New results on semigroups of analytic functions OSCAR BLASCO Departamento An´ alisis Matem´ atico Universidad Valencia 2013 AHA Granada, 23 May 2013 www.uv.es/oblasco Oscar Blasco New results on semigroups of analytic functions
References The basic definitions New results on semigroups of analytic functions A theorem with proof Contents References 1 Oscar Blasco New results on semigroups of analytic functions
References The basic definitions New results on semigroups of analytic functions A theorem with proof Contents References 1 The basic definitions 2 Oscar Blasco New results on semigroups of analytic functions
References The basic definitions New results on semigroups of analytic functions A theorem with proof Contents References 1 The basic definitions 2 New results on semigroups of analytic functions 3 Oscar Blasco New results on semigroups of analytic functions
References The basic definitions New results on semigroups of analytic functions A theorem with proof Contents References 1 The basic definitions 2 New results on semigroups of analytic functions 3 A theorem with proof 4 Oscar Blasco New results on semigroups of analytic functions
References The basic definitions New results on semigroups of analytic functions A theorem with proof The papers and their authors BCDMPS Semigroups of composition operators and integral operators in spaces of analytic functions Ann. Acad. Scient. Fennicae Math. 38 (2013), 1-23. BCDMS Semigroups of composition operators in BMOA and the extension of a theorem of Sarason Int. Eq. Oper. Theory 61 (2008), 45-62. Authors: B=Oscar Blasco, M=Josep Martinez (Univ. Valencia) C= Manuel Contreras, D= Santiago Diaz-Madrigal (Univ. Sevilla) S= Aristomenis Siskakis (Univ. Thessaloniki, Grecia) P=Michael Papadimitrakis (Univ. Crete) Oscar Blasco New results on semigroups of analytic functions
References The basic definitions New results on semigroups of analytic functions A theorem with proof Semigroups of analytic functions A (one-parameter) semigroup of analytic functions is any continuous homomorphism Φ : ( R + , +) → { f ∈ H ∞ ( D ) : � f � ∞ ≤ 1 } , that is t �→ Φ( t ) = ϕ t from the additive semigroup of nonnegative real numbers into the composition semigroup of all analytic functions which map D into D . Oscar Blasco New results on semigroups of analytic functions
References The basic definitions New results on semigroups of analytic functions A theorem with proof Semigroups of analytic functions A (one-parameter) semigroup of analytic functions is any continuous homomorphism Φ : ( R + , +) → { f ∈ H ∞ ( D ) : � f � ∞ ≤ 1 } , that is t �→ Φ( t ) = ϕ t from the additive semigroup of nonnegative real numbers into the composition semigroup of all analytic functions which map D into D . Φ = ( ϕ t ) consists of ϕ t ∈ H ( D ) with ϕ t ( D ) ⊂ D and satisfying ϕ 0 is the identity in D , 1 ϕ t + s = ϕ t ◦ ϕ s , for all t , s ≥ 0 , 2 ϕ t ( z ) → ϕ 0 ( z ) = z , as t → 0, z ∈ D . 3 Oscar Blasco New results on semigroups of analytic functions
References The basic definitions New results on semigroups of analytic functions A theorem with proof Semigroups of analytic functions A (one-parameter) semigroup of analytic functions is any continuous homomorphism Φ : ( R + , +) → { f ∈ H ∞ ( D ) : � f � ∞ ≤ 1 } , that is t �→ Φ( t ) = ϕ t from the additive semigroup of nonnegative real numbers into the composition semigroup of all analytic functions which map D into D . Φ = ( ϕ t ) consists of ϕ t ∈ H ( D ) with ϕ t ( D ) ⊂ D and satisfying ϕ 0 is the identity in D , 1 ϕ t + s = ϕ t ◦ ϕ s , for all t , s ≥ 0 , 2 ϕ t ( z ) → ϕ 0 ( z ) = z , as t → 0, z ∈ D . 3 Examples: φ t ( z ) = e − t z (Dilation semigroup) φ t ( z ) = e it z (Rotation semigroup) φ t ( z ) = e − t z +(1 − e − t ) Oscar Blasco New results on semigroups of analytic functions
References The basic definitions New results on semigroups of analytic functions A theorem with proof Generators of analytic semigroups (E. Berkson, H. Porta (1978)) The infinitesimal generator of ( ϕ t ) is the function ϕ t ( z ) − z = ∂ϕ t G ( z ) := l´ ım ∂ t ( z ) | t =0 , z ∈ D . t t → 0 + Oscar Blasco New results on semigroups of analytic functions
References The basic definitions New results on semigroups of analytic functions A theorem with proof Generators of analytic semigroups (E. Berkson, H. Porta (1978)) The infinitesimal generator of ( ϕ t ) is the function ϕ t ( z ) − z = ∂ϕ t G ( z ) := l´ ım ∂ t ( z ) | t =0 , z ∈ D . t t → 0 + G ( ϕ t ( z )) = ∂ϕ t ( z ) = G ( z ) ∂ϕ t ( z ) , z ∈ D , t ≥ 0 . (1) ∂ t ∂ z Oscar Blasco New results on semigroups of analytic functions
References The basic definitions New results on semigroups of analytic functions A theorem with proof Generators of analytic semigroups (E. Berkson, H. Porta (1978)) The infinitesimal generator of ( ϕ t ) is the function ϕ t ( z ) − z = ∂ϕ t G ( z ) := l´ ım ∂ t ( z ) | t =0 , z ∈ D . t t → 0 + G ( ϕ t ( z )) = ∂ϕ t ( z ) = G ( z ) ∂ϕ t ( z ) , z ∈ D , t ≥ 0 . (1) ∂ t ∂ z G has a unique representation G ( z ) = ( bz − 1)( z − b ) P ( z ) , z ∈ D where b ∈ D ( called the Denjoy-Wolff point of the semigroup) and P ∈ H ( D ) with Re P ( z ) ≥ 0 for all z ∈ D . Oscar Blasco New results on semigroups of analytic functions
References The basic definitions New results on semigroups of analytic functions A theorem with proof Generators of analytic semigroups (E. Berkson, H. Porta (1978)) The infinitesimal generator of ( ϕ t ) is the function ϕ t ( z ) − z = ∂ϕ t G ( z ) := l´ ım ∂ t ( z ) | t =0 , z ∈ D . t t → 0 + G ( ϕ t ( z )) = ∂ϕ t ( z ) = G ( z ) ∂ϕ t ( z ) , z ∈ D , t ≥ 0 . (1) ∂ t ∂ z G has a unique representation G ( z ) = ( bz − 1)( z − b ) P ( z ) , z ∈ D where b ∈ D ( called the Denjoy-Wolff point of the semigroup) and P ∈ H ( D ) with Re P ( z ) ≥ 0 for all z ∈ D . G ( z ) = − z for the dilation semigroup ( b = 0, P ( z ) = 1) G ( z ) = iz for the rotation semigroup ( b = 0, P ( z ) = − i ) G ( z ) = − ( z − 1) for φ t ( z ) = e − t z +1 − e − t ( b = 1, P ( z ) = 1 1 − z ) Oscar Blasco New results on semigroups of analytic functions
References The basic definitions New results on semigroups of analytic functions A theorem with proof Semigroups of operators Each semigroup of analytic functions gives rise to a semigroup ( C t ) consisting of composition operators on H ( D ) via composition C t ( f ) := f ◦ ϕ t , f ∈ H ( D ) . Oscar Blasco New results on semigroups of analytic functions
References The basic definitions New results on semigroups of analytic functions A theorem with proof Semigroups of operators Each semigroup of analytic functions gives rise to a semigroup ( C t ) consisting of composition operators on H ( D ) via composition C t ( f ) := f ◦ ϕ t , f ∈ H ( D ) . Given a Banach space X ⊂ H ( D ) and a semigroup ( ϕ t ) , we say that ( ϕ t ) generates a semigroup of operators on X if ( C t ) is a C 0 -semigroup of bounded operators in X , i.e. Oscar Blasco New results on semigroups of analytic functions
References The basic definitions New results on semigroups of analytic functions A theorem with proof Semigroups of operators Each semigroup of analytic functions gives rise to a semigroup ( C t ) consisting of composition operators on H ( D ) via composition C t ( f ) := f ◦ ϕ t , f ∈ H ( D ) . Given a Banach space X ⊂ H ( D ) and a semigroup ( ϕ t ) , we say that ( ϕ t ) generates a semigroup of operators on X if ( C t ) is a C 0 -semigroup of bounded operators in X , i.e. C t ( f ) ∈ X for all t ≥ 0 and for every f ∈ X l´ ım t → 0 + � C t ( f ) − f � X = 0 . Oscar Blasco New results on semigroups of analytic functions
References The basic definitions New results on semigroups of analytic functions A theorem with proof Semigroups of operators Each semigroup of analytic functions gives rise to a semigroup ( C t ) consisting of composition operators on H ( D ) via composition C t ( f ) := f ◦ ϕ t , f ∈ H ( D ) . Given a Banach space X ⊂ H ( D ) and a semigroup ( ϕ t ) , we say that ( ϕ t ) generates a semigroup of operators on X if ( C t ) is a C 0 -semigroup of bounded operators in X , i.e. C t ( f ) ∈ X for all t ≥ 0 and for every f ∈ X l´ ım t → 0 + � C t ( f ) − f � X = 0 . Given a semigroup ( ϕ t ) and a Banach space X contained in H ( D ) we denote by [ ϕ t , X ] the maximal closed linear subspace of X such that ( ϕ t ) generates a semigroups of operators on it. Oscar Blasco New results on semigroups of analytic functions
References The basic definitions New results on semigroups of analytic functions A theorem with proof Previous results on semigroups of analytic functions Theorem Every semigroup of analytic functions generates a semigroup of 1 operators on the Hardy spaces H p (1 ≤ p < ∞ ) , the Bergman spaces A p (1 ≤ p < ∞ ) and the Dirichlet space, i.e. [ ϕ t , X ] = X in these cases. Oscar Blasco New results on semigroups of analytic functions
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