A Katznelson-Tzafriri theorem for measures
David Seifert
Operator Semigroups meet Complex Analysis,
Harmonic Analysis and Mathematical Physics
Herrnhut, 6 June 2013
Background
Preliminaries
A result
Open problems
. . . or “The Sofa in Kraków”
David Seifert
A Katznelson-Tzafriri theorem for measures
University of Oxford
Background
Preliminaries
A result
Open problems
. . . or “The Sofa in Kraków”
David Seifert
A Katznelson-Tzafriri theorem for measures
University of Oxford
Background
Preliminaries
A result
Open problems
Overview
1 Background
2 Preliminaries
3 A result
4 Open problems
David Seifert
A Katznelson-Tzafriri theorem for measures
University of Oxford
Background
Preliminaries
A result
Open problems
Overview
1 Background
2 Preliminaries
3 A result
4 Open problems
David Seifert
A Katznelson-Tzafriri theorem for measures
University of Oxford
Background
Preliminaries
A result
Open problems
The Katznelson-Tzafriri theorem
Throughout, let X be a complex Banach space and T a bounded
C0 -semigroup on X with generator A. Given a ∈ L1 (R), s ∈ R and
x ∈ X , define
Z
Z
−ist
b
a(s) =
e a(t) dt and b
a(T )x =
T (t)x a(t) dt.
R
David Seifert
A Katznelson-Tzafriri theorem for measures
R+
University of Oxford
Background
Preliminaries
A result
Open problems
The Katznelson-Tzafriri theorem
Throughout, let X be a complex Banach space and T a bounded
C0 -semigroup on X with generator A. Given a ∈ L1 (R), s ∈ R and
x ∈ X , define
Z
Z
−ist
b
a(s) =
e a(t) dt and b
a(T )x =
T (t)x a(t) dt.
R
R+
Theorem (Esterle et al. ’92, Vũ ’92)
Suppose iσ(A) ∩ R is of spectral synthesis and let a ∈ L1 (R+ ) be
such that b
a vanishes on iσ(A) ∩ R. Then kT (t)b
a(T )k → 0 as
t → ∞.
David Seifert
A Katznelson-Tzafriri theorem for measures
University of Oxford
Background
Preliminaries
A result
Open problems
The Katznelson-Tzafriri theorem
Throughout, let X be a complex Banach space and T a bounded
C0 -semigroup on X with generator A. Given a ∈ L1 (R), s ∈ R and
x ∈ X , define
Z
Z
−ist
b
a(s) =
e a(t) dt and b
a(T )x =
T (t)x a(t) dt.
R
R+
Theorem (Esterle et al. ’92, Vũ ’92)
Suppose iσ(A) ∩ R is of spectral synthesis and let a ∈ L1 (R+ ) be
such that b
a vanishes on iσ(A) ∩ R. Then kT (t)b
a(T )k → 0 as
t → ∞.
Original result was for power-bounded operators
(Katznelson-Tzafriri ’86), but true also for more general semigroup
representations (Batty-Vũ ’92).
Many applications.
David Seifert
A Katznelson-Tzafriri theorem for measures
University of Oxford
Background
Preliminaries
A result
Open problems
Towards the problem
Example
Have kT (t)AR(1, A)2 k → 0 as t → ∞ if (and only if)
iσ(A) ∩ R ⊂ {0}.
David Seifert
A Katznelson-Tzafriri theorem for measures
University of Oxford
Background
Preliminaries
A result
Open problems
Towards the problem
Example
Have kT (t)AR(1, A)2 k → 0 as t → ∞ if (and only if)
iσ(A) ∩ R ⊂ {0}.
Related result:
Theorem (Arendt-Prüß ’92)
Suppose T is eventually differentiable. Then kAT (t)k → 0 as
t → ∞ if (and only if) iσ(A) ∩ R ⊂ {0}.
David Seifert
A Katznelson-Tzafriri theorem for measures
University of Oxford
Background
Preliminaries
A result
Open problems
Towards the problem
Example
Have kT (t)AR(1, A)2 k → 0 as t → ∞ if (and only if)
iσ(A) ∩ R ⊂ {0}.
Related result:
Theorem (Arendt-Prüß ’92)
Suppose T is eventually differentiable. Then kAT (t)k → 0 as
t → ∞ if (and only if) iσ(A) ∩ R ⊂ {0}.
Consider instead kT (t)AR(1, A)k for large t and general T .
Problem: AR(1, A) 6= b
a(T ) for a ∈ L1 (R+ ).
David Seifert
A Katznelson-Tzafriri theorem for measures
University of Oxford
Background
Preliminaries
A result
Open problems
Towards the problem
Example
Have kT (t)AR(1, A)2 k → 0 as t → ∞ if (and only if)
iσ(A) ∩ R ⊂ {0}.
Related result:
Theorem (Arendt-Prüß ’92)
Suppose T is eventually differentiable. Then kAT (t)k → 0 as
t → ∞ if (and only if) iσ(A) ∩ R ⊂ {0}.
Consider instead kT (t)AR(1, A)k for large t and general T .
Problem: AR(1, A) 6= b
a(T ) for a ∈ L1 (R+ ).
Aim
Extend the K-T theorem to bounded Borel measures on R+ .
David Seifert
A Katznelson-Tzafriri theorem for measures
University of Oxford
Background
Preliminaries
A result
Open problems
Overview
1 Background
2 Preliminaries
3 A result
4 Open problems
David Seifert
A Katznelson-Tzafriri theorem for measures
University of Oxford
Background
Preliminaries
A result
Open problems
Growth bounds
Given a family F of B(X )-valued functions on R+ , define
ωF (T ) := inf{ω(T − S) : S ∈ F}.
David Seifert
A Katznelson-Tzafriri theorem for measures
University of Oxford
Background
Preliminaries
A result
Open problems
Growth bounds
Given a family F of B(X )-valued functions on R+ , define
ωF (T ) := inf{ω(T − S) : S ∈ F}.
Three important cases:
If F consists of all K(X )-valued functions, write ωess (T ) for
ωF (T ). This is the essential growth bound of T .
If F = C (R+ ; B(X )), write δ(T ) for ωF (T ). This is the
critical growth bound of T .
If F consists of functions which are analytic on a sector
containing (0, ∞), write ζ(T ) for ωF (T ). This is the
non-analytic growth bound of T .
David Seifert
A Katznelson-Tzafriri theorem for measures
University of Oxford
Background
Preliminaries
A result
Open problems
Growth bounds
Given a family F of B(X )-valued functions on R+ , define
ωF (T ) := inf{ω(T − S) : S ∈ F}.
Three important cases:
If F consists of all K(X )-valued functions, write ωess (T ) for
ωF (T ). This is the essential growth bound of T .
If F = C (R+ ; B(X )), write δ(T ) for ωF (T ). This is the
critical growth bound of T .
If F consists of functions which are analytic on a sector
containing (0, ∞), write ζ(T ) for ωF (T ). This is the
non-analytic growth bound of T .
Remark
Have δ(T ) ≤ ζ(T ) ≤ ωess (T ) ≤ ω(T ). On Hilbert space
δ(T ) = ζ(T ), but open whether this is always true.
David Seifert
A Katznelson-Tzafriri theorem for measures
University of Oxford
Background
Preliminaries
A result
Open problems
Properties of δ(T ) and ζ(T )
The critical growth bound δ(T ) satisfies
ω(T ) = max{δ(T ), s(A)}.
This is of interest when δ(T ) < ω(T ).
David Seifert
A Katznelson-Tzafriri theorem for measures
University of Oxford
Background
Preliminaries
A result
Open problems
Properties of δ(T ) and ζ(T )
The critical growth bound δ(T ) satisfies
ω(T ) = max{δ(T ), s(A)}.
This is of interest when δ(T ) < ω(T ).
The non-analytic growth bound ζ(T ) has the following property:
Theorem (Batty-Srivastava ’03)
The following are equivalent:
(i) ζ(T ) < 0;
(ii) There exists R > 0 such that {is : |s| ≥ R} ⊂ ρ(A) and the
map s 7→ φ(s)R(is, A) (for an appropriate smooth φ) is the
Fourier transform of some element of L1s (R; B(X )).
David Seifert
A Katznelson-Tzafriri theorem for measures
University of Oxford
Background
Preliminaries
A result
Open problems
Overview
1 Background
2 Preliminaries
3 A result
4 Open problems
David Seifert
A Katznelson-Tzafriri theorem for measures
University of Oxford
Background
Preliminaries
A result
Open problems
Katznelson-Tzafriri for measures
Theorem
Suppose iσ(A) ∩ R is of spectral synthesis and that ζ(T ) < 0. Let
µ ∈ M(R+ ) be such that µ
b vanishes on iσ(A) ∩ R. Then
kT (t)b
µ(T )k → 0 as t → ∞.
David Seifert
A Katznelson-Tzafriri theorem for measures
University of Oxford
Background
Preliminaries
A result
Open problems
Katznelson-Tzafriri for measures
Theorem
Suppose iσ(A) ∩ R is of spectral synthesis and that ζ(T ) < 0. Let
µ ∈ M(R+ ) be such that µ
b vanishes on iσ(A) ∩ R. Then
kT (t)b
µ(T )k → 0 as t → ∞.
Corollary
For iσ(A) ∩ R and µ ∈ M(R+ ) as above, have kT (t)b
µ(T )k → 0 as
t → ∞ provided any of the following hold:
(i) X is a Hilbert space and sup|s|≥R kR(is, A)k < ∞ for
sufficiently large R > 0;
(ii) T is quasi-compact, i.e. ωess (T ) < 0;
(iii) T has Lp -resolvent for some p ∈ (1, ∞);
(iv) T is eventually differentiable.
David Seifert
A Katznelson-Tzafriri theorem for measures
University of Oxford
Background
Preliminaries
A result
Open problems
Proof outline
1. Since ζ(T ) < 0, can choose ϕ ∈ S(R) such that ϕ
b has
compact support and equals 1 in a neighbourhood of
iσ(A) ∩ R. Define the measures µ0 = µ ∗ ϕ and
µ1 = µ ∗ (δ0 − ϕ) on R, and note that, for t ≥ 0,
T (t)b
µ(T ) = F0 (t) + F1 (t), where
Z
Fj (t)x =
T (s + t)x dµj (s) (x ∈ X , j = 0, 1).
R
2. Observe that µ0 is absolutely continuous, so kF0 (t)k → 0 as
t → ∞ by the usual method (spectral synthesis, Parseval,
Riemann-Lebesgue).
3. Use ζ(T ) < 0 to obtain that s 7→ (1 − ϕ(−s))R(is,
b
A) is the
1
Fourier transform of an element of Ls (R; B(X )), and show
that in fact F1 ∈ L1s (R; B(X )). Gives kF1 (t)k → 0 as t → ∞.
David Seifert
A Katznelson-Tzafriri theorem for measures
University of Oxford
Background
Preliminaries
A result
Open problems
Overview
1 Background
2 Preliminaries
3 A result
4 Open problems
David Seifert
A Katznelson-Tzafriri theorem for measures
University of Oxford
Background
Preliminaries
A result
Open problems
Open problems
Several interesting questions remain open:
1. When does the converse hold?
2. Can the assumption ζ(T ) < 0 be weakened to δ(T ) < 0?
3. Is the spectral synthesis assumption on iσ(A) ∩ R needed
when X is a Hilbert space?
4. How to obtain a good estimate on the rate of decay in the
case of one-point boundary spectrum?
David Seifert
A Katznelson-Tzafriri theorem for measures
University of Oxford