Modern Aspects of Complex Analysis and Its Applications Wayne Smith - PowerPoint PPT Presentation

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 D . A one-parameter semigroup { ϕ t } t ≥ 0 of analytic functions on D is a family of analytic functions ϕ t : D → D that satisfies the following three conditions: 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 D . A one-parameter semigroup { ϕ t } t ≥ 0 of analytic functions on D is a family of analytic functions ϕ t : D → D that satisfies the following three conditions: (SG1) ϕ 0 is the identity, i.e. ϕ 0 ( z ) = z , z ∈ D ; 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 D . A one-parameter semigroup { ϕ t } t ≥ 0 of analytic functions on D is a family of analytic functions ϕ t : D → D that satisfies the following three conditions: (SG1) ϕ 0 is the identity, i.e. ϕ 0 ( z ) = z , z ∈ D ; (SG2) ϕ s + t = ϕ s ◦ ϕ t , for all t , s ≥ 0; 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 D . A one-parameter semigroup { ϕ t } t ≥ 0 of analytic functions on D is a family of analytic functions ϕ t : D → D that satisfies the following three conditions: (SG1) ϕ 0 is the identity, i.e. ϕ 0 ( z ) = z , z ∈ D ; (SG2) ϕ s + t = ϕ s ◦ ϕ t , for all t , s ≥ 0; (SG3) the mapping ( t , z ) → ϕ t ( z ) is continuous on [0 , ∞ ) × D . 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 D . A one-parameter semigroup { ϕ t } t ≥ 0 of analytic functions on D is a family of analytic functions ϕ t : D → D that satisfies the following three conditions: (SG1) ϕ 0 is the identity, i.e. ϕ 0 ( z ) = z , z ∈ D ; (SG2) ϕ s + t = ϕ s ◦ ϕ t , for all t , s ≥ 0; (SG3) the mapping ( t , z ) → ϕ t ( z ) is continuous on [0 , ∞ ) × D . The trivial case is that ϕ t ( z ) = z for all t ≥ 0. Otherwise, we say that { ϕ t } is nontrivial. Wayne Smith Composition Semigroups on BMOA and H ∞

Examples of semigroups of analytic functions Wayne Smith Composition Semigroups on BMOA and H ∞

Examples of semigroups of analytic functions ϕ t ( z ) = e − ct z , where Re c ≥ 0. Wayne Smith Composition Semigroups on BMOA and H ∞

Examples of semigroups of analytic functions ϕ t ( z ) = e − ct z , where Re c ≥ 0. Rotation and Shrinking, common fixed point point 0 Wayne Smith Composition Semigroups on BMOA and H ∞

Examples of semigroups of analytic functions ϕ t ( z ) = e − ct z , where Re c ≥ 0. Rotation and Shrinking, common fixed point point 0 Wayne Smith Composition Semigroups on BMOA and H ∞

Examples of semigroups of analytic functions ϕ t ( z ) = e − ct z , where Re c ≥ 0. Rotation and Shrinking, common fixed point point 0 ϕ t ( z ) = 1 + e − t ( z − 1). Wayne Smith Composition Semigroups on BMOA and H ∞

Examples of semigroups of analytic functions ϕ t ( z ) = e − ct z , where Re c ≥ 0. Rotation and Shrinking, common fixed point point 0 ϕ t ( 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 ∞

Examples of semigroups of analytic functions ϕ t ( z ) = e − ct z , where Re c ≥ 0. Rotation and Shrinking, common fixed point point 0 ϕ t ( 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 ∞

Examples of semigroups of analytic functions ϕ t ( z ) = e − ct z , where Re c ≥ 0. Rotation and Shrinking, common fixed point point 0 ϕ t ( 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 ∞

Structure of semigroups of analytic functions; Berkson and Porta 1978 Wayne Smith Composition Semigroups on BMOA and H ∞

Structure of semigroups of analytic functions; Berkson and Porta 1978 Every nontrivial semigroup of analytic functions { ϕ t } 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 ∞

Structure of semigroups of analytic functions; Berkson and Porta 1978 Every nontrivial semigroup of analytic functions { ϕ t } 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 ∞

Structure of semigroups of analytic functions; Berkson and Porta 1978 Every nontrivial semigroup of analytic functions { ϕ t } 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 ϕ t ( z ) = h − 1 ( e − ct h ( 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 ∞

Structure of semigroups of analytic functions; Berkson and Porta 1978 Every nontrivial semigroup of analytic functions { ϕ t } 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 ϕ t ( z ) = h − 1 ( e − ct h ( 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 ϕ t ( 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 ∞

Composition semigroups Wayne Smith Composition Semigroups on BMOA and H ∞

Composition semigroups Associated with the semigroup { ϕ t } is the composition semigroup of linear operators { C t } , where C t ( f ) = f ◦ ϕ t for f ∈ H ( D ). Wayne Smith Composition Semigroups on BMOA and H ∞

Composition semigroups Associated with the semigroup { ϕ t } is the composition semigroup of linear operators { C t } , where C t ( f ) = f ◦ ϕ t for f ∈ H ( D ). If C t is a bounded operator on some Banach space X ⊂ H ( D ) for all t ≥ 0, we say that the semigroup { ϕ t } acts on X . Wayne Smith Composition Semigroups on BMOA and H ∞

Composition semigroups Associated with the semigroup { ϕ t } is the composition semigroup of linear operators { C t } , where C t ( f ) = f ◦ ϕ t for f ∈ H ( D ). If C t is a bounded operator on some Banach space X ⊂ H ( D ) for all t ≥ 0, we say that the semigroup { ϕ t } acts on X . If in addition the strong continuity condition t → 0 + � f ◦ ϕ t − f � X = 0 lim holds for all f ∈ X , then it is said that { ϕ t } is strongly continuous on X . Wayne Smith Composition Semigroups on BMOA and H ∞

Space of strong continuity Wayne Smith Composition Semigroups on BMOA and H ∞

Space of strong continuity Denote by [ ϕ t , X ] the maximal closed subspace of X on which { C t } is strongly continuous. Wayne Smith Composition Semigroups on BMOA and H ∞

Space of strong continuity Denote by [ ϕ t , X ] the maximal closed subspace of X on which { C t } is strongly continuous. Theorem (O. Blasco, M. Contreras, S. D´ ıaz-Madrigal, J. Mart´ ınez, M. Papadimitrakis, and A. Siskakis) Let { ϕ t } t ≥ 0 be a semigroup with generator G and X a Banach space of analytic functions which contains the constant functions and such that sup 0 ≤ t ≤ 1 � C t � < ∞ . Then [ ϕ t , X ] = { f ∈ X : Gf ′ ∈ X } . Wayne Smith Composition Semigroups on BMOA and H ∞

Space of strong continuity Denote by [ ϕ t , X ] the maximal closed subspace of X on which { C t } is strongly continuous. Theorem (O. Blasco, M. Contreras, S. D´ ıaz-Madrigal, J. Mart´ ınez, M. Papadimitrakis, and A. Siskakis) Let { ϕ t } t ≥ 0 be a semigroup with generator G and X a Banach space of analytic functions which contains the constant functions and such that sup 0 ≤ t ≤ 1 � C t � < ∞ . Then [ ϕ t , X ] = { f ∈ X : Gf ′ ∈ X } . ϕ t ( z ) − z Here G ( z ) = lim is the infinitesimal generator of t t → 0 + { ϕ t } t ≥ 0 . Wayne Smith Composition Semigroups on BMOA and H ∞