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Modern Aspects of Complex Analysis and Its Applications Wayne Smith Composition Semigroups on BMOA and H Semigroups of analytic functions on the Disk Let D denote the unit disk { z : | z | < 1 } and H ( D ) the set of analytic functions on
Modern Aspects of Complex Analysis and Its Applications
Wayne Smith Composition Semigroups on BMOA and H∞
Let D denote the unit disk {z : |z| < 1} and H(D) the set of analytic functions on D. A one-parameter semigroup {ϕt}t≥0 of analytic functions on D is a family of analytic functions ϕt : D → D that satisfies the following three conditions:
Wayne Smith Composition Semigroups on BMOA and H∞
Let D denote the unit disk {z : |z| < 1} and H(D) the set of analytic functions on D. A one-parameter semigroup {ϕt}t≥0 of analytic functions on D is a family of analytic functions ϕt : D → D that satisfies the following three conditions: (SG1) ϕ0 is the identity, i.e. ϕ0(z) = z, z ∈ D;
Wayne Smith Composition Semigroups on BMOA and H∞
Let D denote the unit disk {z : |z| < 1} and H(D) the set of analytic functions on D. A one-parameter semigroup {ϕt}t≥0 of analytic functions on D is a family of analytic functions ϕt : D → D that satisfies the following three conditions: (SG1) ϕ0 is the identity, i.e. ϕ0(z) = z, z ∈ D; (SG2) ϕs+t = ϕs ◦ ϕt, for all t, s ≥ 0;
Wayne Smith Composition Semigroups on BMOA and H∞
Let D denote the unit disk {z : |z| < 1} and H(D) the set of analytic functions on D. A one-parameter semigroup {ϕt}t≥0 of analytic functions on D is a family of analytic functions ϕt : D → D that satisfies the following three conditions: (SG1) ϕ0 is the identity, i.e. ϕ0(z) = z, z ∈ D; (SG2) ϕs+t = ϕs ◦ ϕt, for all t, s ≥ 0; (SG3) the mapping (t, z) → ϕt(z) is continuous on [0, ∞) × D.
Wayne Smith Composition Semigroups on BMOA and H∞
Let D denote the unit disk {z : |z| < 1} and H(D) the set of analytic functions on D. A one-parameter semigroup {ϕt}t≥0 of analytic functions on D is a family of analytic functions ϕt : D → D that satisfies the following three conditions: (SG1) ϕ0 is the identity, i.e. ϕ0(z) = z, z ∈ D; (SG2) ϕs+t = ϕs ◦ ϕt, for all t, s ≥ 0; (SG3) the mapping (t, z) → ϕt(z) is continuous on [0, ∞) × D. The trivial case is that ϕt(z) = z for all t ≥ 0. Otherwise, we say that {ϕt} is nontrivial.
Wayne Smith Composition Semigroups on BMOA and H∞
Wayne Smith Composition Semigroups on BMOA and H∞
ϕt(z) = e−ctz, where Re c ≥ 0.
Wayne Smith Composition Semigroups on BMOA and H∞
ϕt(z) = e−ctz, where Re c ≥ 0. Rotation and Shrinking, common fixed point point 0
Wayne Smith Composition Semigroups on BMOA and H∞
ϕt(z) = e−ctz, where Re c ≥ 0. Rotation and Shrinking, common fixed point point 0
Wayne Smith Composition Semigroups on BMOA and H∞
ϕt(z) = e−ctz, where Re c ≥ 0. Rotation and Shrinking, common fixed point point 0 ϕt(z) = 1 + e−t(z − 1).
Wayne Smith Composition Semigroups on BMOA and H∞
ϕt(z) = e−ctz, where Re c ≥ 0. Rotation and Shrinking, common fixed point point 0 ϕt(z) = 1 + e−t(z − 1). Shrinking disks all tangent to unit circle at 1, common fixed point 1:
Wayne Smith Composition Semigroups on BMOA and H∞
ϕt(z) = e−ctz, where Re c ≥ 0. Rotation and Shrinking, common fixed point point 0 ϕt(z) = 1 + e−t(z − 1). Shrinking disks all tangent to unit circle at 1, common fixed point 1:
Wayne Smith Composition Semigroups on BMOA and H∞
ϕt(z) = e−ctz, where Re c ≥ 0. Rotation and Shrinking, common fixed point point 0 ϕt(z) = 1 + e−t(z − 1). Shrinking disks all tangent to unit circle at 1, common fixed point 1: An unlimited variety of such examples is easily constructed:
Wayne Smith Composition Semigroups on BMOA and H∞
Wayne Smith Composition Semigroups on BMOA and H∞
Every nontrivial semigroup of analytic functions {ϕt}t≥0 has a unique common fixed point b with |ϕ′
t(b)| ≤ 1 for all t ≥ 0, called
the Denjoy-Wolff point of the semigroup.
Wayne Smith Composition Semigroups on BMOA and H∞
Every nontrivial semigroup of analytic functions {ϕt}t≥0 has a unique common fixed point b with |ϕ′
t(b)| ≤ 1 for all t ≥ 0, called
the Denjoy-Wolff point of the semigroup. Under a normalization, the Denjoy-Wolff point b may be assumed to be 0 or 1.
Wayne Smith Composition Semigroups on BMOA and H∞
Every nontrivial semigroup of analytic functions {ϕt}t≥0 has a unique common fixed point b with |ϕ′
t(b)| ≤ 1 for all t ≥ 0, called
the Denjoy-Wolff point of the semigroup. Under a normalization, the Denjoy-Wolff point b may be assumed to be 0 or 1. If b = 0, then ϕt(z) = h−1(e−cth(z)), where h is a univalent function from D onto a spirallike domain Ω, h(0) = 0, Re c ≥ 0, and we−ct ∈ Ω for each w ∈ Ω, t ≥ 0.
Wayne Smith Composition Semigroups on BMOA and H∞
Every nontrivial semigroup of analytic functions {ϕt}t≥0 has a unique common fixed point b with |ϕ′
t(b)| ≤ 1 for all t ≥ 0, called
the Denjoy-Wolff point of the semigroup. Under a normalization, the Denjoy-Wolff point b may be assumed to be 0 or 1. If b = 0, then ϕt(z) = h−1(e−cth(z)), where h is a univalent function from D onto a spirallike domain Ω, h(0) = 0, Re c ≥ 0, and we−ct ∈ Ω for each w ∈ Ω, t ≥ 0. If b = 1, then ϕt(z) = h−1(h(z) + ct), where h : D → Ω is a Riemann map, Ω is close-to-convex, h(0) = 0, Re c ≥ 0, and w + ct ∈ Ω for each w ∈ Ω, t ≥ 0.
Wayne Smith Composition Semigroups on BMOA and H∞
Wayne Smith Composition Semigroups on BMOA and H∞
Associated with the semigroup {ϕt} is the composition semigroup
Wayne Smith Composition Semigroups on BMOA and H∞
Associated with the semigroup {ϕt} is the composition semigroup
If Ct is a bounded operator on some Banach space X ⊂ H(D) for all t ≥ 0, we say that the semigroup {ϕt} acts on X.
Wayne Smith Composition Semigroups on BMOA and H∞
Associated with the semigroup {ϕt} is the composition semigroup
If Ct is a bounded operator on some Banach space X ⊂ H(D) for all t ≥ 0, we say that the semigroup {ϕt} acts on X. If in addition the strong continuity condition lim
t→0+ f ◦ ϕt − f X = 0
holds for all f ∈ X, then it is said that {ϕt} is strongly continuous
Wayne Smith Composition Semigroups on BMOA and H∞
Wayne Smith Composition Semigroups on BMOA and H∞
Denote by [ϕt, X] the maximal closed subspace of X on which {Ct} is strongly continuous.
Wayne Smith Composition Semigroups on BMOA and H∞
Denote by [ϕt, X] the maximal closed subspace of X on which {Ct} is strongly continuous. Theorem (O. Blasco, M. Contreras, S. D´ ıaz-Madrigal, J. Mart´ ınez,
Let {ϕt}t≥0 be a semigroup with generator G and X a Banach space of analytic functions which contains the constant functions and such that sup0≤t≤1 Ct < ∞. Then [ϕt, X] = {f ∈ X : Gf ′ ∈ X}.
Wayne Smith Composition Semigroups on BMOA and H∞
Denote by [ϕt, X] the maximal closed subspace of X on which {Ct} is strongly continuous. Theorem (O. Blasco, M. Contreras, S. D´ ıaz-Madrigal, J. Mart´ ınez,
Let {ϕt}t≥0 be a semigroup with generator G and X a Banach space of analytic functions which contains the constant functions and such that sup0≤t≤1 Ct < ∞. Then [ϕt, X] = {f ∈ X : Gf ′ ∈ X}. Here G(z) = lim
t→0+
ϕt(z) − z t is the infinitesimal generator of {ϕt}t≥0.
Wayne Smith Composition Semigroups on BMOA and H∞
Denote by [ϕt, X] the maximal closed subspace of X on which {Ct} is strongly continuous. Theorem (O. Blasco, M. Contreras, S. D´ ıaz-Madrigal, J. Mart´ ınez,
Let {ϕt}t≥0 be a semigroup with generator G and X a Banach space of analytic functions which contains the constant functions and such that sup0≤t≤1 Ct < ∞. Then [ϕt, X] = {f ∈ X : Gf ′ ∈ X}. Here G(z) = lim
t→0+
ϕt(z) − z t is the infinitesimal generator of {ϕt}t≥0. This convergence holds uniformly on compact subsets on D so G ∈ H(D). G has a representation G(z) = (bz − 1)(z − b)P(z), z ∈ D, where b is the Denjoy-Wolff point of {ϕt}t≥0, P ∈ H(D) with Re P(z) ≥ 0 for all z ∈ D.
Wayne Smith Composition Semigroups on BMOA and H∞
Some known results: (i) If X ∈ {Hp (1 ≤ p < ∞), Ap (1 ≤ p < ∞), D, B0, VMOA} and {ϕt} is any semigroup, then [ϕt, X] = X;
Wayne Smith Composition Semigroups on BMOA and H∞
Some known results: (i) If X ∈ {Hp (1 ≤ p < ∞), Ap (1 ≤ p < ∞), D, B0, VMOA} and {ϕt} is any semigroup, then [ϕt, X] = X; (ii) For every nontrivial semigroup {ϕt}, [ϕt, H∞] H∞ and [ϕt, B] B;
Wayne Smith Composition Semigroups on BMOA and H∞
Some known results: (i) If X ∈ {Hp (1 ≤ p < ∞), Ap (1 ≤ p < ∞), D, B0, VMOA} and {ϕt} is any semigroup, then [ϕt, X] = X; (ii) For every nontrivial semigroup {ϕt}, [ϕt, H∞] H∞ and [ϕt, B] B; (iii) For every semigroup {ϕt}, VMOA [ϕt, BMOA] and B0 [ϕt, B].
Wayne Smith Composition Semigroups on BMOA and H∞
Some known results: (i) If X ∈ {Hp (1 ≤ p < ∞), Ap (1 ≤ p < ∞), D, B0, VMOA} and {ϕt} is any semigroup, then [ϕt, X] = X; (ii) For every nontrivial semigroup {ϕt}, [ϕt, H∞] H∞ and [ϕt, B] B; (iii) For every semigroup {ϕt}, VMOA [ϕt, BMOA] and B0 [ϕt, B]. (Berkson and Porta; Siskakis; Blasco, Contreras, D´ ıaz-Madrigal, Mart´ ınez, and Siskakis)
Wayne Smith Composition Semigroups on BMOA and H∞
Statement (ii) can be proved using a functional analytic argument based on H∞ and the Bloch space having the Dunford-Pettis property.
Wayne Smith Composition Semigroups on BMOA and H∞
Statement (ii) can be proved using a functional analytic argument based on H∞ and the Bloch space having the Dunford-Pettis property. (X has the D-P property if every weakly compact operator T : X → Y takes weakly compact sets in X to norm-compact sets in Y .)
Wayne Smith Composition Semigroups on BMOA and H∞
Statement (ii) can be proved using a functional analytic argument based on H∞ and the Bloch space having the Dunford-Pettis property. (X has the D-P property if every weakly compact operator T : X → Y takes weakly compact sets in X to norm-compact sets in Y .) The space BMOA does not have the Dunford-Pettis property, and the corresponding statement had remained open.
Wayne Smith Composition Semigroups on BMOA and H∞
Statement (ii) can be proved using a functional analytic argument based on H∞ and the Bloch space having the Dunford-Pettis property. (X has the D-P property if every weakly compact operator T : X → Y takes weakly compact sets in X to norm-compact sets in Y .) The space BMOA does not have the Dunford-Pettis property, and the corresponding statement had remained open. Theorem (Anderson, Jovovic, S) Suppose H∞ ⊆ X ⊆ B. Then [ϕt, X] X. In particular, [ϕt, BMOA] BMOA.
Wayne Smith Composition Semigroups on BMOA and H∞
Wayne Smith Composition Semigroups on BMOA and H∞
The theorem is an easy consequence of Proposition Given any nontrivial semigroup {ϕt}, there exists f ∈ H∞ such that lim inf
t→0 f ◦ ϕt − f B ≥ 1.
Wayne Smith Composition Semigroups on BMOA and H∞
The theorem is an easy consequence of Proposition Given any nontrivial semigroup {ϕt}, there exists f ∈ H∞ such that lim inf
t→0 f ◦ ϕt − f B ≥ 1.
proof of theorem:
Wayne Smith Composition Semigroups on BMOA and H∞
The theorem is an easy consequence of Proposition Given any nontrivial semigroup {ϕt}, there exists f ∈ H∞ such that lim inf
t→0 f ◦ ϕt − f B ≥ 1.
proof of theorem: Each test function f in the proposition is in H∞, and hence in X from the hypothesis that H∞ ⊆ X.
Wayne Smith Composition Semigroups on BMOA and H∞
The theorem is an easy consequence of Proposition Given any nontrivial semigroup {ϕt}, there exists f ∈ H∞ such that lim inf
t→0 f ◦ ϕt − f B ≥ 1.
proof of theorem: Each test function f in the proposition is in H∞, and hence in X from the hypothesis that H∞ ⊆ X. Since X ⊆ B, the Closed Graph Theorem shows that · B · X and bounding the Bloch norm away from 0 bounds the X norm as well.
Wayne Smith Composition Semigroups on BMOA and H∞
The theorem is an easy consequence of Proposition Given any nontrivial semigroup {ϕt}, there exists f ∈ H∞ such that lim inf
t→0 f ◦ ϕt − f B ≥ 1.
proof of theorem: Each test function f in the proposition is in H∞, and hence in X from the hypothesis that H∞ ⊆ X. Since X ⊆ B, the Closed Graph Theorem shows that · B · X and bounding the Bloch norm away from 0 bounds the X norm as well. Thus it follows from the proposition that f / ∈ [ϕt, X], and so [ϕt, X] X.
Wayne Smith Composition Semigroups on BMOA and H∞
Wayne Smith Composition Semigroups on BMOA and H∞
Let {ϕt} be a nontrivial semigroup, and consider the case that the corresponding Denjoy-Wolff point is 0.
Wayne Smith Composition Semigroups on BMOA and H∞
Let {ϕt} be a nontrivial semigroup, and consider the case that the corresponding Denjoy-Wolff point is 0. Then ϕt(z) = h−1(e−cth(z)), where h : D → Ω, h(0) = 0, Re c ≥ 0, and Ω is spiral-like.
Wayne Smith Composition Semigroups on BMOA and H∞
Let {ϕt} be a nontrivial semigroup, and consider the case that the corresponding Denjoy-Wolff point is 0. Then ϕt(z) = h−1(e−cth(z)), where h : D → Ω, h(0) = 0, Re c ≥ 0, and Ω is spiral-like. Consider the case that Re c > 0.
Wayne Smith Composition Semigroups on BMOA and H∞
Let {ϕt} be a nontrivial semigroup, and consider the case that the corresponding Denjoy-Wolff point is 0. Then ϕt(z) = h−1(e−cth(z)), where h : D → Ω, h(0) = 0, Re c ≥ 0, and Ω is spiral-like. Consider the case that Re c > 0. Choose w0 ∈ ∂Ω such that |w0| = inf{|w| : w ∈ ∂Ω}.
Wayne Smith Composition Semigroups on BMOA and H∞
Let {ϕt} be a nontrivial semigroup, and consider the case that the corresponding Denjoy-Wolff point is 0. Then ϕt(z) = h−1(e−cth(z)), where h : D → Ω, h(0) = 0, Re c ≥ 0, and Ω is spiral-like. Consider the case that Re c > 0. Choose w0 ∈ ∂Ω such that |w0| = inf{|w| : w ∈ ∂Ω}. Then [0, w0) ⊂ Ω,
Wayne Smith Composition Semigroups on BMOA and H∞
Let {ϕt} be a nontrivial semigroup, and consider the case that the corresponding Denjoy-Wolff point is 0. Then ϕt(z) = h−1(e−cth(z)), where h : D → Ω, h(0) = 0, Re c ≥ 0, and Ω is spiral-like. Consider the case that Re c > 0. Choose w0 ∈ ∂Ω such that |w0| = inf{|w| : w ∈ ∂Ω}. Then [0, w0) ⊂ Ω, so w0 is the principal point of an accessible prime end,
Wayne Smith Composition Semigroups on BMOA and H∞
Let {ϕt} be a nontrivial semigroup, and consider the case that the corresponding Denjoy-Wolff point is 0. Then ϕt(z) = h−1(e−cth(z)), where h : D → Ω, h(0) = 0, Re c ≥ 0, and Ω is spiral-like. Consider the case that Re c > 0. Choose w0 ∈ ∂Ω such that |w0| = inf{|w| : w ∈ ∂Ω}. Then [0, w0) ⊂ Ω, so w0 is the principal point of an accessible prime end, and hence there is γ0 ∈ ∂D such that limr→1− h(rγ0) = w0.
Wayne Smith Composition Semigroups on BMOA and H∞
Let {ϕt} be a nontrivial semigroup, and consider the case that the corresponding Denjoy-Wolff point is 0. Then ϕt(z) = h−1(e−cth(z)), where h : D → Ω, h(0) = 0, Re c ≥ 0, and Ω is spiral-like. Consider the case that Re c > 0. Choose w0 ∈ ∂Ω such that |w0| = inf{|w| : w ∈ ∂Ω}. Then [0, w0) ⊂ Ω, so w0 is the principal point of an accessible prime end, and hence there is γ0 ∈ ∂D such that limr→1− h(rγ0) = w0. Thus, lim
r→1− ϕt(rγ0) = h−1(e−ctw0) ∈ D,
t > 0.
Wayne Smith Composition Semigroups on BMOA and H∞
Let {ϕt} be a nontrivial semigroup, and consider the case that the corresponding Denjoy-Wolff point is 0. Then ϕt(z) = h−1(e−cth(z)), where h : D → Ω, h(0) = 0, Re c ≥ 0, and Ω is spiral-like. Consider the case that Re c > 0. Choose w0 ∈ ∂Ω such that |w0| = inf{|w| : w ∈ ∂Ω}. Then [0, w0) ⊂ Ω, so w0 is the principal point of an accessible prime end, and hence there is γ0 ∈ ∂D such that limr→1− h(rγ0) = w0. Thus, lim
r→1− ϕt(rγ0) = h−1(e−ctw0) ∈ D,
t > 0.
Wayne Smith Composition Semigroups on BMOA and H∞
Since ϕt is univalent and bounded, ϕt ∈ D ⊂ B0.
Wayne Smith Composition Semigroups on BMOA and H∞
Since ϕt is univalent and bounded, ϕt ∈ D ⊂ B0. Hence lim
r→1− |ϕ′ t(rγ0)|(1 − r) = 0.
Wayne Smith Composition Semigroups on BMOA and H∞
Since ϕt is univalent and bounded, ϕt ∈ D ⊂ B0. Hence lim
r→1− |ϕ′ t(rγ0)|(1 − r) = 0.
Let f be an infinite interpolating Blaschke product with zeros all
Wayne Smith Composition Semigroups on BMOA and H∞
Since ϕt is univalent and bounded, ϕt ∈ D ⊂ B0. Hence lim
r→1− |ϕ′ t(rγ0)|(1 − r) = 0.
Let f be an infinite interpolating Blaschke product with zeros all
Then lim sup
r→1− |f ′(rγ0)|(1 − r) ≥ δ, for some δ > 0.
Wayne Smith Composition Semigroups on BMOA and H∞
Since ϕt is univalent and bounded, ϕt ∈ D ⊂ B0. Hence lim
r→1− |ϕ′ t(rγ0)|(1 − r) = 0.
Let f be an infinite interpolating Blaschke product with zeros all
Then lim sup
r→1− |f ′(rγ0)|(1 − r) ≥ δ, for some δ > 0.
Also, by continuity of f ′ on D, lim
r→1− |f ′(ϕt(rγ0))| = |f ′(h−1(e−ctw0))| < ∞.
Wayne Smith Composition Semigroups on BMOA and H∞
Since ϕt is univalent and bounded, ϕt ∈ D ⊂ B0. Hence lim
r→1− |ϕ′ t(rγ0)|(1 − r) = 0.
Let f be an infinite interpolating Blaschke product with zeros all
Then lim sup
r→1− |f ′(rγ0)|(1 − r) ≥ δ, for some δ > 0.
Also, by continuity of f ′ on D, lim
r→1− |f ′(ϕt(rγ0))| = |f ′(h−1(e−ctw0))| < ∞.
Thus, for all fixed t > 0,
Wayne Smith Composition Semigroups on BMOA and H∞
Since ϕt is univalent and bounded, ϕt ∈ D ⊂ B0. Hence lim
r→1− |ϕ′ t(rγ0)|(1 − r) = 0.
Let f be an infinite interpolating Blaschke product with zeros all
Then lim sup
r→1− |f ′(rγ0)|(1 − r) ≥ δ, for some δ > 0.
Also, by continuity of f ′ on D, lim
r→1− |f ′(ϕt(rγ0))| = |f ′(h−1(e−ctw0))| < ∞.
Thus, for all fixed t > 0, f ◦ ϕt − f B ≥ lim sup
r→1− |f ′(ϕt(rγ0))ϕ′ t(rγ0) − f ′(rγ0)|(1 − r)
≥ δ.
Wayne Smith Composition Semigroups on BMOA and H∞
Since ϕt is univalent and bounded, ϕt ∈ D ⊂ B0. Hence lim
r→1− |ϕ′ t(rγ0)|(1 − r) = 0.
Let f be an infinite interpolating Blaschke product with zeros all
Then lim sup
r→1− |f ′(rγ0)|(1 − r) ≥ δ, for some δ > 0.
Also, by continuity of f ′ on D, lim
r→1− |f ′(ϕt(rγ0))| = |f ′(h−1(e−ctw0))| < ∞.
Thus, for all fixed t > 0, f ◦ ϕt − f B ≥ lim sup
r→1− |f ′(ϕt(rγ0))ϕ′ t(rγ0) − f ′(rγ0)|(1 − r)
≥ δ. Replacing f by f /δ gives f ◦ ϕt − f B ≥ 1.
Wayne Smith Composition Semigroups on BMOA and H∞
Wayne Smith Composition Semigroups on BMOA and H∞
From (SG1) and (SG3) we have the pointwise convergence ϕt(z) → z as t → 0+.
Wayne Smith Composition Semigroups on BMOA and H∞
From (SG1) and (SG3) we have the pointwise convergence ϕt(z) → z as t → 0+. This is easily be extended to uniform convergence on compact subsets of D.
Wayne Smith Composition Semigroups on BMOA and H∞
From (SG1) and (SG3) we have the pointwise convergence ϕt(z) → z as t → 0+. This is easily be extended to uniform convergence on compact subsets of D. It was recently observed by P. Gumenyuk that this extends to uniform convergence on all of D for every semigroup {ϕt}. Theorem ( Gumenyuk; Anderson, Jovovic, S) For every semigroup {ϕt}, lim
t→0+ ϕt(z) − zH∞ = 0.
Wayne Smith Composition Semigroups on BMOA and H∞
Wayne Smith Composition Semigroups on BMOA and H∞
Corollary Let X be a Banach space that contains H∞, and let XP be the closure of the polynomials in X. For all semigroups {ϕt}, XP ⊂ [ϕt, X].
Wayne Smith Composition Semigroups on BMOA and H∞
Corollary Let X be a Banach space that contains H∞, and let XP be the closure of the polynomials in X. For all semigroups {ϕt}, XP ⊂ [ϕt, X]. This provides a unified proof of some of the known results mentioned above:
Wayne Smith Composition Semigroups on BMOA and H∞
Corollary Let X be a Banach space that contains H∞, and let XP be the closure of the polynomials in X. For all semigroups {ϕt}, XP ⊂ [ϕt, X]. This provides a unified proof of some of the known results mentioned above: [ϕt, Hp] = Hp and [ϕt, Ap] = Ap, all {ϕt}.
Wayne Smith Composition Semigroups on BMOA and H∞
Corollary Let X be a Banach space that contains H∞, and let XP be the closure of the polynomials in X. For all semigroups {ϕt}, XP ⊂ [ϕt, X]. This provides a unified proof of some of the known results mentioned above: [ϕt, Hp] = Hp and [ϕt, Ap] = Ap, all {ϕt}. VMOA [ϕt, BMOA] and B0 [ϕt, B], all {ϕt}.
Wayne Smith Composition Semigroups on BMOA and H∞
Corollary Let X be a Banach space that contains H∞, and let XP be the closure of the polynomials in X. For all semigroups {ϕt}, XP ⊂ [ϕt, X]. This provides a unified proof of some of the known results mentioned above: [ϕt, Hp] = Hp and [ϕt, Ap] = Ap, all {ϕt}. VMOA [ϕt, BMOA] and B0 [ϕt, B], all {ϕt}.
Wayne Smith Composition Semigroups on BMOA and H∞
Corollary Let X be a Banach space that contains H∞, and let XP be the closure of the polynomials in X. For all semigroups {ϕt}, XP ⊂ [ϕt, X]. This provides a unified proof of some of the known results mentioned above: [ϕt, Hp] = Hp and [ϕt, Ap] = Ap, all {ϕt}. VMOA [ϕt, BMOA] and B0 [ϕt, B], all {ϕt}. And hence also that VMOA = [ϕt, VMOA] and B0 = [ϕt, B0], all {ϕt}.
Wayne Smith Composition Semigroups on BMOA and H∞
Corollary Let X be a Banach space that contains H∞, and let XP be the closure of the polynomials in X. For all semigroups {ϕt}, XP ⊂ [ϕt, X]. This provides a unified proof of some of the known results mentioned above: [ϕt, Hp] = Hp and [ϕt, Ap] = Ap, all {ϕt}. VMOA [ϕt, BMOA] and B0 [ϕt, B], all {ϕt}. And hence also that VMOA = [ϕt, VMOA] and B0 = [ϕt, B0], all {ϕt}. It also establishes that the natural extension to H∞ is valid:
Wayne Smith Composition Semigroups on BMOA and H∞
Corollary Let X be a Banach space that contains H∞, and let XP be the closure of the polynomials in X. For all semigroups {ϕt}, XP ⊂ [ϕt, X]. This provides a unified proof of some of the known results mentioned above: [ϕt, Hp] = Hp and [ϕt, Ap] = Ap, all {ϕt}. VMOA [ϕt, BMOA] and B0 [ϕt, B], all {ϕt}. And hence also that VMOA = [ϕt, VMOA] and B0 = [ϕt, B0], all {ϕt}. It also establishes that the natural extension to H∞ is valid: The disk algebra A satisfies A ⊂ [ϕt, H∞], all {ϕt}.
Wayne Smith Composition Semigroups on BMOA and H∞
Wayne Smith Composition Semigroups on BMOA and H∞
This time we consider the case that the Denjoy-Wolff point of {ϕt} is the point 1 on the unit circle.
Wayne Smith Composition Semigroups on BMOA and H∞
This time we consider the case that the Denjoy-Wolff point of {ϕt} is the point 1 on the unit circle. Then there is c ∈ C with Re c ≥ 0 and univalent h : D → Ω, where Ω close-to-convex, such that ϕt(z) = h−1(h(z) + ct).
Wayne Smith Composition Semigroups on BMOA and H∞
This time we consider the case that the Denjoy-Wolff point of {ϕt} is the point 1 on the unit circle. Then there is c ∈ C with Re c ≥ 0 and univalent h : D → Ω, where Ω close-to-convex, such that ϕt(z) = h−1(h(z) + ct). If c = 0, the result is trivial. So assume c = 0.
Wayne Smith Composition Semigroups on BMOA and H∞
This time we consider the case that the Denjoy-Wolff point of {ϕt} is the point 1 on the unit circle. Then there is c ∈ C with Re c ≥ 0 and univalent h : D → Ω, where Ω close-to-convex, such that ϕt(z) = h−1(h(z) + ct). If c = 0, the result is trivial. So assume c = 0. Suppose ϕt(z) does not converge uniformly to z in D.
Wayne Smith Composition Semigroups on BMOA and H∞
This time we consider the case that the Denjoy-Wolff point of {ϕt} is the point 1 on the unit circle. Then there is c ∈ C with Re c ≥ 0 and univalent h : D → Ω, where Ω close-to-convex, such that ϕt(z) = h−1(h(z) + ct). If c = 0, the result is trivial. So assume c = 0. Suppose ϕt(z) does not converge uniformly to z in D. Then there exist some δ > 0 and infinite sequences {tn}, tn → 0+ and {zn} ⊂ D such that δ ≤ |ϕtn(zn) − zn|, n ≥ 1.
Wayne Smith Composition Semigroups on BMOA and H∞
This time we consider the case that the Denjoy-Wolff point of {ϕt} is the point 1 on the unit circle. Then there is c ∈ C with Re c ≥ 0 and univalent h : D → Ω, where Ω close-to-convex, such that ϕt(z) = h−1(h(z) + ct). If c = 0, the result is trivial. So assume c = 0. Suppose ϕt(z) does not converge uniformly to z in D. Then there exist some δ > 0 and infinite sequences {tn}, tn → 0+ and {zn} ⊂ D such that δ ≤ |ϕtn(zn) − zn|, n ≥ 1. Letting wn = h(zn) ∈ Ω,
Wayne Smith Composition Semigroups on BMOA and H∞
This time we consider the case that the Denjoy-Wolff point of {ϕt} is the point 1 on the unit circle. Then there is c ∈ C with Re c ≥ 0 and univalent h : D → Ω, where Ω close-to-convex, such that ϕt(z) = h−1(h(z) + ct). If c = 0, the result is trivial. So assume c = 0. Suppose ϕt(z) does not converge uniformly to z in D. Then there exist some δ > 0 and infinite sequences {tn}, tn → 0+ and {zn} ⊂ D such that δ ≤ |ϕtn(zn) − zn|, n ≥ 1. Letting wn = h(zn) ∈ Ω, |ϕtn(zn) − zn| = |h−1(h(zn) + ctn) − h−1(h(zn))| = |h−1(wn + ctn) − h−1(wn)|.
Wayne Smith Composition Semigroups on BMOA and H∞
Wayne Smith Composition Semigroups on BMOA and H∞
The points wn + ctn and wn are endpoints of a line segment in Ω which pulls back to the Jordan arc ηn = {h−1(wn + ct) : 0 ≤ t ≤ tn} ⊂ D.
Wayne Smith Composition Semigroups on BMOA and H∞
The points wn + ctn and wn are endpoints of a line segment in Ω which pulls back to the Jordan arc ηn = {h−1(wn + ct) : 0 ≤ t ≤ tn} ⊂ D. Since tn → 0 and Ω is compact in the Riemann sphere, we may pass to a subsequence of {wn} and assume the line segment [wn, wn + ctn] = h(ηn) → w0 ∈ Ω ∪ {∞}.
Wayne Smith Composition Semigroups on BMOA and H∞
The points wn + ctn and wn are endpoints of a line segment in Ω which pulls back to the Jordan arc ηn = {h−1(wn + ct) : 0 ≤ t ≤ tn} ⊂ D. Since tn → 0 and Ω is compact in the Riemann sphere, we may pass to a subsequence of {wn} and assume the line segment [wn, wn + ctn] = h(ηn) → w0 ∈ Ω ∪ {∞}. However, diam ηn ≥ |h−1(wn + ctn) − h−1(wn)| = |ϕtn(zn) − zn| ≥ δ, contradicting the fact that univalent functions do not have Koebe arcs.
Wayne Smith Composition Semigroups on BMOA and H∞
The points wn + ctn and wn are endpoints of a line segment in Ω which pulls back to the Jordan arc ηn = {h−1(wn + ct) : 0 ≤ t ≤ tn} ⊂ D. Since tn → 0 and Ω is compact in the Riemann sphere, we may pass to a subsequence of {wn} and assume the line segment [wn, wn + ctn] = h(ηn) → w0 ∈ Ω ∪ {∞}. However, diam ηn ≥ |h−1(wn + ctn) − h−1(wn)| = |ϕtn(zn) − zn| ≥ δ, contradicting the fact that univalent functions do not have Koebe arcs. Therefore, |ϕt(z) − z| → 0 uniformly in D as t → 0+.
Wayne Smith Composition Semigroups on BMOA and H∞
Wayne Smith Composition Semigroups on BMOA and H∞