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

• perator-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) ∂t = −C(t)u(t), u(s) = xs ∈ X, s ∈ [0, T] =: I.

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) =IdX,

for

t ∈ I, U(t, r)U(r, s) =U(t, s),

for

t, r, s ∈ I

with

s ≤ r ≤ t,

If C(t) = C then T(t − s) := U(t, s) is a semigroup generated by C.

Artur Stephan

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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) ∂t = −C(t)u(t), u(s) = xs ∈ X, s ∈ [0, T] =: I.

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) =IdX,

for

t ∈ I, U(t, r)U(r, s) =U(t, s),

for

t, r, s ∈ I

with

s ≤ r ≤ t,

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 Cn(t) [Kato ’70].

Question Un(t, s) → U(t, s)?

• 2. Lifting time-dependent problem to autonomous problem as a Cauchy problem on

Lp(I, X)[Howland, Evans, Neidhardt] → Extension problem

Artur Stephan

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On evolution semigroups and Trotter product operator-norm estimates

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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 Lp(I, X), p ∈]1, ∞[.

Artur Stephan

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On evolution semigroups and Trotter product operator-norm estimates

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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 Lp(I, X), p ∈]1, ∞[. One-to-one-correspondence: Evolution generator K and solution operators U(t, s):

(e−τKf)(t) = (U(τ)f)(t) := U(t, t − τ)χI(t − τ)f(t − τ), f ∈ Lp(I, X).

Question: How do we find the right evolution generator? Simplest evolution generator D0 =

∂ ∂t , dom(D0) = {f ∈ W 1,p(I, X) : f(0) = 0}:

(e−τD0f)(t) = χI(t − τ)f(t − τ), f ∈ Lp(I, X).

corresponds to U(t, s) = IdX.

Operators {C(t)}t∈I on X define induced multiplication operator (C, dom(C)) on

Lp(I, X) by (Cf)(t) := C(t)f(t), for t ∈ I, dom(C) := {f ∈ Lp(I, X) : f(t) ∈ dom(C(t)), t → C(t)f(t) ∈ Lp(I, X)}.

Define ˜

K = D0 + C, dom( ˜ K) = dom(D0) ∩ dom(C) ⊂ Lp(I, X)

Under suitable conditions K = ˜

K is the right evolution generator

Artur Stephan

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On evolution semigroups and Trotter product operator-norm estimates

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Workshop on Operator Theory and Krein Spaces, Wien, December 19-22, 2019 · Page 3 (15)

Looking for evolution generators

Define (−D0 generates the shift-semigroup)

˜ K = D0 + C, dom( ˜ K) = dom(D0) ∩ dom(C) ⊂ Lp(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 Lp(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)χI(t − τ)f(t − τ) is a semigroup and

∂ ∂τ U(τ) = − ∂sU(·, · − τ)χI(· − τ)f(· − τ) + U(·, · − τ)∂τ(χI(· − τ)f(· − τ)) ⇒ ∂ ∂τ U(τ)|τ=0 = C(·)f + D0f

Problem: D0 is a bad operator. Need some good properties of C for the perturbation and

extension problem!

Artur Stephan

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On evolution semigroups and Trotter product operator-norm estimates

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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

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On evolution semigroups and Trotter product operator-norm estimates

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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(Aα) ⊂ dom(B(t)) and the
• perator 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

||U(t, 0) −

n

• j=1

e−t/nAe−t/nB(jt/n)|| = O ln(n) n

• .

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(Aα) ⊂ dom(B(t)) and the
• perator 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

||U(t, 0) −

n

• j=1

e−t/nAe−t/nB(jt/n)|| = O ln(n) n

• .

Question: Is this an operator-norm convergence of a Trotter product formula?

Artur Stephan

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On evolution semigroups and Trotter product operator-norm estimates

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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) = u0 ∈ 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−tCx = lim

n→∞[e−t/nAe−t/nB]nx,

with uniform convergence compact intervals [0, T].

Artur Stephan

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On evolution semigroups and Trotter product operator-norm estimates

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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) = u0 ∈ 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−tCx = lim

n→∞[e−t/nAe−t/nB]nx,

with uniform convergence compact intervals [0, T]. Aim:

Want to apply Trotter product on Lp(I, X) for A and B Want to show that the Trotter product converges in operator-norm which can be estimated

Artur Stephan

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On evolution semigroups and Trotter product operator-norm estimates

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Workshop on Operator Theory and Krein Spaces, Wien, December 19-22, 2019 · Page 6 (15)

Particular example: C(t) = A + B(t)

Assumption: C(t) = A + B(t), dom(C(t)) = dom(A) ∩ dom(B(t)) where A is a generator of a holomorphic contraction semigroup, 0 ∈ ρ(A) B(t) small perturbation: ∃α ∈]0, 1[: ess supt∈IB(t)A−α < ∞. Why generators of holomorphic semigroups? Compare to the talk of Valentin! Possible to define fractional powers Estimate Aαe−tA ≤ t−α Since A is time-independent, its semigroup commutes with the semigroup of D0. Define

K0 as the generator of e−τD0e−τAf = e−τAχI(· − τ)f(· − τ)

If A = A(t) this would not work! Note: D0 + A not necessarily closed and dom(K0) dom(D0) ∩ dom(A). We do

not assume maximal parabolic regularity which means K0 = D0 + A.

Artur Stephan

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On evolution semigroups and Trotter product operator-norm estimates

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Workshop on Operator Theory and Krein Spaces, Wien, December 19-22, 2019 · Page 7 (15)

Particular example: C(t) = A + B(t)

Assumption: C(t) = A + B(t), dom(C(t)) = dom(A) ∩ dom(B(t)) where A is a generator of a holomorphic contraction semigroup, 0 ∈ ρ(A) B(t) small perturbation: ∃α ∈]0, 1[: ess supt∈IB(t)A−α < ∞. Why generators of holomorphic semigroups? Compare to the talk of Valentin! Possible to define fractional powers Estimate Aαe−tA ≤ t−α Since A is time-independent, its semigroup commutes with the semigroup of D0. Define

K0 as the generator of e−τD0e−τAf = e−τAχI(· − τ)f(· − τ)

If A = A(t) this would not work! Note: D0 + A not necessarily closed and dom(K0) dom(D0) ∩ dom(A). We do

not assume maximal parabolic regularity which means K0 = D0 + A.

Idea: Investigate K0 + B where K0 = D0 + A is a sum of two operators

→ Trotter product formula

Artur Stephan

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On evolution semigroups and Trotter product operator-norm estimates

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Workshop on Operator Theory and Krein Spaces, Wien, December 19-22, 2019 · Page 7 (15)

Assumptions on A and B(t) Assumptions: (A1) The operator A is a generator of a holomorphic contraction semigroup and 0 ∈ ̺(A). (A2) The operators {B(t), dom(B(t))}t∈I are contraction generators for a.e. t ∈ I. It holds dom(A) ⊂ dom(B(t)) for a.e. t ∈ I and for all x ∈ dom(A) the function

t → B(t)x is in Lp(I, X).

(A3) There is a α ∈ (0, 1) such that for t ∈ I it holds dom(Aα) ⊂ dom(B(t)) and

sup

t∈I

||B(t)A−α|| < ∞.

(A4) For the adjoint operators, it holds dom(A∗) ⊂ dom(B(t)∗) and

sup

t∈I

||B(t)∗(A∗)−1|| < ∞,

(A5) It holds the following estimate for some β ∈ (α, 1)

||A−1(B(t) − B(s))A−α|| ≤ Lβ|t − s|β, t, s ∈ I.

Artur Stephan

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On evolution semigroups and Trotter product operator-norm estimates

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Workshop on Operator Theory and Krein Spaces, Wien, December 19-22, 2019 · Page 8 (15)

Theorem (Error estimate in operator norm in Lp(I, X) – Neidhardt-AS-Zagrebnov ) Let A and {B(t)}t∈I satisfying the above assumptions. Then, it holds:

||(e−τB/ne−τK0/n)n − e−τ(B+K0)|| = O(n−(β−α)).

Artur Stephan

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On evolution semigroups and Trotter product operator-norm estimates

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Theorem (Error estimate in operator norm in Lp(I, X) – Neidhardt-AS-Zagrebnov ) Let A and {B(t)}t∈I satisfying the above assumptions. Then, it holds:

||(e−τB/ne−τK0/n)n − e−τ(B+K0)|| = O(n−(β−α)).

Idea of proof: Let σ = τ

n . Decompose:

(e−τB/ne−τK0/n)n − e−τ(B+K0) = (e−σBe−σK0)n − (e−σ(B+K0))n = T (σ)n − U(σ)n =

n−1

• k=0

T (σ)n−k−1(T (σ) − U(σ))U(σ)k = =

n−1

• k=0

T (σ)n−k−1AA−1(T (σ) − U(σ))A−αAαU(σ)k.

Then, estimate each term separately. Works, since A dominates B(t), i.e. A dominates B.

Artur Stephan

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On evolution semigroups and Trotter product operator-norm estimates

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Workshop on Operator Theory and Krein Spaces, Wien, December 19-22, 2019 · Page 9 (15)

Theorem (Error estimate in operator norm in Lp(I, X) – Neidhardt-AS-Zagrebnov ) Let A and {B(t)}t∈I satisfying the above assumptions. Then, it holds:

||(e−τB/ne−τK0/n)n − e−τ(B+K0)|| = O(n−(β−α)).

Idea of proof: Let σ = τ

n . Decompose:

(e−τB/ne−τK0/n)n − e−τ(B+K0) = (e−σBe−σK0)n − (e−σ(B+K0))n = T (σ)n − U(σ)n =

n−1

• k=0

T (σ)n−k−1(T (σ) − U(σ))U(σ)k = =

n−1

• k=0

T (σ)n−k−1AA−1(T (σ) − U(σ))A−αAαU(σ)k.

Then, estimate each term separately. Works, since A dominates B(t), i.e. A dominates B. Question: What does this mean for the propagators?

Artur Stephan

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On evolution semigroups and Trotter product operator-norm estimates

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Workshop on Operator Theory and Krein Spaces, Wien, December 19-22, 2019 · Page 9 (15)

Equivalence convergence of Trotter product and approximation of solution operators

K0 be the generator of an evolution semigroup {U0(τ)}τ≥0, B be a multiplication

• perator induced by a measurable family {B(t)}t∈I of generators of contraction
• semigroups. Then

(U0(τ)f)(t) = U0(t, t − τ)χI(t − τ)f(t − τ), f ∈ Lp(I, X).

We define for j ∈ {1, 2, . . . , n}, n ∈ N, (t, s) ∈ ∆

Gj(t, s; n) := U0(s + j (t−s)

n

, s + (j − 1) (t−s)

n

)e− (t−s)

n

B

• s+(j−1) (t−s)

n

• and we set

Un(t, s) :=

n ←

• j=1

Gj(t, s; n), n ∈ N, (t, s) ∈ ∆,

Hence

• e−τK0/ne−τB/nn

f

• (t) = Un(t, t − τ)χI(t − τ)f(t − τ)

Proposition (Neidhardt-AS-Zagrebnov)

sup

τ≥0

• e−τK−
• e−τK0/ne−τB/nn
• B(Lp(I,X))

= ess sup(t,s)∈∆U(t, s)−Un(t, s)B(X)

Artur Stephan

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On evolution semigroups and Trotter product operator-norm estimates

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Theorem (Error estimate for propagators – Neidhardt-AS-Zagrebnov) Assume the same as in Theorem (Error estimate in operator in Lp(I, X)). Then

sup

(t,s)∈∆

||Un(t, s) − U(t, s)|| = O(n−(β−α)),

where the approximating propagator Un(t, s) has the form

Un(t, s) =

n

• j=1

e− t−s

n B(s+(n+1−j) t−s n )e− t−s n A.

Compare to result of [Ichinose-Tamura ’98]:

||U(t, 0) − Un(t, 0)|| = O ln(n) n

• .

Artur Stephan

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On evolution semigroups and Trotter product operator-norm estimates

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Workshop on Operator Theory and Krein Spaces, Wien, December 19-22, 2019 · Page 11 (15)

Optimality?

Summarizing: K0 = D0 + A

||(e−τB/ne−τK0/n)n − e−τ(B+K0)|| = O(n−(β−α)).

if

||A−1(B(t) − B(s))A−α|| ≤ Lβ|t − s|β, t, s ∈ I.

If B : I → B(X) is Hölder continuous with power β, then we may choose α = 0 and

||(e−τB/ne−τK0/n)n − e−τ(B+K0)|| = O(n−γ).

for any γ ∈]0, β[.

If A = 0, we also conclude

||(e−τB/ne−τD0/n)n − e−τ(B+D0)|| = O(n−γ).

for any γ ∈]0, β[.

In the following: Is this optimal?

Artur Stephan

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On evolution semigroups and Trotter product operator-norm estimates

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Workshop on Operator Theory and Krein Spaces, Wien, December 19-22, 2019 · Page 12 (15)

Easy case which is fully explicit: Bounded perturbations of the shift-semigroup

Let X = C, I = [0, 1] B multiplication operator induced by bounded measurable q : [0, 1] → R. Then, q

induces a bounded multiplication operator Q on the Banach space Lp(I):

(Qf)(t) = q(t)f(t), f ∈ Lp(I)

We assume that q ≥ 0. Hence Q generates on Lp(I) a contraction semigroup. The Trotter product formula in the strong topology follows immediately

• e−τD0/ne−τQ/nn

f → e−τ(D0+Q)f, f ∈ Lp(I),

• e−τ(D0+Q)f
• (t) = e−

t

t−τ q(y)dyf(t − τ)χI(t − τ).

Hence the solution operator is U(t, s) = e−

t

s q(y)dy

• e−τD0/ne−τQ/nn

f

• (t) = Un(t, t − τ)χI(t − τ)f(t − τ) .

where

Un(t, s) = e− t−s

n n−1

k=0 q(s+k t−s

n ). Artur Stephan

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On evolution semigroups and Trotter product operator-norm estimates

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Workshop on Operator Theory and Krein Spaces, Wien, December 19-22, 2019 · Page 13 (15)

From operator theory to integral approximation Proposition (Neidhardt-AS-Zagrebnov) Let q ∈ L∞(I) be non-negative. Then, as n → ∞,

sup

τ≥0

• e−τ(D0+Q) −
• e−τD0/ne−τQ/nn
• B(Lp(I))

≈

• ess sup(t,s)∈∆
• t

s

q(y)dy − t − s n

n−1

• k=0

q(s + k t−s

n )

• Artur Stephan

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On evolution semigroups and Trotter product operator-norm estimates

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From operator theory to integral approximation Proposition (Neidhardt-AS-Zagrebnov) Let q ∈ L∞(I) be non-negative. Then, as n → ∞,

sup

τ≥0

• e−τ(D0+Q) −
• e−τD0/ne−τQ/nn
• B(Lp(I))

≈

• ess sup(t,s)∈∆
• t

s

q(y)dy − t − s n

n−1

• k=0

q(s + k t−s

n )

• So let us compute the Riemann sum of an L∞- function!

Artur Stephan

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On evolution semigroups and Trotter product operator-norm estimates

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Operator-norm convergence depends on time regularity Theorem (Neidhardt-AS-Zagrebnov)

• 1. If q : [0, 1] → R is Hölder continuous with β ∈]0, 1], then

sup

τ≥0

• e−τ(D0+Q) −
• e−τD0/ne−τQ/nn
• B(Lp(I)) = O(n−β)
• 2. If q : I → R is continuous and non-negative, then
• e−τ(D0+Q) −
• e−τD0/ne−τQ/nn
• B(Lp(I)) = o(1).
• 3. Let δn > 0 be a sequence with δn → 0 as n → ∞. Then there exists a continuous

function q : I = [0, 1] → R such that

sup

τ≥0

• e−τ(D0+Q) −
• e−τD0/ne−τQ/nn
• B(Lp(I)) ≈ ˜

δn, ˜ δn/δn → ∞

• 4. There is a non-negative function q ∈ L∞([0, 1]) such that

lim sup

n→∞

sup

τ≥0

• e−τ(D0+Q) −
• e−τD0/ne−τQ/nn
• B(Lp(I)) > 0.

Artur Stephan

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On evolution semigroups and Trotter product operator-norm estimates

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Workshop on Operator Theory and Krein Spaces, Wien, December 19-22, 2019 · Page 15 (15)

Operator-norm convergence depends on time regularity Theorem (Neidhardt-AS-Zagrebnov)

• 1. If q : [0, 1] → R is Hölder continuous with β ∈]0, 1], then

sup

τ≥0

• e−τ(D0+Q) −
• e−τD0/ne−τQ/nn
• B(Lp(I)) = O(n−β)
• 2. If q : I → R is continuous and non-negative, then
• e−τ(D0+Q) −
• e−τD0/ne−τQ/nn
• B(Lp(I)) = o(1).
• 3. Let δn > 0 be a sequence with δn → 0 as n → ∞. Then there exists a continuous

function q : I = [0, 1] → R such that

sup

τ≥0

• e−τ(D0+Q) −
• e−τD0/ne−τQ/nn
• B(Lp(I)) ≈ ˜

δn, ˜ δn/δn → ∞

• 4. There is a non-negative function q ∈ L∞([0, 1]) such that

lim sup

n→∞

sup

τ≥0

• e−τ(D0+Q) −
• e−τD0/ne−τQ/nn
• B(Lp(I)) > 0.

THANK YOU FOR YOUR ATTENTION!

Artur Stephan

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On evolution semigroups and Trotter product operator-norm estimates

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Workshop on Operator Theory and Krein Spaces, Wien, December 19-22, 2019 · Page 15 (15)