing to prove. If kφk∞ = 1, then by Theorem 3.5 we have
that limr→1 u∗ χ{|φ∗ |>r} ∂D = 0 if and only if lim|φ(z)|→1 u(z) = 0. So we may assume that kφk∞ = kψk∞ = 1.
Proof.
772
K.J. Izuchi, Y. Izuchi, S. Ohno
Suppose that conditions (c1 )–(c4 ) hold. To prove (d1 ), suppose that |φ(zn )| → 1 and lim inf n→∞ ρ(φ(zn ), ψ(zn )) > 0. We have |u(zn )|ρ(φ(zn ), ψ(zn )) ≤ |u(zn )|d∞ (φ(zn ), ψ(zn )) = sup
u(zn ) f(φ(zn )) − f(ψ(zn ))
f∈ball(H ∞ )
Z = sup u∗ (eiθ ) (f ◦ φ)∗ (eiθ ) − (f ◦ψ)∗ (eiθ ) Pzn (eiθ ) dm(eiθ ) because u ∈ H ∞ ∞ f∈ball(H ) ∂D Z ≤ sup |u∗ (eiθ )| f(φ∗ (eiθ )) − f(ψ ∗ (eiθ )) Pzn (eiθ ) dm(eiθ ) by condition (c1 ) f∈ball(H ∞ )
{|φ∗ |<1}∩{|ψ ∗ |<1}∩Ω
Z ≤2
{|φ∗ |<1}∩{|ψ ∗ |<1}∩Ω
|u∗ (eiθ )|ρ φ∗ (eiθ ), ψ ∗ (eiθ ) Pzn (eiθ ) dm(eiθ ).
Since |φ(zn )| → 1, by Lemma 3.4 we have that Z Pzn (eiθ ) dm(eiθ ) → 1
as
{|φ∗ |>r}
n→∞
R for every 0 < r < 1. Since {|φ∗ | > r} = {|φ∗ | = 1}∪{r < |φ∗ | < 1}, by condition (c1 ) again {r<|φ∗ |<1} Pzn (eiθ ) dm(eiθ ) → 1 as n → ∞. Hence for each r, we have Z lim sup |u(zn )|ρ(φ(zn ), ψ(zn )) ≤ lim sup 2 |u∗ (eiθ )|ρ φ∗ (eiθ ), ψ ∗ (eiθ ) Pzn (eiθ ) dm(eiθ ). {r<|φ∗ |<1}∩{|ψ ∗ |<1}∩Ω
n→∞
n→∞
By condition (c2 ), for any ε > 0 there exists a constant r0 , 0 < r0 < 1, such that sup 2|u∗ (eiθ )|ρ φ∗ (eiθ ), ψ ∗ (eiθ ) < ε, where sup is taken eiθ ∈ {r0 < |φ∗ | < 1} ∩ {|ψ ∗ | < 1} ∩ Ω. Hence lim sup |u(zn )|ρ(φ(zn ), ψ(zn )) ≤ ε, n→∞
so we get lim |u(zn )|ρ(φ(zn ), ψ(zn )) = 0.
n→∞
Since lim inf ρ(φ(zn ), ψ(zn )) > 0, we obtain (d1 ). Similarly we may prove (d2 ). n→∞
Next, we shall prove (d3 ). Suppose that |φ(zn )| → 1 and |ψ(zn )| → 1. By Lemma 3.4, for each r we have R R P (eiθ ) dm(eiθ ) → 1 and {|ψ ∗ |>r} Pzn (eiθ ) dm(eiθ ) → 1 as n → ∞. Hence {|φ∗ |>r} zn Z
Pzn (eiθ ) dm(eiθ ) → 1
{|φ∗ |>r}∩{|ψ ∗ |>r}
as
n → ∞.
Take ε > 0 arbitrarily. By (c1 ) and (c4 ), there is a constant r0 , 0 < r0 < 1, such that
∗
(u − v ∗ )χ({|φ∗ |>r }∩{|ψ ∗ |>r }) < ε. 0 0 ∂D We have Z |(u − v)(zn )| = u∗ (eiθ ) − v ∗ (eiθ ) Pzn (eiθ ) dm(eiθ ) Z∂D ≤ u∗ (eiθ ) − v ∗ (eiθ ) Pzn (eiθ ) dm(eiθ ) {|φ∗ |>r0 }∩{|ψ ∗ |>r0 } Z + u∗ (eiθ ) − v ∗ (eiθ ) Pzn (eiθ ) dm(eiθ ) ({|φ∗ |>r0 }∩{|ψ ∗ |>r0 })c
Z < ε + ku − vk∞
Pzn (eiθ ) dm(eiθ ).
({|φ∗ |>r0 }∩{|ψ ∗ |>r0 })c
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Boundary vs. interior conditions associated with weighted composition operators
Hence lim supn→∞ |(u − v)(zn )| ≤ ε. Since ε is arbitrary, we get (d3 ). The fact that (d1 )–(d3 ) imply (c1 )–(c4 ) follows from Theorem 3.9. When u = v = 1 in Theorem 4.5, we have
Corollary 4.6. Let φ, ψ ∈ S(D) and φ 6= ψ. Then the following conditions are equivalent: (i) There hold: (e1 ) |φ∗ | < 1 and |ψ ∗ | < 1 a.e. on ∂D. (e2 ) lim sup ρ φ∗ (eiθ ), ψ ∗ (eiθ ) = 0, where sup is taken over eiθ ∈ {r < |φ∗ | < 1} ∩ {|ψ ∗ | < 1}. r→1 (e3 ) lim sup ρ φ∗ (eiθ ), ψ ∗ (eiθ ) = 0, where sup is taken over eiθ ∈ {|φ∗ | < 1} ∩ {r < |ψ ∗ | < 1}. r→1
(ii)
lim ρ(φ(z), ψ(z)) = lim ρ(φ(z), ψ(z)) = 0.
|φ(z)|→1
|ψ(z)|→1
In [10, Theorem 4.4], the authors proved that condition (i) is equivalent to Cφ −Cψ being compact on h∞ . In [14, Theorem 3], MacCluer, Ohno and Zhao proved that condition (ii) is equivalent to Cφ − Cψ being compact on H ∞ .
Lemma 4.7. Let u ∈ H ∞ and φ ∈ S(D). Then the following conditions are equivalent: (i) Mu Cφ is compact on H ∞ . (ii) Mu Cφ is compact on h∞ . First let us show (i) ⇒ (ii). If kφk∞ < 1, then Cφ is compact on h∞ . So we may assume that kφk∞ = 1. Suppose that Mu Cφ is compact on H ∞ . By [3], u(zn ) → 0 if |φ(zn )| → 1. By Lemma 3.3 and Theorem 3.5, Mu Cφ is compact on h∞ . The implication (ii) ⇒ (i) is trivial.
Proof.
Corollary 4.8. Let u, v ∈ H ∞ and φ, ψ ∈ S(D). Then the following conditions are equivalent: (i) Mu Cφ − Mv Cψ is compact on H ∞ . (ii) Mu Cφ − Mv Cψ is compact on h∞ . Suppose that Mu Cφ − Mv Cψ is compact on H ∞ . If φ = ψ, then Mu Cφ − Mv Cψ = Mu−v Cφ . By Lemma 4.7, Mu−v Cφ is compact on h∞ , so we may assume φ 6= ψ. Suppose that v = 0. Then by Lemma 4.7, Mu Cφ is compact on h∞ , so we may assume that u 6= 0 and v 6= 0. In the other cases, by Lemmas 4.3, 4.4 and Theorem 4.5 we get the assertion.
Proof.
5.
Essential norms
Recall that the essential norm of a bounded linear operator T on h∞ is defined as kT kh∞ ,e = inf kT − K k : K is compact on h∞ . Let u, v ∈ L∞ (∂D) and φ, ψ ∈ S(D) with φ 6= ψ. In [10, Theorem 4.3], we presented a formula for the essential norm of Mu Cφ − Mv Cψ on h∞ . In the special case of u = v = 1, we have the following.
774
K.J. Izuchi, Y. Izuchi, S. Ohno
Theorem 5.1. Let φ, ψ ∈ S(D) and φ 6= ψ. Put
f1 = lim 2 χ({|φ∗ |=1}∩{|ψ ∗ |>R}) ∂D , R→1
f3 = lim 2 χ({|φ∗ |>r}∩{|ψ ∗ |=1}) ∂D , r→1
f5 = lim lim χ({r<|φ∗ |<1}∩{|ψ ∗ |≤R}) ∂D ,
f2 = lim χ({|φ∗ |=1}∩{|ψ ∗ |≤R}) ∂D , R→1
f4 = lim χ({|φ∗ |≤r}∩{|ψ ∗ |=1}) ∂D , r→1
f6 = lim lim χ({|φ∗ |≤r}∩{R<|ψ ∗ |<1}) ∂D ,
R→1 r→1
r→1 R→1
and
f7 = lim sup Pφ∗ (eiθ ) − Pψ ∗ (eiθ ) 1 , r→1
where sup is taken over
eiθ
∗
∈ {r < |φ | < 1} ∩ {r < |ψ ∗ | < 1}. Then kCφ − Cψ kh∞ ,e = max1≤j≤7 fj on h∞ .
For φ, ψ ∈ S(D), let λ∞ (φ, ψ) = lim sup kPφ(z) − Pψ(z) k1 . |(φψ)(z)|→1
In [2, Theorem 6.1], Choa, Izuchi and Ohno presented a formula for kCφ − Cψ kh∞ ,e , namely
Theorem 5.2. Let φ, ψ ∈ S(D) and φ 6= ψ. Then the value kCφ − Cψ kh∞ ,e takes either 0 or 1 or λ∞ (φ, ψ). To prove Theorem 5.2, the argument was done on the maximal ideal space of H ∞ . In this paper we prove Theorem 5.2 applying Theorem 5.1, it will serve for better understanding of Theorem 5.2.
Proof of Theorem 5.2.
We study several cases separately. Trivially the values of f1 and f3 are either 0 or 2, the values of f2 , f4 , f5 and f6 are either 0 or 1 and the value of f7 varies in [0, 1]. Case 1. Suppose that max {kφk∞ , kψk∞ } < 1. Then both Cφ and Cψ are compact and so Cφ − Cψ is also compact and kCφ − Cψ kh∞ ,e = 0. Case 2. Suppose that max {kφk∞ , kψk∞ } = 1 and kφψk∞ < 1. Then {|(φψ)∗ | > r} = ∅, so {|φ∗ | > r}∩{|ψ ∗ | > r} = ∅ for 0 < r < 1 sufficiently close to 1. We have f1 = f3 = f7 = 0. If m({|φ∗ | = 1}) > 0, then f2 = 1, and if m({|ψ ∗ | = 1}) > 0, then f4 = 1. Suppose that m({|φ∗ | = 1}) = m({|ψ ∗ | = 1}) = 0, then f2 = f4 = 0. If kφk∞ = 1, then f5 = 1 and m({r < |φ∗ | < 1}) > 0 for every 0 < r < 1, so by Theorem 5.1 we have kCφ − Cψ kh∞ ,e = 1, and if kψk∞ = 1, then f6 = 1 and kCφ − Cψ kh∞ ,e = 1. Case 3. Suppose that kφψk∞ = 1 and m {|φ∗ | = 1} ∩ {|ψ ∗ | = 1} > 0, then f1 = 2, so by Theorem 5.1 we have kCφ − Cψ kh∞ ,e = 2. We shall show that λ∞ (φ, ψ) = 2. Since φ 6= ψ, φ∗ 6= ψ ∗ a.e. on {|φ∗ | = 1} ∩ {|ψ ∗ | = 1}. Then ∗ iθ ∗ iθ ∗ iθ ∗ iθ there exists a point eiθ ∈ {|φ∗ | = 1} ∩ {|ψ ∗ | = 1} such = 1 and
that φ (e ) 6= ψ (e ). We have |φ (e )| = |ψ (e )| iθ iθ iθ
limt→1 ρ φ(te ), ψ(te ) = 1. By Lemma 3.6, limr→1 Pφ(reiθ ) − Pψ(reiθ ) 1 = 2. We have |(φψ)(te )| → |(φψ)∗ (eiθ )| = 1 as t → 1, so we get λ∞ (φ, ψ) = 2. We note that if there exists a point eiθ ∈ {|φ∗ | = 1} ∩ {|ψ ∗ | = 1} such that φ∗ (eiθ ) 6= ψ ∗ (eiθ ), then kCφ − Cψ kh∞ ,e = 2 = λ∞ (φ, ψ) without any other assumptions. Case 4. Suppose that kφψk∞ = 1 and m {|φ∗ | = 1}∩{|ψ ∗ | = 1} = 0. In this case, we study either m({|φ∗ | = 1}) > 0 or m({|ψ ∗ | = 1}) > 0. Subcase 4.1.
Moreover we assume that m {|φ∗ | = 1} ∩ {|ψ ∗ | > R} > 0
for every 0 < R < 1. Then we have f1 = 2, so by Theorem 5.1 we have kCφ − Cψ kh∞ ,e = 2. We shall show that λ∞ (φ, ψ) = 2. By our assumptions, m {|φ∗ | = 1} ∩ {R < |ψ ∗ | < 1} > 0 for every 0 < R < 1. Take a sequence {eiθR }R such that eiθR ∈ {|φ∗ | = 1} ∩ {R < |ψ ∗ | < 1}.
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Boundary vs. interior conditions associated with weighted composition operators
Then |φ∗ (eiθR )| = 1 and R < |ψ ∗ (eiθR )| < 1. Take 0 < tR < 1 such that R < |φ∗ (tR eiθR )|, R < |ψ ∗ (tR eiθR )| < 1 and ρ φ∗ (tR eiθR ), ψ ∗ (tR eiθR ) → 1
as R → 1.
Therefore λ∞ (φ, ψ) = 2. Subcase 4.2.
Similarly, if
m {|φ∗ | > r} ∩ {|ψ ∗ | = 1} > 0
for every 0 < r < 1, then we have kCφ − Cψ kh∞ ,e = 2 = λ∞ (φ, ψ). Subcase 4.3.
Suppose that m {|φ∗ | = 1} ∩ {|ψ ∗ | > r} = 0,
m {|φ∗ | > r} ∩ {|ψ ∗ | = 1} = 0
for 0 < r < 1 sufficiently close to 1. Then f1 = f3 = 0. By Theorem 3.10, we have f7 = λ∞ (φ, ψ). If m({|φ∗ | = 1}) > 0, then f2 = 1, and if m({|ψ ∗ | = 1}) > 0, then f4 = 1, so we have kCφ − Cψ kh∞ ,e = max {1, λ∞ (φ, ψ)}. Case 5. Suppose that kφψk∞ = 1 and m({|φ∗ | = 1}) = m({|ψ ∗ | = 1}) = 0. Then f1 = f2 = f3 = f4 = 0 and by Theorem 3.10 we have f7 = λ∞ (φ, ψ). Moreover if either f5 = 1 or f6 = 1, then kCφ − Cψ kh∞ ,e = max {1, λ∞ (φ, ψ)}. If f5 = f6 = 0, then we have kCφ − Cψ kh∞ ,e = λ∞ (φ, ψ) and only in this case it may occur that 0 ≤ kCφ − Cψ kh∞ ,e < 1. By the proof of Theorem 5.2, we have the following.
Corollary 5.3. Let φ, ψ ∈ S(D) and φ 6= ψ. Then kCφ − Cψ kh∞ ,e < 1 if and only if kφψk∞ = 1, m({|φ∗ | = 1}) = m({|ψ ∗ | = 1}) = 0, f5 = f6 = 0 and λ∞ (φ, ψ) < 1.
Remark 5.4. We have that f5 = 0 if and only if |ψ ∗ (eiθn )| → 1 for every sequence {eiθn }n in Lφ∗ ∩ Lψ ∗ satisfying 0 < |φ∗ (eiθn )| < 1 and |φ∗ (eiθn )| → 1. For, suppose that f5 = 0. Since
lim χ{r<|φ∗ |<1}∩{|ψ ∗ |≤R} ∂D
r→1
increases in R, we have
lim χ{r<|φ∗ |<1}∩{|ψ ∗ |≤R} ∂D = 0
for every 0 < R < 1. For each fixed R, χ{r<|φ∗ |<1}∩{|ψ ∗ |≤R} ∂D decreases in r, so there exists a number rR , 0 < rR < 1, such that m {r < |φ∗ | < 1} ∩ {|ψ ∗ | ≤ R} = 0 r→1
for r, rR ≤ r < 1. This shows that if {eiθn }n is a sequence in Lφ∗ ∩ Lψ ∗ such that 0 < |φ∗ (eiθn )| < 1 and |φ∗ (eiθn )| → 1, then |ψ ∗ (eiθn )| → 1. Also if the above condition holds, then we may check that f5 = 0.
Remark 5.5. Suppose that m({|φ∗ | = 1}) = 0. If there exists a point eiθ0 ∈ Lφ∗ ∩ Lψ ∗ satisfying |φ∗ (eiθ0 )| = 1 and |ψ ∗ (eiθ0 )| < 1, then f5 = 1. For, by Lemma 2.1 there are a sequence of open subarcs {In }n of ∂D centered at eiθ0 and a sequence of measurable subsets {En }n such that En ⊂ In ∩ Lφ∗ ∩ Lψ ∗ , m(In ) → 0, m(En )/m(In ) → 1, and lim sup |φ∗ (eiθ ) − φ∗ (eiθ0 )| = lim sup |ψ ∗ (eiθ ) − ψ ∗ (eiθ0 )| = 0.
n→∞ iθ e ∈En
n→∞ iθ e ∈En
Take a number R0 satisfying |ψ ∗ (eiθ0 )| < R0 < 1. For any number ε satisfying 0 < ε < R0 − |ψ ∗ (eiθ0 )|, we have En ⊂ {|φ∗ | > 1 − ε}
and
En ⊂ {|ψ ∗ | < R0 }
for sufficiently large n. For such a large n, since m({|φ∗ | = 1}) = 0 we may assume that En ⊂ {1 − ε < |φ∗ | < 1}. This shows that f5 = 1.
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K.J. Izuchi, Y. Izuchi, S. Ohno
Acknowledgements We would like to thank the referee for carefully reading and for helpful comments that improved our manuscript. The first author is partially supported by Grant-in-Aid for Scientific Research, Japan Society for the Promotion of Science (No. 24540164). The third author is partially supported by Grant-in-Aid for Scientific Research, Japan Society for the Promotion of Science (No. 24540190).
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