On evolution semigroups and Trotter product operator-norm estimates - PowerPoint PPT Presentation

Weierstrass Institute for Applied Analysis and Stochastics On evolution semigroups and Trotter product operator-norm estimates Artur Stephan joint work with Hagen Neidhardt and Valentin Zagrebnov Workshop on Operator Theory and Krein Spaces dedicated in memory to Hagen Neidhardt Mohrenstrasse 39 · 10117 Berlin · Germany · Tel. +49 30 20372 0 · www.wias-berlin.de Workshop on Operator Theory and Krein Spaces, Wien, December 19-22, 2019

Non-autonomous abstract Cauchy problems On a Banach space X , solve ∂u ( t ) = − C ( t ) u ( t ) , u ( s ) = x s ∈ X, s ∈ [0 , T ] =: I . ∂t � Question: existence of solution operator U ( t, s ) s.t. u ( t ) = U ( t, s ) u ( s ) is a solution � Solution operator is a strongly continuous, uniformly bounded family of bounded operators { U ( t, s ) } ( t,s ) ∈ ∆ , ∆ := { ( t, s ) ∈ I × I : 0 ≤ s ≤ t ≤ T } , with U ( t, t ) =Id X , t ∈ I , for U ( t, r ) U ( r, s ) = U ( t, s ) , t, r, s ∈ I s ≤ r ≤ t, for with � If C ( t ) = C then T ( t − s ) := U ( t, s ) is a semigroup generated by C . Artur Stephan On evolution semigroups and Trotter product operator-norm estimates · · Workshop on Operator Theory and Krein Spaces, Wien, December 19-22, 2019 · Page 2 (15)

Non-autonomous abstract Cauchy problems On a Banach space X , solve ∂u ( t ) = − C ( t ) u ( t ) , u ( s ) = x s ∈ X, s ∈ [0 , T ] =: I . ∂t � Question: existence of solution operator U ( t, s ) s.t. u ( t ) = U ( t, s ) u ( s ) is a solution � Solution operator is a strongly continuous, uniformly bounded family of bounded operators { U ( t, s ) } ( t,s ) ∈ ∆ , ∆ := { ( t, s ) ∈ I × I : 0 ≤ s ≤ t ≤ T } , with U ( t, t ) =Id X , t ∈ I , for U ( t, r ) U ( r, s ) = U ( t, s ) , t, r, s ∈ I s ≤ r ≤ t, for with � If C ( t ) = C then T ( t − s ) := U ( t, s ) is a semigroup generated by C . � How to solve such evolution equations? Two ways: 1. Approximation of C ( t ) by piecewise constant operators C n ( t ) [Kato ’70]. Question U n ( t, s ) → U ( t, s ) ? 2. Lifting time-dependent problem to autonomous problem as a Cauchy problem on L p ( I , X ) [Howland, Evans, Neidhardt] → Extension problem Artur Stephan On evolution semigroups and Trotter product operator-norm estimates · · Workshop on Operator Theory and Krein Spaces, Wien, December 19-22, 2019 · Page 2 (15)

Time-dependent problem to autonomous problem [Howland, Hagen’s PhD thesis 1979] General Idea: The non-autonomous Cauchy problem in X can be formulated as an autonomous Cauchy problem in the Banach space L p ( I , X ) , p ∈ ]1 , ∞ [ . Artur Stephan On evolution semigroups and Trotter product operator-norm estimates · · Workshop on Operator Theory and Krein Spaces, Wien, December 19-22, 2019 · Page 3 (15)

Time-dependent problem to autonomous problem [Howland, Hagen’s PhD thesis 1979] General Idea: The non-autonomous Cauchy problem in X can be formulated as an autonomous Cauchy problem in the Banach space L p ( I , X ) , p ∈ ]1 , ∞ [ . One-to-one-correspondence: Evolution generator K and solution operators U ( t, s ) : (e − τ K f )( t ) = ( U ( τ ) f )( t ) := U ( t, t − τ ) χ I ( t − τ ) f ( t − τ ) , f ∈ L p ( I , X ) . � Question: How do we find the right evolution generator? ∂t , dom( D 0 ) = { f ∈ W 1 ,p ( I , X ) : f (0) = 0 } : ∂ � Simplest evolution generator D 0 = (e − τ D 0 f )( t ) = χ I ( t − τ ) f ( t − τ ) , f ∈ L p ( I , X ) . corresponds to U ( t, s ) = Id X . � Operators { C ( t ) } t ∈ I on X define induced multiplication operator ( C , dom( C )) on L p ( I , X ) by ( C f )( t ) := C ( t ) f ( t ) , for t ∈ I, dom( C ) := { f ∈ L p ( I , X ) : f ( t ) ∈ dom( C ( t )) , t �→ C ( t ) f ( t ) ∈ L p ( I , X ) } . � Define ˜ K = D 0 + C , dom( ˜ K ) = dom( D 0 ) ∩ dom( C ) ⊂ L p ( I , X ) � Under suitable conditions K = ˜ K is the right evolution generator Artur Stephan On evolution semigroups and Trotter product operator-norm estimates · · Workshop on Operator Theory and Krein Spaces, Wien, December 19-22, 2019 · Page 3 (15)

Looking for evolution generators � Define ( − D 0 generates the shift-semigroup) K = D 0 + C , dom( ˜ ˜ K ) = dom( D 0 ) ∩ dom( C ) ⊂ L p ( I , X ) , � Usually it is hard to answer, whether an extension of ˜ K is an evolution generator � Fact: If ˜ K is an evolution operator (has a good domain), is closable in L p ( I , X ) with closure K which is a generator, then the non-autonomous Cauchy problem has a unique solution operator on I . � ( U ( τ ) f )( t ) = U ( t, t − τ ) χ I ( t − τ ) f ( t − τ ) is a semigroup and ∂ ∂τ U ( τ ) = − ∂ s U ( · , · − τ ) χ I ( · − τ ) f ( · − τ ) + U ( · , · − τ ) ∂ τ ( χ I ( · − τ ) f ( · − τ )) ⇒ ∂ ∂τ U ( τ ) | τ =0 = C ( · ) f + D 0 f � Problem: D 0 is a bad operator. Need some good properties of C for the perturbation and extension problem! Artur Stephan On evolution semigroups and Trotter product operator-norm estimates · · Workshop on Operator Theory and Krein Spaces, Wien, December 19-22, 2019 · Page 4 (15)

Approximation of solution operators � Assume, we know that there is a solution. How can the solution operator be approximated? � Assumption: C ( t ) = A + B ( t ) (e.g. A = − ∆ and B ( t ) is a time-dependent potential) Artur Stephan On evolution semigroups and Trotter product operator-norm estimates · · Workshop on Operator Theory and Krein Spaces, Wien, December 19-22, 2019 · Page 5 (15)

Approximation of solution operators � Assume, we know that there is a solution. How can the solution operator be approximated? � Assumption: C ( t ) = A + B ( t ) (e.g. A = − ∆ and B ( t ) is a time-dependent potential) Theorem (T.Ichinose, H.Tamura ’98) For given positive self-adjoint operators A and B ( t ) on the Hilbert space H satisfying: 1. There is α ∈ [0 , 1) , independent of t ∈ I , such that dom( A α ) ⊂ dom( B ( t )) and the operator B ( t ) A − α : H → H is uniformly bounded, and 2. There is a constant L > 0 such that || A − α ( B ( t ) − B ( s )) A − α || ≤ L | t − s | Then, C ( t ) = A + B ( t ) with domain dom( C ( t )) = dom( A ) and generates contraction propagators { U ( t, s ) } 0 ≤ s ≤ t ≤ T which can be uniformly estimated by n � ln( n ) � � e − t/nA e − t/nB ( jt/n ) || = O || U ( t, 0) − . n j =1 Artur Stephan On evolution semigroups and Trotter product operator-norm estimates · · Workshop on Operator Theory and Krein Spaces, Wien, December 19-22, 2019 · Page 5 (15)

Approximation of solution operators � Assume, we know that there is a solution. How can the solution operator be approximated? � Assumption: C ( t ) = A + B ( t ) (e.g. A = − ∆ and B ( t ) is a time-dependent potential) Theorem (T.Ichinose, H.Tamura ’98) For given positive self-adjoint operators A and B ( t ) on the Hilbert space H satisfying: 1. There is α ∈ [0 , 1) , independent of t ∈ I , such that dom( A α ) ⊂ dom( B ( t )) and the operator B ( t ) A − α : H → H is uniformly bounded, and 2. There is a constant L > 0 such that || A − α ( B ( t ) − B ( s )) A − α || ≤ L | t − s | Then, C ( t ) = A + B ( t ) with domain dom( C ( t )) = dom( A ) and generates contraction propagators { U ( t, s ) } 0 ≤ s ≤ t ≤ T which can be uniformly estimated by n � ln( n ) � � e − t/nA e − t/nB ( jt/n ) || = O || U ( t, 0) − . n j =1 Question: Is this an operator-norm convergence of a Trotter product formula? Artur Stephan On evolution semigroups and Trotter product operator-norm estimates · · Workshop on Operator Theory and Krein Spaces, Wien, December 19-22, 2019 · Page 5 (15)

The Trotter product formula in the strong topology u ( t ) = − Au ( t ) − Bu ( t ) , u (0) = u 0 ∈ X, t ∈ I ˙ Theorem (Classical Trotter product formula, Trotter ’59) Let A and B be two generators on X generating contraction semigroups. If the sum C = A + B is a generator, then its semigroup is given by the Trotter product formula e − tC x = lim n →∞ [e − t/nA e − t/nB ] n x, with uniform convergence compact intervals [0 , T ] . Artur Stephan On evolution semigroups and Trotter product operator-norm estimates · · Workshop on Operator Theory and Krein Spaces, Wien, December 19-22, 2019 · Page 6 (15)

The Trotter product formula in the strong topology u ( t ) = − Au ( t ) − Bu ( t ) , u (0) = u 0 ∈ X, t ∈ I ˙ Theorem (Classical Trotter product formula, Trotter ’59) Let A and B be two generators on X generating contraction semigroups. If the sum C = A + B is a generator, then its semigroup is given by the Trotter product formula e − tC x = lim n →∞ [e − t/nA e − t/nB ] n x, with uniform convergence compact intervals [0 , T ] . Aim: � Want to apply Trotter product on L p ( I , X ) for A and B � Want to show that the Trotter product converges in operator-norm which can be estimated Artur Stephan On evolution semigroups and Trotter product operator-norm estimates · · Workshop on Operator Theory and Krein Spaces, Wien, December 19-22, 2019 · Page 6 (15)