Blum-Hanson type theorems for weak mixing
Joanna Kułaga
Faculty of Mathematics and Computer Science
Nicolaus Copernicus University
Blaubeuren, 7-13.06.2009
Joanna Kułaga
Blum-Hanson type theorems for weak mixing
Convergence in density
Setting
We consider sequences (xn ) in the Banach space X .
Goal
We want to compare different types of convergence, including
convergence in weak topology.
Definition
(ω)n ⊂ Ω converges in density to ω ∈ Ω if
lim
E63n→∞
ωn = ω
for some E ⊂ N of zero density.
Notation: D − limn→∞ ωn = ω.
Joanna Kułaga
Blum-Hanson type theorems for weak mixing
Weak Mixing (jjj)’ and Uniform Weak Mixing (jjj)
Definition - weak mixing (jjj)’
∃x∈X ∀x ∗ ∈X ∗ D − lim hxn , x ∗ i = hx, x ∗ i.
n→∞
Remarks
If X ∗ separable ⇒ equivalent to convergence in density.
If X reflexive ⇒ equivalent to convergence in density.
For bounded sequences equivalent to
a) ∀ε>0 d {k ∈P
N : |hxk − x, x ∗ i| > ε} = 0
n
1
b) limn→∞ n+1 k=0 |hxk − x, x ∗ i| = 0
Definition - uniform weak mixing (jjj)
1 Pn
∗
limn→∞ supkx ∗ k≤1 n+1
k=0 |hxk − x, x i| = 0
Joanna Kułaga
Blum-Hanson type theorems for weak mixing
Cesàro averages
1
n+1
Pn
j=0 xkj
Types of convergence
In the norm topology:
(j)1
(jj)1
∀{kj }∈N+ ∀{kj }∈N+
to the same limit
Weakly:
(j)01
(jj)01
∀{kj }∈N+ ∀{kj }∈N+
to the same limit
(j)2
∀{kj }∈Nrd
(jj)2
∀{kj }∈Nrd
to the same limit
(j)02
∀{kj }∈Nrd
(jj)02
∀{kj }∈Nrd
to the same limit
Remark
(jj)1 - Blum-Hanson Property with respect to N+
(jj)2 - Blum-Hanson Property with respect to Nrd
Joanna Kułaga
Blum-Hanson type theorems for weak mixing
Relationships for (bounded) sequences in a Banach space
Diagram:
Remarks:
convergence in the norm ⇒ convergence in the weak topology
Joanna Kułaga
Blum-Hanson type theorems for weak mixing
Relationships for (bounded) sequences in a Banach space
Diagram:
Joanna Kułaga
Blum-Hanson type theorems for weak mixing
Relationships for (bounded) sequences in a Banach space
Diagram:
Remarks:
Follows from the implication Nrd ⊂ N+ .
Joanna Kułaga
Blum-Hanson type theorems for weak mixing
Relationships for (bounded) sequences in a Banach space
Diagram:
Remarks:
Take {kj }, {lj } ∈ N+ (Nrd ) satisfy:
1 Pn
1 Pn
j=0 xkj → x0 and n+1
j=0 xlj → y0 .
n+1
Construct {sj } ∈ N+ (Nrd ) such that
convergent.
1
n+1
Pn
j=0 xkj
is not
Easier for N+ than for Nrd .
Joanna Kułaga
Blum-Hanson type theorems for weak mixing
Relationships for (bounded) sequences in a Banach space
Diagram:
Remarks:
(jj)02 ⇒ (jjj)0 holds for bounded sequences.
Main idea:
Ap = A ∪ {p, 2p, . . . } ∈ Nrd , take p big enough to obtain a
contradiction.
Joanna Kułaga
Blum-Hanson type theorems for weak mixing
Relationships for (bounded) sequences in a Banach space
Diagram:
Remarks:
(jjj)0 ⇒ (jj)01 is quite straightforward.
We will see a similar reasoning later.
Joanna Kułaga
Blum-Hanson type theorems for weak mixing
Relationships for (bounded) sequences in a Banach space
Diagram:
Remarks:
(jjj) ⇒ (j)1 can be shown in the same way as the previous
implication (jjj)0 ⇒ (jj)01 .
Joanna Kułaga
Blum-Hanson type theorems for weak mixing
Relationships for (bounded) sequences in a Banach space
Diagram:
Remarks:
(jj)2 ⇒ (jjj) more difficult. Holds for bounded sequences.
Joanna Kułaga
Blum-Hanson type theorems for weak mixing
Relationships for (bounded) sequences in a Banach space
Diagram:
Question:
When are the conditions in both rows equivalent?
Answer:
X - Banach space
x ∈ X fixed
T : X → X power bounded (supn kT n k < ∞) linear operator
sequence: xn = T n x
Joanna Kułaga
Blum-Hanson type theorems for weak mixing
Orbits of power bounded operators
Diagram (L. K. Jones and M. Lin, 1976):
First step: (jjj)0 ⇒ (jjj)
Joanna Kułaga
Blum-Hanson type theorems for weak mixing
Orbits of power bounded operators
Proof of (jjj)0 ⇒ (jjj)
Assume kT k ≤ 1 (otherwise we construct an equivalent norm).
B := {x ∗ : kx ∗ k ≤ 1}
B - compact Hausdorff in w ∗ -topology
T ∗ B ⊂ B, T ∗ is w ∗ -continuous on B
⇒ A : C (B) → C (B), Af = f ◦ T ∗ contraction
g (x ∗ ) := |hx ∗ , xi|
1 PN
k
0
k=1 A g → 0 pointwise (by (jjj) )
N
For µ ∈ C (B)∗ s.t. A∗ µ = µ we have:
P
hg , µi = hg , A∗ µi = hAg , µi = h n1 nk=1 Ak g , µi → 0.
Hence hg , µi = 0 and g ∈ ran(I − A).
P
j g → 0 (≡ (jjj)).
Therefore N1 N
A
j=1
Joanna Kułaga
Blum-Hanson type theorems for weak mixing
Orbits of power bounded operators
Proof of (jjj)0 ⇒ (jjj)
Assume kT k ≤ 1 (otherwise we construct an equivalent norm).
B := {x ∗ : kx ∗ k ≤ 1}
B - compact Hausdorff in w ∗ -topology
T ∗ B ⊂ B, T ∗ is w ∗ -continuous on B
⇒ A : C (B) → C (B), Af = f ◦ T ∗ contraction
g (x ∗ ) := |hx ∗ , xi|
1 PN
k
0
k=1 A g → 0 pointwise (by (jjj) )
N
For µ ∈ C (B)∗ s.t. A∗ µ = µ we have:
P
hg , µi = hg , A∗ µi = hAg , µi = h n1 nk=1 Ak g , µi → 0.
Hence hg , µi = 0 and g ∈ ran(I − A).
P
j g → 0 (≡ (jjj)).
Therefore N1 N
A
j=1
Joanna Kułaga
Blum-Hanson type theorems for weak mixing
Orbits of power bounded operators
Proof of (jjj)0 ⇒ (jjj)
Assume kT k ≤ 1 (otherwise we construct an equivalent norm).
B := {x ∗ : kx ∗ k ≤ 1}
B - compact Hausdorff in w ∗ -topology
T ∗ B ⊂ B, T ∗ is w ∗ -continuous on B
⇒ A : C (B) → C (B), Af = f ◦ T ∗ contraction
g (x ∗ ) := |hx ∗ , xi|
1 PN
k
0
k=1 A g → 0 pointwise (by (jjj) )
N
For µ ∈ C (B)∗ s.t. A∗ µ = µ we have:
P
hg , µi = hg , A∗ µi = hAg , µi = h n1 nk=1 Ak g , µi → 0.
Hence hg , µi = 0 and g ∈ ran(I − A).
P
j g → 0 (≡ (jjj)).
Therefore N1 N
A
j=1
Joanna Kułaga
Blum-Hanson type theorems for weak mixing
Orbits of power bounded operators
Proof of (jjj)0 ⇒ (jjj)
Assume kT k ≤ 1 (otherwise we construct an equivalent norm).
B := {x ∗ : kx ∗ k ≤ 1}
B - compact Hausdorff in w ∗ -topology
T ∗ B ⊂ B, T ∗ is w ∗ -continuous on B
⇒ A : C (B) → C (B), Af = f ◦ T ∗ contraction
g (x ∗ ) := |hx ∗ , xi|
1 PN
k
0
k=1 A g → 0 pointwise (by (jjj) )
N
For µ ∈ C (B)∗ s.t. A∗ µ = µ we have:
P
hg , µi = hg , A∗ µi = hAg , µi = h n1 nk=1 Ak g , µi → 0.
Hence hg , µi = 0 and g ∈ ran(I − A).
P
j g → 0 (≡ (jjj)).
Therefore N1 N
A
j=1
Joanna Kułaga
Blum-Hanson type theorems for weak mixing
Orbits of power bounded operators
Proof of (jjj)0 ⇒ (jjj)
Assume kT k ≤ 1 (otherwise we construct an equivalent norm).
B := {x ∗ : kx ∗ k ≤ 1}
B - compact Hausdorff in w ∗ -topology
T ∗ B ⊂ B, T ∗ is w ∗ -continuous on B
⇒ A : C (B) → C (B), Af = f ◦ T ∗ contraction
g (x ∗ ) := |hx ∗ , xi|
1 PN
k
0
k=1 A g → 0 pointwise (by (jjj) )
N
For µ ∈ C (B)∗ s.t. A∗ µ = µ we have:
P
hg , µi = hg , A∗ µi = hAg , µi = h n1 nk=1 Ak g , µi → 0.
Hence hg , µi = 0 and g ∈ ran(I − A).
P
j g → 0 (≡ (jjj)).
Therefore N1 N
A
j=1
Joanna Kułaga
Blum-Hanson type theorems for weak mixing
Orbits of power bounded operators
Proof of (jjj)0 ⇒ (jjj)
Assume kT k ≤ 1 (otherwise we construct an equivalent norm).
B := {x ∗ : kx ∗ k ≤ 1}
B - compact Hausdorff in w ∗ -topology
T ∗ B ⊂ B, T ∗ is w ∗ -continuous on B
⇒ A : C (B) → C (B), Af = f ◦ T ∗ contraction
g (x ∗ ) := |hx ∗ , xi|
1 PN
k
0
k=1 A g → 0 pointwise (by (jjj) )
N
For µ ∈ C (B)∗ s.t. A∗ µ = µ we have:
P
hg , µi = hg , A∗ µi = hAg , µi = h n1 nk=1 Ak g , µi → 0.
Hence hg , µi = 0 and g ∈ ran(I − A).
P
j g → 0 (≡ (jjj)).
Therefore N1 N
A
j=1
Joanna Kułaga
Blum-Hanson type theorems for weak mixing
Orbits of power bounded operators
Proof of (jjj)0 ⇒ (jjj)
Assume kT k ≤ 1 (otherwise we construct an equivalent norm).
B := {x ∗ : kx ∗ k ≤ 1}
B - compact Hausdorff in w ∗ -topology
T ∗ B ⊂ B, T ∗ is w ∗ -continuous on B
⇒ A : C (B) → C (B), Af = f ◦ T ∗ contraction
g (x ∗ ) := |hx ∗ , xi|
1 PN
k
0
k=1 A g → 0 pointwise (by (jjj) )
N
For µ ∈ C (B)∗ s.t. A∗ µ = µ we have:
P
hg , µi = hg , A∗ µi = hAg , µi = h n1 nk=1 Ak g , µi → 0.
Hence hg , µi = 0 and g ∈ ran(I − A).
P
j g → 0 (≡ (jjj)).
Therefore N1 N
A
j=1
Joanna Kułaga
Blum-Hanson type theorems for weak mixing
Orbits of power bounded operators
Proof of (jjj)0 ⇒ (jjj)
Assume kT k ≤ 1 (otherwise we construct an equivalent norm).
B := {x ∗ : kx ∗ k ≤ 1}
B - compact Hausdorff in w ∗ -topology
T ∗ B ⊂ B, T ∗ is w ∗ -continuous on B
⇒ A : C (B) → C (B), Af = f ◦ T ∗ contraction
g (x ∗ ) := |hx ∗ , xi|
1 PN
k
0
k=1 A g → 0 pointwise (by (jjj) )
N
For µ ∈ C (B)∗ s.t. A∗ µ = µ we have:
P
hg , µi = hg , A∗ µi = hAg , µi = h n1 nk=1 Ak g , µi → 0.
Hence hg , µi = 0 and g ∈ ran(I − A).
P
j g → 0 (≡ (jjj)).
Therefore N1 N
A
j=1
Joanna Kułaga
Blum-Hanson type theorems for weak mixing
Orbits of power bounded operators
Proof of (jjj)0 ⇒ (jjj)
Assume kT k ≤ 1 (otherwise we construct an equivalent norm).
B := {x ∗ : kx ∗ k ≤ 1}
B - compact Hausdorff in w ∗ -topology
T ∗ B ⊂ B, T ∗ is w ∗ -continuous on B
⇒ A : C (B) → C (B), Af = f ◦ T ∗ contraction
g (x ∗ ) := |hx ∗ , xi|
1 PN
k
0
k=1 A g → 0 pointwise (by (jjj) )
N
For µ ∈ C (B)∗ s.t. A∗ µ = µ we have:
P
hg , µi = hg , A∗ µi = hAg , µi = h n1 nk=1 Ak g , µi → 0.
Hence hg , µi = 0 and g ∈ ran(I − A).
P
j g → 0 (≡ (jjj)).
Therefore N1 N
A
j=1
Joanna Kułaga
Blum-Hanson type theorems for weak mixing
Orbits of power bounded operators
Proof of (jjj)0 ⇒ (jjj)
Assume kT k ≤ 1 (otherwise we construct an equivalent norm).
B := {x ∗ : kx ∗ k ≤ 1}
B - compact Hausdorff in w ∗ -topology
T ∗ B ⊂ B, T ∗ is w ∗ -continuous on B
⇒ A : C (B) → C (B), Af = f ◦ T ∗ contraction
g (x ∗ ) := |hx ∗ , xi|
1 PN
k
0
k=1 A g → 0 pointwise (by (jjj) )
N
For µ ∈ C (B)∗ s.t. A∗ µ = µ we have:
P
hg , µi = hg , A∗ µi = hAg , µi = h n1 nk=1 Ak g , µi → 0.
Hence hg , µi = 0 and g ∈ ran(I − A).
P
j g → 0 (≡ (jjj)).
Therefore N1 N
A
j=1
Joanna Kułaga
Blum-Hanson type theorems for weak mixing
Orbits of power bounded operators
Proof of (jjj)0 ⇒ (jjj)
Assume kT k ≤ 1 (otherwise we construct an equivalent norm).
B := {x ∗ : kx ∗ k ≤ 1}
B - compact Hausdorff in w ∗ -topology
T ∗ B ⊂ B, T ∗ is w ∗ -continuous on B
⇒ A : C (B) → C (B), Af = f ◦ T ∗ contraction
g (x ∗ ) := |hx ∗ , xi|
1 PN
k
0
k=1 A g → 0 pointwise (by (jjj) )
N
For µ ∈ C (B)∗ s.t. A∗ µ = µ we have:
P
hg , µi = hg , A∗ µi = hAg , µi = h n1 nk=1 Ak g , µi → 0.
Hence hg , µi = 0 and g ∈ ran(I − A).
P
j g → 0 (≡ (jjj)).
Therefore N1 N
A
j=1
Joanna Kułaga
Blum-Hanson type theorems for weak mixing
Orbits of power bounded operators
We have:
Second step: (jjj) ⇒ (jj)1
Joanna Kułaga
Blum-Hanson type theorems for weak mixing
Orbits of power bounded operators
Proof of (jjj) ⇒ (jj)1
n
n
X
1
1 X kj
kj T x = sup h
T x, x ∗ i
n + 1
∗ k≤1 n + 1
kx
j=0
j=0
n
1 X kj
≤ sup
hT x, x ∗ i
kx ∗ k≤1 n + 1
j=0
kn kn + 1 1 X
k
≤ sup
hT x, x ∗ i
kx ∗ k≤1 n + 1 kn + 1
k=0
kn 1 X
k
∗ ≤ M sup
hT x, x i → 0.
kx ∗ k≤1 kn + 1
k=0
Joanna Kułaga
Blum-Hanson type theorems for weak mixing
Orbits of power bounded operators
Proof of (jjj) ⇒ (jj)1
n
n
X
1
1 X kj
kj T x = sup h
T x, x ∗ i
n + 1
∗ k≤1 n + 1
kx
j=0
j=0
n
1 X kj
≤ sup
hT x, x ∗ i
kx ∗ k≤1 n + 1
j=0
kn kn + 1 1 X
k
≤ sup
hT x, x ∗ i
kx ∗ k≤1 n + 1 kn + 1
k=0
kn 1 X
k
∗ ≤ M sup
hT x, x i → 0.
kx ∗ k≤1 kn + 1
k=0
Joanna Kułaga
Blum-Hanson type theorems for weak mixing
Orbits of power bounded operators
Proof of (jjj) ⇒ (jj)1
n
n
X
1
1 X kj
kj T x = sup h
T x, x ∗ i
n + 1
∗ k≤1 n + 1
kx
j=0
j=0
n
1 X kj
≤ sup
hT x, x ∗ i
kx ∗ k≤1 n + 1
j=0
kn kn + 1 1 X
k
≤ sup
hT x, x ∗ i
kx ∗ k≤1 n + 1 kn + 1
k=0
kn 1 X
k
∗ ≤ M sup
hT x, x i → 0.
kx ∗ k≤1 kn + 1
k=0
Joanna Kułaga
Blum-Hanson type theorems for weak mixing
Orbits of power bounded operators
Proof of (jjj) ⇒ (jj)1
n
n
X
1
1 X kj
kj T x = sup h
T x, x ∗ i
n + 1
∗ k≤1 n + 1
kx
j=0
j=0
n
1 X kj
≤ sup
hT x, x ∗ i
kx ∗ k≤1 n + 1
j=0
kn kn + 1 1 X
k
≤ sup
hT x, x ∗ i
kx ∗ k≤1 n + 1 kn + 1
k=0
kn 1 X
k
∗ ≤ M sup
hT x, x i → 0.
kx ∗ k≤1 kn + 1
k=0
Joanna Kułaga
Blum-Hanson type theorems for weak mixing
Orbits of power bounded operators
Proof of (jjj) ⇒ (jj)1
n
n
X
1
1 X kj
kj T x = sup h
T x, x ∗ i
n + 1
∗ k≤1 n + 1
kx
j=0
j=0
n
1 X kj
≤ sup
hT x, x ∗ i
kx ∗ k≤1 n + 1
j=0
kn kn + 1 1 X
k
≤ sup
hT x, x ∗ i
kx ∗ k≤1 n + 1 kn + 1
k=0
kn 1 X
k
∗ ≤ M sup
hT x, x i → 0.
kx ∗ k≤1 kn + 1
k=0
Joanna Kułaga
Blum-Hanson type theorems for weak mixing
Orbits of power bounded operators
We have:
The only missing implication (jj)02 ⇒ (jjj)0 :
Joanna Kułaga
Blum-Hanson type theorems for weak mixing
Orbits of power bounded operators
Proof of (jj)02 ⇒ (jjj)0 for bounded sequences:
suppose hxn , x ∗ i 6→ 0 in density
Joanna Kułaga
Blum-Hanson type theorems for weak mixing
Orbits of power bounded operators
Proof of (jj)02 ⇒ (jjj)0 for bounded sequences:
suppose hxn , x ∗ i 6→ 0 in density
either Rehxn , x ∗ i 6→ 0 or Imhxn , x ∗ i 6→ 0 in density
Joanna Kułaga
Blum-Hanson type theorems for weak mixing
Orbits of power bounded operators
Proof of (jj)02 ⇒ (jjj)0 for bounded sequences:
suppose hxn , x ∗ i 6→ 0 in density
WLOG Re(hxn , x ∗ i) 6→ 0
Joanna Kułaga
Blum-Hanson type theorems for weak mixing
Orbits of power bounded operators
Proof of (jj)02 ⇒ (jjj)0 for bounded sequences:
suppose hxn , x ∗ i 6→ 0 in density
WLOG Re(hxn , x ∗ i) 6→ 0
either (Rehxn , x ∗ i)+ 6→ 0 or (Rehxn , x ∗ i)− 6→ 0 in density
Joanna Kułaga
Blum-Hanson type theorems for weak mixing
Orbits of power bounded operators
Proof of (jj)02 ⇒ (jjj)0 for bounded sequences:
suppose hxn , x ∗ i 6→ 0 in density
WLOG Re(hxn , x ∗ i) 6→ 0
WLOG (Rehxn , x ∗ i)+ 6→ 0
Joanna Kułaga
Blum-Hanson type theorems for weak mixing
Orbits of power bounded operators
Proof of (jj)02 ⇒ (jjj)0 for bounded sequences:
suppose hxn , x ∗ i 6→ 0 in density
WLOG Re(hxn , x ∗ i) 6→ 0
WLOG (Rehxn , x ∗ i)+ 6→ 0
∃ε>0 d ∗ {k ∈ N : Re(hxk , x ∗ i)+ > ε} > 0
Joanna Kułaga
Blum-Hanson type theorems for weak mixing
Orbits of power bounded operators
Proof of (jj)02 ⇒ (jjj)0 for bounded sequences:
suppose hxn , x ∗ i 6→ 0 in density
WLOG Re(hxn , x ∗ i) 6→ 0
WLOG (Rehxn , x ∗ i)+ 6→ 0
∃ε>0 d ∗ {k ∈ N : Re(hxk , x ∗ i)+ > ε} > 0
A := {k ∈ N : Re(hxk , x ∗ i)+ > ε}, Ap := A ∪ {p, 2p, . . . }
Joanna Kułaga
Blum-Hanson type theorems for weak mixing
Orbits of power bounded operators
Proof of (jj)02 ⇒ (jjj)0 for bounded sequences:
suppose hxn , x ∗ i 6→ 0 in density
WLOG Re(hxn , x ∗ i) 6→ 0
WLOG (Rehxn , x ∗ i)+ 6→ 0
∃ε>0 d ∗ {k ∈ N : Re(hxk , x ∗ i)+ > ε} > 0
A := {k ∈ N : Re(hxk , x ∗ i)+ > ε}, Ap := A ∪ {p, 2p, . . . }
P
by (jj)02 ∃nε ∀n>nε #(Ap 1∩[0,n]) k∈Ap ∩[0,n] Rehxk , x ∗ i ≤ 2ε
Joanna Kułaga
Blum-Hanson type theorems for weak mixing
Orbits of power bounded operators
Proof of (jj)02 ⇒ (jjj)0 for bounded sequences:
suppose hxn , x ∗ i 6→ 0 in density
WLOG Re(hxn , x ∗ i) 6→ 0
WLOG (Rehxn , x ∗ i)+ 6→ 0
∃ε>0 d ∗ {k ∈ N : Re(hxk , x ∗ i)+ > ε} > 0
A := {k ∈ N : Re(hxk , x ∗ i)+ > ε}, Ap := A ∪ {p, 2p, . . . }
P
by (jj)02 ∃nε ∀n>nε #(Ap 1∩[0,n]) k∈Ap ∩[0,n] Rehxk , x ∗ i ≤ 2ε
P
Rehxk , x ∗ i
k∈A
P p ∩[0,n]
P
≥ k∈A∩[0,n] Rehxk , x ∗ i − 1≤l≤ n |hxk , x ∗ i|
p
Joanna Kułaga
Blum-Hanson type theorems for weak mixing
Orbits of power bounded operators
Proof of (jj)02 ⇒ (jjj)0 for bounded sequences:
suppose hxn , x ∗ i 6→ 0 in density
WLOG Re(hxn , x ∗ i) 6→ 0
WLOG (Rehxn , x ∗ i)+ 6→ 0
∃ε>0 d ∗ {k ∈ N : Re(hxk , x ∗ i)+ > ε} > 0
A := {k ∈ N : Re(hxk , x ∗ i)+ > ε}, Ap := A ∪ {p, 2p, . . . }
P
by (jj)02 ∃nε ∀n>nε #(Ap 1∩[0,n]) k∈Ap ∩[0,n] Rehxk , x ∗ i ≤ 2ε
P
Rehxk , x ∗ i
k∈A
P p ∩[0,n]
P
≥ k∈A∩[0,n] Rehxk , x ∗ i − 1≤l≤ n |hxk , x ∗ i|
≥ ε#(A ∩ [0, n]) − pn
(assume sup kxk k ≤ 1, kx ∗ k ≤ 1)
Joanna Kułaga
p
Blum-Hanson type theorems for weak mixing
Orbits of power bounded operators
Proof of (jj)02 ⇒ (jjj)0 for bounded sequences:
suppose hxn , x ∗ i 6→ 0 in density
WLOG Re(hxn , x ∗ i) 6→ 0
WLOG (Rehxn , x ∗ i)+ 6→ 0
∃ε>0 d ∗ {k ∈ N : Re(hxk , x ∗ i)+ > ε} > 0
A := {k ∈ N : Re(hxk , x ∗ i)+ > ε}, Ap := A ∪ {p, 2p, . . . }
P
by (jj)02 ∃nε ∀n>nε #(Ap 1∩[0,n]) k∈Ap ∩[0,n] Rehxk , x ∗ i ≤ 2ε
P
n
∗
k∈Ap ∩[0,n] Rehxk , x i ≥ ε#(A ∩ [0, n]) − p
Joanna Kułaga
Blum-Hanson type theorems for weak mixing
Orbits of power bounded operators
Proof of (jj)02 ⇒ (jjj)0 for bounded sequences:
suppose hxn , x ∗ i 6→ 0 in density
WLOG Re(hxn , x ∗ i) 6→ 0
WLOG (Rehxn , x ∗ i)+ 6→ 0
∃ε>0 d ∗ {k ∈ N : Re(hxk , x ∗ i)+ > ε} > 0
A := {k ∈ N : Re(hxk , x ∗ i)+ > ε}, Ap := A ∪ {p, 2p, . . . }
P
by (jj)02 ∃nε ∀n>nε #(Ap 1∩[0,n]) k∈Ap ∩[0,n] Rehxk , x ∗ i ≤ 2ε
P
n
∗
k∈Ap ∩[0,n] Rehxk , x i ≥ ε#(A ∩ [0, n]) − p
#(Ap ∩ [0, n]) ≤ #(A ∩ [0, n]) + #({lp; 1 ≤ l ≤ pn })
Joanna Kułaga
Blum-Hanson type theorems for weak mixing
Orbits of power bounded operators
Proof of (jj)02 ⇒ (jjj)0 for bounded sequences:
suppose hxn , x ∗ i 6→ 0 in density
WLOG Re(hxn , x ∗ i) 6→ 0
WLOG (Rehxn , x ∗ i)+ 6→ 0
∃ε>0 d ∗ {k ∈ N : Re(hxk , x ∗ i)+ > ε} > 0
A := {k ∈ N : Re(hxk , x ∗ i)+ > ε}, Ap := A ∪ {p, 2p, . . . }
P
by (jj)02 ∃nε ∀n>nε #(Ap 1∩[0,n]) k∈Ap ∩[0,n] Rehxk , x ∗ i ≤ 2ε
P
n
∗
k∈Ap ∩[0,n] Rehxk , x i ≥ ε#(A ∩ [0, n]) − p
#(Ap ∩ [0, n]) ≤ #(A ∩ [0, n]) + #({lp; 1 ≤ l ≤ pn })
≤ #(A ∩ [0, n]) + pn
Joanna Kułaga
Blum-Hanson type theorems for weak mixing
Orbits of power bounded operators
Proof of (jj)02 ⇒ (jjj)0 for bounded sequences:
suppose hxn , x ∗ i 6→ 0 in density
WLOG Re(hxn , x ∗ i) 6→ 0
WLOG (Rehxn , x ∗ i)+ 6→ 0
∃ε>0 d ∗ {k ∈ N : Re(hxk , x ∗ i)+ > ε} > 0
A := {k ∈ N : Re(hxk , x ∗ i)+ > ε}, Ap := A ∪ {p, 2p, . . . }
P
by (jj)02 ∃nε ∀n>nε #(Ap 1∩[0,n]) k∈Ap ∩[0,n] Rehxk , x ∗ i ≤ 2ε
P
n
∗
k∈Ap ∩[0,n] Rehxk , x i ≥ ε#(A ∩ [0, n]) − p
#(Ap ∩ [0, n]) ≤ #(A ∩ [0, n]) +
Joanna Kułaga
n
p
Blum-Hanson type theorems for weak mixing
Orbits of power bounded operators
Proof of (jj)02 ⇒ (jjj)0 for bounded sequences:
suppose hxn , x ∗ i 6→ 0 in density
WLOG Re(hxn , x ∗ i) 6→ 0
WLOG (Rehxn , x ∗ i)+ 6→ 0
∃ε>0 d ∗ {k ∈ N : Re(hxk , x ∗ i)+ > ε} > 0
A := {k ∈ N : Re(hxk , x ∗ i)+ > ε}, Ap := A ∪ {p, 2p, . . . }
P
by (jj)02 ∃nε ∀n>nε #(Ap 1∩[0,n]) k∈Ap ∩[0,n] Rehxk , x ∗ i ≤ 2ε
P
n
∗
k∈Ap ∩[0,n] Rehxk , x i ≥ ε#(A ∩ [0, n]) − p
#(Ap ∩ [0, n]) ≤ #(A ∩ [0, n]) + pn
P
ε
1
∗
k∈Ap ∩[0,n] Rehxk , x i
2 ≥ #(Ap )∩[0,n]
Joanna Kułaga
Blum-Hanson type theorems for weak mixing
Orbits of power bounded operators
Proof of (jj)02 ⇒ (jjj)0 for bounded sequences:
suppose hxn , x ∗ i 6→ 0 in density
WLOG Re(hxn , x ∗ i) 6→ 0
WLOG (Rehxn , x ∗ i)+ 6→ 0
∃ε>0 d ∗ {k ∈ N : Re(hxk , x ∗ i)+ > ε} > 0
A := {k ∈ N : Re(hxk , x ∗ i)+ > ε}, Ap := A ∪ {p, 2p, . . . }
P
by (jj)02 ∃nε ∀n>nε #(Ap 1∩[0,n]) k∈Ap ∩[0,n] Rehxk , x ∗ i ≤ 2ε
P
n
∗
k∈Ap ∩[0,n] Rehxk , x i ≥ ε#(A ∩ [0, n]) − p
#(Ap ∩ [0, n]) ≤ #(A ∩ [0, n]) + pn
P
ε
1
∗
k∈Ap ∩[0,n] Rehxk , x i
2 ≥ #(Ap )∩[0,n]
≥
ε#(A∩[0,n])− pn
#(A∩[0,n])+ pn
Joanna Kułaga
Blum-Hanson type theorems for weak mixing
Orbits of power bounded operators
Proof of (jj)02 ⇒ (jjj)0 for bounded sequences:
suppose hxn , x ∗ i 6→ 0 in density
WLOG Re(hxn , x ∗ i) 6→ 0
WLOG (Rehxn , x ∗ i)+ 6→ 0
∃ε>0 d ∗ {k ∈ N : Re(hxk , x ∗ i)+ > ε} > 0
A := {k ∈ N : Re(hxk , x ∗ i)+ > ε}, Ap := A ∪ {p, 2p, . . . }
P
by (jj)02 ∃nε ∀n>nε #(Ap 1∩[0,n]) k∈Ap ∩[0,n] Rehxk , x ∗ i ≤ 2ε
P
n
∗
k∈Ap ∩[0,n] Rehxk , x i ≥ ε#(A ∩ [0, n]) − p
#(Ap ∩ [0, n]) ≤ #(A ∩ [0, n]) + pn
P
ε
1
∗
k∈Ap ∩[0,n] Rehxk , x i
2 ≥ #(Ap )∩[0,n]
≥
ε#(A∩[0,n])− pn
#(A∩[0,n])+ pn
=ε−
(1+ε) pn
#(A∩[0,n])+ pn
Joanna Kułaga
Blum-Hanson type theorems for weak mixing
Orbits of power bounded operators
Proof of (jj)02 ⇒ (jjj)0 for bounded sequences:
ε
2
≤
(1+ε) pn
#(A∩[0,n])+ pn
for n > nε
Joanna Kułaga
Blum-Hanson type theorems for weak mixing
Orbits of power bounded operators
Proof of (jj)02 ⇒ (jjj)0 for bounded sequences:
ε
2
≤
(1+ε) pn
#(A∩[0,n])+ pn
for n > nε
take 0 < δ < d ∗ (A)
Joanna Kułaga
Blum-Hanson type theorems for weak mixing
Orbits of power bounded operators
Proof of (jj)02 ⇒ (jjj)0 for bounded sequences:
ε
2
≤
(1+ε) pn
#(A∩[0,n])+ pn
for n > nε
take 0 < δ < d ∗ (A)
for infinitely many n:
1
n+1 #(A
Joanna Kułaga
∩ [0, n]) > δ
Blum-Hanson type theorems for weak mixing
Orbits of power bounded operators
Proof of (jj)02 ⇒ (jjj)0 for bounded sequences:
ε
2
≤
(1+ε) pn
#(A∩[0,n])+ pn
for n > nε
take 0 < δ < d ∗ (A)
for infinitely many n:
1
n+1 #(A
∩ [0, n]) > δ
⇔ #(A ∩ [0, n]) > δ(n + 1) > δn
Joanna Kułaga
Blum-Hanson type theorems for weak mixing
Orbits of power bounded operators
Proof of (jj)02 ⇒ (jjj)0 for bounded sequences:
ε
2
≤
(1+ε) pn
#(A∩[0,n])+ pn
for n > nε
take 0 < δ < d ∗ (A)
for infinitely many n:
1
n+1 #(A
∩ [0, n]) > δ
⇔ #(A ∩ [0, n]) > δ(n + 1) > δn
hence for infinitely many n > nε :
Joanna Kułaga
ε
2
≤
(1+ε) pn
nδ+ pn
Blum-Hanson type theorems for weak mixing
Orbits of power bounded operators
Proof of (jj)02 ⇒ (jjj)0 for bounded sequences:
ε
2
≤
(1+ε) pn
#(A∩[0,n])+ pn
for n > nε
take 0 < δ < d ∗ (A)
for infinitely many n:
1
n+1 #(A
∩ [0, n]) > δ
⇔ #(A ∩ [0, n]) > δ(n + 1) > δn
hence for infinitely many n > nε :
Joanna Kułaga
ε
2
≤
(1+ε) pn
nδ+ pn
=
(1+ε) p1
δ+ p1
Blum-Hanson type theorems for weak mixing
Orbits of power bounded operators
Proof of (jj)02 ⇒ (jjj)0 for bounded sequences:
ε
2
≤
(1+ε) pn
#(A∩[0,n])+ pn
for n > nε
take 0 < δ < d ∗ (A)
for infinitely many n:
1
n+1 #(A
∩ [0, n]) > δ
⇔ #(A ∩ [0, n]) > δ(n + 1) > δn
hence for infinitely many n > nε :
ε
2
≤
(1+ε) pn
nδ+ pn
=
(1+ε) p1
δ+ p1
p is arbitrary ⇒ contradition!
Joanna Kułaga
Blum-Hanson type theorems for weak mixing
Orbits for power bounded opearors
Theorem
For a power bounded linear operator T : X → X on a Banach space
the following 10 conditions are equivalent:
Joanna Kułaga
Blum-Hanson type theorems for weak mixing