[PPT] - Dynamics of Disjoint Hypercyclic Operators: Hypercyclicity vs. PowerPoint Presentation

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Introduction Dynamics of Hypercyclic Operators Results

Dynamics of Disjoint Hypercyclic Operators: Hypercyclicity vs. Disjoint Hypercyclicity

Rebecca Sanders Department of Mathematics, Statistics, and Computer Sci. Marquette University April 11-13, 2014

Sanders Disjoint Hypercyclic Operators

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Introduction Dynamics of Hypercyclic Operators Results

Throughout this talk, let

X = separable,∞-dimensional Banach space
B(X) = algebra of operators T : X −

→ X

Sanders Disjoint Hypercyclic Operators

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Introduction Dynamics of Hypercyclic Operators Results

Throughout this talk, let

X = separable,∞-dimensional Banach space
B(X) = algebra of operators T : X −

→ X Definition. An operator T ∈ B(X) is hypercyclic if there is a vector x ∈ X for which its orbit Orb(T, x) = {T nx : n ≥ 0} is dense in X.

Sanders Disjoint Hypercyclic Operators

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Introduction Dynamics of Hypercyclic Operators Results

Throughout this talk, let

X = separable,∞-dimensional Banach space
B(X) = algebra of operators T : X −

→ X Definition. An operator T ∈ B(X) is hypercyclic if there is a vector x ∈ X for which its orbit Orb(T, x) = {T nx : n ≥ 0} is dense in X. HC(T) = set of hypercyclic vectors for T

Sanders Disjoint Hypercyclic Operators

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Introduction Dynamics of Hypercyclic Operators Results

Definition. Operators T1, T2, . . . , TN ∈ B(X) with N ≥ 2 are disjoint hypercyclic or d-hypercyclic if there is a vector x ∈ X for which (x, x, . . . , x) ∈ HC(T1 ⊕ T2 ⊕ · · · ⊕ TN)

Sanders Disjoint Hypercyclic Operators

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Introduction Dynamics of Hypercyclic Operators Results

Definition. Operators T1, T2, . . . , TN ∈ B(X) with N ≥ 2 are disjoint hypercyclic or d-hypercyclic if there is a vector x ∈ X for which (x, x, . . . , x) ∈ HC(T1 ⊕ T2 ⊕ · · · ⊕ TN) d-HC(T1, T2, . . . , TN) = set d-hypercyclic vectors for T1, . . . , TN

Sanders Disjoint Hypercyclic Operators

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Introduction Dynamics of Hypercyclic Operators Results

Definition. Operators T1, T2, . . . , TN ∈ B(X) with N ≥ 2 are disjoint hypercyclic or d-hypercyclic if there is a vector x ∈ X for which (x, x, . . . , x) ∈ HC(T1 ⊕ T2 ⊕ · · · ⊕ TN) d-HC(T1, T2, . . . , TN) = set d-hypercyclic vectors for T1, . . . , TN Definition. The operators T1, T2, . . . , TN are densely d-hypercyclic if the set d-HC(T1, T2, . . . , TN) is dense in X.

Sanders Disjoint Hypercyclic Operators

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Introduction Dynamics of Hypercyclic Operators Results

Goal Show several of the standard dynamical properties of hypercyclic operators fail to hold true for disjoint hypercyclic

perators.

Sanders Disjoint Hypercyclic Operators

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Introduction Dynamics of Hypercyclic Operators Results

Goal Show several of the standard dynamical properties of hypercyclic operators fail to hold true for disjoint hypercyclic

perators.
Remark. Restrict our attention for d-hypercyclicity to two
perators T1, T2.

Sanders Disjoint Hypercyclic Operators

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Dynamics of Hypercyclic Operators

Sanders Disjoint Hypercyclic Operators

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Introduction Dynamics of Hypercyclic Operators Results

Dynamics of Hypercyclic Operators Fact 1. The following statements are equivalent: (1) T is hypercyclic. (2) HC(T) is a dense Gδ set. (3) T is topologically transitive; that is, for any nonempty, open sets U, V , there is n ≥ 1 such that U ∩ T −n(V ) = ∅.

Sanders Disjoint Hypercyclic Operators

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Fact 2. The following statements are equivalent: (1) T satisfies the Hypercyclicity Criterion. (2) T is weakly mixing; that is, T ⊕ T is topologically transitive. (3) M

i=1 T is topologically transitive for each M ≥ 1.

Sanders Disjoint Hypercyclic Operators

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Fact 2. The following statements are equivalent: (1) T satisfies the Hypercyclicity Criterion. (2) T is weakly mixing; that is, T ⊕ T is topologically transitive. (3) M

i=1 T is topologically transitive for each M ≥ 1.

(4) T satisfies the Blow-up/Collapse Property; that is, for any nonempty, open sets U, V, W with 0 ∈ W, there is n ≥ 1 such that U ∩ T −n(W) = ∅ and W ∩ T −n(V ) = ∅.

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Introduction Dynamics of Hypercyclic Operators Results

Fact 2. The following statements are equivalent: (1) T satisfies the Hypercyclicity Criterion. (2) T is weakly mixing; that is, T ⊕ T is topologically transitive. (3) M

i=1 T is topologically transitive for each M ≥ 1.

(4) T satisfies the Blow-up/Collapse Property; that is, for any nonempty, open sets U, V, W with 0 ∈ W, there is n ≥ 1 such that U ∩ T −n(W) = ∅ and W ∩ T −n(V ) = ∅.

Remark. If M

i=1 T with M ≥ 2 is hypercyclic, then M+1 i=1 T is

also hypercyclic.

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Defintion. Operators T1, T2 ∈ B(X) are d-topologically transitive if for any nonempty, open sets U, V1, V2, there is n ≥ 1 such that U ∩ T −n

1

(V1) ∩ T −n

2

(V2) = ∅.

Sanders Disjoint Hypercyclic Operators

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Introduction Dynamics of Hypercyclic Operators Results

Defintion. Operators T1, T2 ∈ B(X) are d-topologically transitive if for any nonempty, open sets U, V1, V2, there is n ≥ 1 such that U ∩ T −n

1

(V1) ∩ T −n

2

(V2) = ∅. Theorem A. (B` es, Peris) The following statements are equivalent: (1) T1, T2 are densely d-hypercyclic. (2) d-HC(T1, T2) is a dense Gδ set. (3) T1, T2 are d-topologically transitive.

Sanders Disjoint Hypercyclic Operators

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Questions

1. Does d-hypercyclicity imply dense d-hypercyclicity?

Sanders Disjoint Hypercyclic Operators

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Definition. Operators T1, T2 ∈ B(X) are d-weakly mixing if T1 ⊕ T1, T2 ⊕ T2 are d-topologically transitive.

Sanders Disjoint Hypercyclic Operators

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Definition. Operators T1, T2 ∈ B(X) are d-weakly mixing if T1 ⊕ T1, T2 ⊕ T2 are d-topologically transitive. Disjoint Hypercyclicity Criterion. Operators T1, T2 ∈ B(X) satisfy the d-Hypercyclicity Criterion if there exist sequence (nk)∞

k=1, dense sets X0, X1, X2 of X, and

maps Sm,k : Xm − → X (m = 1, 2) such that T nk

m −

→ 0 pointwise on X0 for m = 1, 2, Sm,k − → 0 pointwise on Xm for m = 1, 2, T nk

m Si,k −

→ δi,mI pointwise on Xi for i, m = 1, 2.

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Disjoint Blow-up/Collapse Property. Operators T1, T2 ∈ B(X) satisfy the Disjoint Blow-up/Collapse Property if for any nonempty, open sets U, V1, V2, W with 0 ∈ W, there is n ≥ 1 for which U ∩ T −n

1

(W) ∩ T −n

2

(W) = ∅ and W ∩ T −n

1

(V1) ∩ T −n

2

(V2) = ∅.

Sanders Disjoint Hypercyclic Operators

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Theorem B. (B` es, Peris)

The two statements are equivalent:

(1) T1, T2 satisfy d-Hypercyclicity Criterion (2) M

i=1 T1, M i=1 T2 are d-topologically transitive for each

M ≥ 1.

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Theorem B. (B` es, Peris)

The two statements are equivalent:

(1) T1, T2 satisfy d-Hypercyclicity Criterion (2) M

i=1 T1, M i=1 T2 are d-topologically transitive for each

M ≥ 1.

If T1, T2 satisfy the Disjoint Blow-up/Collapse Property,

then T1, T2 are d-topologically transitive and so densely d-hypercyclic.

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Theorem B. (B` es, Peris)

The two statements are equivalent:

(1) T1, T2 satisfy d-Hypercyclicity Criterion (2) M

i=1 T1, M i=1 T2 are d-topologically transitive for each

M ≥ 1.

If T1, T2 satisfy the Disjoint Blow-up/Collapse Property,

then T1, T2 are d-topologically transitive and so densely d-hypercyclic.

If T1, T2 are d-weakly mixing, then T1, T2 satisfy the Disjoint

Blow-up/Collapse Property. Hence, d-Hypercyclicity Criterion implies the Disjoint Blow-up/Collapse Property.

Sanders Disjoint Hypercyclic Operators

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Introduction Dynamics of Hypercyclic Operators Results

Questions

1. Does d-hypercyclicity imply dense d-hypercyclicity?

Sanders Disjoint Hypercyclic Operators

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Introduction Dynamics of Hypercyclic Operators Results

Questions

1. Does d-hypercyclicity imply dense d-hypercyclicity?
2. Can we add “T1, T2 are d-weakly mixing” to the first

statement in Theorem B as in Fact 2?

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Questions

1. Does d-hypercyclicity imply dense d-hypercyclicity?
2. Can we add “T1, T2 are d-weakly mixing” to the first

statement in Theorem B as in Fact 2?

3. Are the d-Hypercyclicity Criterion and the Disjoint

Blow-up/Collapse Property equivalent as in Fact 2?

Sanders Disjoint Hypercyclic Operators

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Introduction Dynamics of Hypercyclic Operators Results

Results

Sanders Disjoint Hypercyclic Operators

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Introduction Dynamics of Hypercyclic Operators Results

Results Theorem 1. (Sanders, Shkarin) Let X be a separable, infinite dimensional Banach space. For each M ≥ 1, there are T1, T2 ∈ B(X) such that

M

i=1

T1,

M

i=1

T2 are densely d-hypercyclic and hence d-topologically transitive, but

M+1

i=1

T1,

M+1

i=1

T2 fail to be d-hypercyclic.

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Introduction Dynamics of Hypercyclic Operators Results

Take M = 2 in Theorem 1.

Sanders Disjoint Hypercyclic Operators

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Introduction Dynamics of Hypercyclic Operators Results

Take M = 2 in Theorem 1.

T1, T2 satisfy the d-Hypercyclicity Criterion ⇐

⇒ M

i=1 T1, M i=1 T2 are d-topologically transitive for each M ≥ 1.

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Take M = 2 in Theorem 1.

T1, T2 satisfy the d-Hypercyclicity Criterion ⇐

⇒ M

i=1 T1, M i=1 T2 are d-topologically transitive for each M ≥ 1.

Corollary 1. There are d-weakly mixing operators T1, T2 ∈ B(X) which fail to satisfy the d-Hypercyclicity Criterion.

Sanders Disjoint Hypercyclic Operators

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If T1, T2 are d-weakly mixing, then T1, T2 ∈ B(X) satisfy the

Disjoint Blow-up/Collapse Property.

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Introduction Dynamics of Hypercyclic Operators Results

If T1, T2 are d-weakly mixing, then T1, T2 ∈ B(X) satisfy the

Disjoint Blow-up/Collapse Property. Corollary 2. There are operators T1, T2 ∈ B(X) which satisfy the Blow-up/Collapse Property but fail to satisfy the d-Hypercyclicity Criterion.

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If T1, T2 are d-weakly mixing, then T1, T2 ∈ B(X) satisfy the

Disjoint Blow-up/Collapse Property. Corollary 2. There are operators T1, T2 ∈ B(X) which satisfy the Blow-up/Collapse Property but fail to satisfy the d-Hypercyclicity Criterion. Take M = 1 in Theorem 1.

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If T1, T2 are d-weakly mixing, then T1, T2 ∈ B(X) satisfy the

Disjoint Blow-up/Collapse Property. Corollary 2. There are operators T1, T2 ∈ B(X) which satisfy the Blow-up/Collapse Property but fail to satisfy the d-Hypercyclicity Criterion. Take M = 1 in Theorem 1. Corollary 3. There are densely d-hypercyclic operators T1, T2 ∈ B(X) which fail to be d-weakly mixing.

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Hypercyclicity vs d-Hypercyclicity T1, T2 satisfy d-Hypercyclicity ⇒ T1, T2 satisfy Disjoint Blow-up Criterion

/Collapse Property
M

i=1 T1, M i=1 T2 d-top. trans.

⇓ ⇑ ր ւ? T1, T2 d-weakly mixing

Sanders Disjoint Hypercyclic Operators

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Introduction Dynamics of Hypercyclic Operators Results

Hypercyclicity vs d-Hypercyclicity T1, T2 satisfy d-Hypercyclicity ⇒ T1, T2 satisfy Disjoint Blow-up Criterion

/Collapse Property
M

i=1 T1, M i=1 T2 d-top. trans.

⇓ ⇑ ր ւ? T1, T2 d-weakly mixing Open Question. Does every separable, infinite dimensional Banach space X admit operators T1, T2 which satisfy the Disjoint Blow-up/Collapse Property but fail to be d-weakly mixing?

Sanders Disjoint Hypercyclic Operators

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Outline of Proof for Theorem 1

Sanders Disjoint Hypercyclic Operators

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Outline of Proof for Theorem 1 Step 1. Construct T1, T2 ∈ B(X).

Sanders Disjoint Hypercyclic Operators

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Outline of Proof for Theorem 1 Step 1. Construct T1, T2 ∈ B(X). Let T1 ∈ B(X) which satisfies the Hypercyclicity Criterion. Select any vector (f1, . . . , fM, g1 . . . , gM) ∈ HC 2M

i=1

T1

Sanders

Disjoint Hypercyclic Operators

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Outline of Proof for Theorem 1 Step 1. Construct T1, T2 ∈ B(X). Let T1 ∈ B(X) which satisfies the Hypercyclicity Criterion. Select any vector (f1, . . . , fM, g1 . . . , gM) ∈ HC 2M

i=1

T1

By Hahn-Banach Theorem, there are λ1, . . . , λM ∈ X∗ such

that λi(fj) =

1,

if i = j 0, if i = j and λi(gj) = 0.

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Define L : X − → X by Lx = x + M

i=1 λi(x)gi. One can show

this operator is invertible, and so we let T2 = L−1T1L.

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Define L : X − → X by Lx = x + M

i=1 λi(x)gi. One can show

this operator is invertible, and so we let T2 = L−1T1L. Step 2. Show M

i=1 T1, M i=1 T2 are densely d-hypercyclic.

Sanders Disjoint Hypercyclic Operators

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Define L : X − → X by Lx = x + M

i=1 λi(x)gi. One can show

this operator is invertible, and so we let T2 = L−1T1L. Step 2. Show M

i=1 T1, M i=1 T2 are densely d-hypercyclic.

Claim. Let A1 = L−1

1 AL1, A2 = L−1 2 AL2. Then (x1, . . . , xM) ∈

d-HC(M

i=1 A1, M i=1 A2) if and only if

(L1x1, . . . , L1xM, L2x1, . . . , L2xM) ∈ HC 2M

i=1

A

.

Sanders Disjoint Hypercyclic Operators

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Define L : X − → X by Lx = x + M

i=1 λi(x)gi. One can show

this operator is invertible, and so we let T2 = L−1T1L. Step 2. Show M

i=1 T1, M i=1 T2 are densely d-hypercyclic.

Claim. Let A1 = L−1

1 AL1, A2 = L−1 2 AL2. Then (x1, . . . , xM) ∈

d-HC(M

i=1 A1, M i=1 A2) if and only if

(L1x1, . . . , L1xM, L2x1, . . . , L2xM) ∈ HC 2M

i=1

A

.

Proof of Claim. Follow directly from definitions of hypercyclic and d-hypercyclic vectors.

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Let U1 ⊕ · · · ⊕ UM be a basic open in M

i=1 X. Define

A : M

i=1 X −

→ B(FM) by A(x1, . . . , xM) = [λi(xj)]i,j=1,...,M.

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Let U1 ⊕ · · · ⊕ UM be a basic open in M

i=1 X. Define

A : M

i=1 X −

→ B(FM) by A(x1, . . . , xM) = [λi(xj)]i,j=1,...,M. Observe that (U1 ⊕ · · · ⊕ UM) ∩ A−1(O) = ∅ where O is the set

f all invertible M × M matrices.

Sanders Disjoint Hypercyclic Operators

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Let U1 ⊕ · · · ⊕ UM be a basic open in M

i=1 X. Define

A : M

i=1 X −

→ B(FM) by A(x1, . . . , xM) = [λi(xj)]i,j=1,...,M. Observe that (U1 ⊕ · · · ⊕ UM) ∩ A−1(O) = ∅ where O is the set

f all invertible M × M matrices. Moreover, since this set is
pen and (f1, . . . , fM) ∈ HC(M

i=1 T1), there is r ≥ 1 with

(T r

1 f1, . . . , T r 1 fM) ∈ (U1 ⊕ · · · ⊕ UM) ∩ A−1(O)

Sanders Disjoint Hypercyclic Operators

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Let U1 ⊕ · · · ⊕ UM be a basic open in M

i=1 X. Define

A : M

i=1 X −

→ B(FM) by A(x1, . . . , xM) = [λi(xj)]i,j=1,...,M. Observe that (U1 ⊕ · · · ⊕ UM) ∩ A−1(O) = ∅ where O is the set

f all invertible M × M matrices. Moreover, since this set is
pen and (f1, . . . , fM) ∈ HC(M

i=1 T1), there is r ≥ 1 with

(T r

1 f1, . . . , T r 1 fM) ∈ (U1 ⊕ · · · ⊕ UM) ∩ A−1(O)

It remains to show (T r

1 f1, . . . , T r 1 fM) ∈ d-HC

M

i=1

T1,

M

i=1

T2

.

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Note that (T r

1 f1, . . . , T r 1 fM, LT1f1, . . . , LT r 1 fM) is the same as

T r

1 f1, . . . , T r 1 fM, T r 1 f1 + M

i=1

λi(T r

1 f1)gi, . . . , T r 1 fM + M

i=1

λi(T r

1 fM)gi

Sanders

Disjoint Hypercyclic Operators

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Note that (T r

1 f1, . . . , T r 1 fM, LT1f1, . . . , LT r 1 fM) is the same as

T r

1 f1, . . . , T r 1 fM, T r 1 f1 + M

i=1

λi(T r

1 f1)gi, . . . , T r 1 fM + M

i=1

λi(T r

1 fM)gi

Since (f1, . . . , fM, g1, . . . , gM) ∈ HC(2M

i=1 T1) and the matrix

A(T r

1 f1, . . . , T r 1 fM) = [λi(T r 1 fj)]i,j=1,...,M is invertible, we get

this vector is in HC(2M

i=1 T1).

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Note that (T r

1 f1, . . . , T r 1 fM, LT1f1, . . . , LT r 1 fM) is the same as

T r

1 f1, . . . , T r 1 fM, T r 1 f1 + M

i=1

λi(T r

1 f1)gi, . . . , T r 1 fM + M

i=1

λi(T r

1 fM)gi

Since (f1, . . . , fM, g1, . . . , gM) ∈ HC(2M

i=1 T1) and the matrix

A(T r

1 f1, . . . , T r 1 fM) = [λi(T r 1 fj)]i,j=1,...,M is invertible, we get

this vector is in HC(2M

i=1 T1). By our Claim, it follows that

(T r

1 f1, . . . , T r 1 fM) ∈ d-HC

M

i=1

T1,

M

i=1

T2

.

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Step 3. Show M+1

i=1 T1, M+1 i=1 T2 fail to be d-hypercyclic.

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Step 3. Show M+1

i=1 T1, M+1 i=1 T2 fail to be d-hypercyclic.

Suppose (x1, . . . , xM+1) ∈ d-HC(M+1

i=1 T1, M+1 i=1 T2). Let

y1, . . . , yM+1 ∈ X be linearly independent.

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Step 3. Show M+1

i=1 T1, M+1 i=1 T2 fail to be d-hypercyclic.

Suppose (x1, . . . , xM+1) ∈ d-HC(M+1

i=1 T1, M+1 i=1 T2). Let

y1, . . . , yM+1 ∈ X be linearly independent. Select (nk)∞

k=1 such

that T nk

1 xj −

→ 0 as k → ∞ and T nk

2 xj −

→ yj as k → ∞ for j = 1, . . . M + 1.

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Using the definition of L, T2, we have T nk

2 xj = L−1T nk 1 xj + M

i=1

λi(xj)L−1T nk

1 gi

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Using the definition of L, T2, we have T nk

2 xj = L−1T nk 1 xj + M

i=1

λi(xj)L−1T nk

1 gi

Using linear algebra, we can show there exists h1, . . . , hM in span{y1, . . . , yM} such that L−1T nk

1 gi −

→ hi as k → ∞ for 1 = 1, . . . , M.

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Using the definition of L, T2, we have T nk

2 xj = L−1T nk 1 xj + M

i=1

λi(xj)L−1T nk

1 gi

Using linear algebra, we can show there exists h1, . . . , hM in span{y1, . . . , yM} such that L−1T nk

1 gi −

→ hi as k → ∞ for 1 = 1, . . . , M. Now, T nk

2 xM+1 −

→ yM+1 as k → ∞, and

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T nk

2 xM+1 = L−1T nk 1 xM+1 + M

i=1

λi(xM+1)L−1T nk

1 gi

− → L−10 +

M

i=1

λi(xM+1)hi =

M

i=1

λi(xM+1)hi ∈ span{y1, . . . , yM}.

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T nk

2 xM+1 = L−1T nk 1 xM+1 + M

i=1

λi(xM+1)L−1T nk

1 gi

− → L−10 +

M

i=1

λi(xM+1)hi =

M

i=1

λi(xM+1)hi ∈ span{y1, . . . , yM}. Therefore, yM+1 =

M

i=1

λi(xM+1)hi ∈ span{y1, . . . , yM} which gives us a contradiction.

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Results

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Results Theorem 2. (Sanders, Shkarin) Let X be a separable, infinite dimensional Banach space. For each M ≥ 1, there are T1, T2 ∈ B(X) and x1, x2, . . . , xM ∈ X such that the set d-HC M

i=1 T1, M i=1 T2

is exactly
(h1, . . . , hM) ∈

M

i=1

span{x1, . . . , xM} : h1, . . . , hM lin. ind.

Sanders

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Take M = 1 in Theorem 2.

d-HC(T1, T2) = span{x1} \ {0}

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Take M = 1 in Theorem 2.

d-HC(T1, T2) = span{x1} \ {0}

Corollary 1. There are d-hypercyclic operators T1, T2 ∈ B(X) for which d-HC(T1, T2) is nowhere dense, and so fail to be densely d-hypercyclic.

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Take M = 1 in Theorem 2.

d-HC(T1, T2) = span{x1} \ {0}

Corollary 1. There are d-hypercyclic operators T1, T2 ∈ B(X) for which d-HC(T1, T2) is nowhere dense, and so fail to be densely d-hypercyclic. Corollary 2. There are d-hypercyclic operators T1, T2 ∈ B(X) which fail to be d-topologically transitive.

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Corollary 3. There are d-hypercyclic operators T1, T2 ∈ B(X) and densely d-hypercyclic operators A1, A2 ∈ B(X) for which d-HC(T1, T2) ∩ d-HC(A1, A2) = ∅.

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Corollary 3. There are d-hypercyclic operators T1, T2 ∈ B(X) and densely d-hypercyclic operators A1, A2 ∈ B(X) for which d-HC(T1, T2) ∩ d-HC(A1, A2) = ∅. Take M = 2 in Theorem 2.

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Corollary 3. There are d-hypercyclic operators T1, T2 ∈ B(X) and densely d-hypercyclic operators A1, A2 ∈ B(X) for which d-HC(T1, T2) ∩ d-HC(A1, A2) = ∅. Take M = 2 in Theorem 2. Corollary 4. There are T1, T2 ∈ B(X) such that T1 ⊕ T1, T2 ⊕ T2 are d-hypercyclic, but T1, T2 fail to be d-weakly mixing.

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Remark. The operators T1, T2 in Theorem 2 satisfy the

property that M

i=1 T1, M i=1 T2 are d-hypercyclic, but

M+1

i=1 T1, M+1 i=1 T2 fail to be d-hypercylic.

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Remark. The operators T1, T2 in Theorem 2 satisfy the

property that M

i=1 T1, M i=1 T2 are d-hypercyclic, but

M+1

i=1 T1, M+1 i=1 T2 fail to be d-hypercylic.

Proof. Let (f1, . . . , fM+1) ∈ d-HC(M+1

i=1 T1, M+1 i=1 T2).

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Remark. The operators T1, T2 in Theorem 2 satisfy the

property that M

i=1 T1, M i=1 T2 are d-hypercyclic, but

M+1

i=1 T1, M+1 i=1 T2 fail to be d-hypercylic.

Proof. Let (f1, . . . , fM+1) ∈ d-HC(M+1

i=1 T1, M+1 i=1 T2). Then

(f1, . . . , fm), (f2, . . . fM+1) are both in d-HC(M

i=1 T1, M i=1 T2).

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Remark. The operators T1, T2 in Theorem 2 satisfy the

property that M

i=1 T1, M i=1 T2 are d-hypercyclic, but

M+1

i=1 T1, M+1 i=1 T2 fail to be d-hypercylic.

Proof. Let (f1, . . . , fM+1) ∈ d-HC(M+1

i=1 T1, M+1 i=1 T2). Then

(f1, . . . , fm), (f2, . . . fM+1) are both in d-HC(M

i=1 T1, M i=1 T2).

Thus, f1, . . . , fM+1 ∈ span{x1, . . . , xM} which is impossible.

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Remark. The operators T1, T2 in Theorem 2 satisfy the

property that M

i=1 T1, M i=1 T2 are d-hypercyclic, but

M+1

i=1 T1, M+1 i=1 T2 fail to be d-hypercylic.

Proof. Let (f1, . . . , fM+1) ∈ d-HC(M+1

i=1 T1, M+1 i=1 T2). Then

(f1, . . . , fm), (f2, . . . fM+1) are both in d-HC(M

i=1 T1, M i=1 T2).

Thus, f1, . . . , fM+1 ∈ span{x1, . . . , xM} which is impossible. Open Question. Does there exist T1, T2 ∈ B(X) whose d-HC(T1, T2) is somewhere dense in X but not dense?

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Outline of Proof for Theorem 2

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Outline of Proof for Theorem 2 Step 1. Provide a sufficient condition for the conclusion of Theorem 2 to hold true.

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Outline of Proof for Theorem 2 Step 1. Provide a sufficient condition for the conclusion of Theorem 2 to hold true. Proposition. Let T1 ∈ B(X) and M ≥ 1. If there exist injective operator R ∈ B(X) and x1, . . . , xM, y1, . . . , yM ∈ X such that (i) (x1, . . . , xM, y1, . . . , yM) ∈ HC 2M

i=1 T

,

(ii) Rxi = yj for i = 1, . . . M, (iii) R(X) ∩ HC(T1) ⊆ span{x1, . . . xM}, then there is T2 ∈ B(X) such that d-HC(M

i=1 T1, M i=1 T2) is

exactly

(h1, . . . , hM) ∈

M

i=1

span{x1, . . . , xM} : h1, . . . , hM lin. ind.

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Disjoint Hypercyclic Operators

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Proof of Proposition. Let A be this set. Select c > R, and set L1 = cI and L2 = cI + R.

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Proof of Proposition. Let A be this set. Select c > R, and set L1 = cI and L2 = cI + R. Note T1 = L−1

1 T1L1 and set

T2 = L−1

2 T1L2.

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Proof of Proposition. Let A be this set. Select c > R, and set L1 = cI and L2 = cI + R. Note T1 = L−1

1 T1L1 and set

T2 = L−1

2 T1L2.

To show A ⊆ d-HC(M

i=1 T1, M i=1 T2), let h1, . . . , hM in

span{x1, . . . , xM} be linearly independent.

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Proof of Proposition. Let A be this set. Select c > R, and set L1 = cI and L2 = cI + R. Note T1 = L−1

1 T1L1 and set

T2 = L−1

2 T1L2.

To show A ⊆ d-HC(M

i=1 T1, M i=1 T2), let h1, . . . , hM in

span{x1, . . . , xM} be linearly independent. By Claim from Theorem 1, it suffice to show (L1h1, . . . , L1hM, L2h1, . . . , L2hM) ∈ HC 2M

i=1

T1

.

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By definition of L1, L2, this vector is (ch1, . . . , chM, ch1 + Rh1, . . . , chM + RhM).

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By definition of L1, L2, this vector is (ch1, . . . , chM, ch1 + Rh1, . . . , chM + RhM). Moreover, this vector is in HC(2M

i=1 T1) if and only if

(h1, . . . , hM, Rh1, . . . , RhM) ∈ HC 2M

i=1

T1

.

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By definition of L1, L2, this vector is (ch1, . . . , chM, ch1 + Rh1, . . . , chM + RhM). Moreover, this vector is in HC(2M

i=1 T1) if and only if

(h1, . . . , hM, Rh1, . . . , RhM) ∈ HC 2M

i=1

T1

.

Now, h1, . . . , hM are linearly independent in span{x1, . . . , xM}. Since R is injective and Rxi = yi, we also have Rh1, . . . , RhM are linearly independent in span{y1, . . . , yM}.

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By definition of L1, L2, this vector is (ch1, . . . , chM, ch1 + Rh1, . . . , chM + RhM). Moreover, this vector is in HC(2M

i=1 T1) if and only if

(h1, . . . , hM, Rh1, . . . , RhM) ∈ HC 2M

i=1

T1

.

Now, h1, . . . , hM are linearly independent in span{x1, . . . , xM}. Since R is injective and Rxi = yi, we also have Rh1, . . . , RhM are linearly independent in span{y1, . . . , yM}. Since (x1, . . . , xM, y1, . . . , yM) ∈ HC(2M

i=1 T1), it follows that

(h1, . . . , hM, Rh1, . . . , RhM) ∈ HC 2M

i=1

T1

.

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To show d-HC(M

i=1 T1, M i=1 T2) ⊆ A, let (h1, . . . , hM) be in

d-HC(M

i=1 T1, M i=1 T2).

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To show d-HC(M

i=1 T1, M i=1 T2) ⊆ A, let (h1, . . . , hM) be in

d-HC(M

i=1 T1, M i=1 T2). Note, h1, . . . , hM must be linearly

independent.

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To show d-HC(M

i=1 T1, M i=1 T2) ⊆ A, let (h1, . . . , hM) be in

d-HC(M

i=1 T1, M i=1 T2). Note, h1, . . . , hM must be linearly

independent. Also,

(h1, . . . , hM, Rh1, . . . , RhM) ∈ HC 2M

i=1

T1

.

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To show d-HC(M

i=1 T1, M i=1 T2) ⊆ A, let (h1, . . . , hM) be in

d-HC(M

i=1 T1, M i=1 T2). Note, h1, . . . , hM must be linearly

independent. Also,

(h1, . . . , hM, Rh1, . . . , RhM) ∈ HC 2M

i=1

T1

.

This implies Rh1, . . . , RhM ∈ R(X) ∩ HC(T1) ⊆ span{y1, . . . , yM}.

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Thus, we can write Rhi = α1,iy1 + · · · + αM,iyM.

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Thus, we can write Rhi = α1,iy1 + · · · + αM,iyM. Also R(α1,ix1 + · · · + αM,ixM) = α1,iy1 + · · · + αM,iyM.

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Thus, we can write Rhi = α1,iy1 + · · · + αM,iyM. Also R(α1,ix1 + · · · + αM,ixM) = α1,iy1 + · · · + αM,iyM. Since R is injective, we get hi = α1,ix1 + · · · + αM,ixM ∈ span{x1, . . . , xM}, and so (h1, . . . hM) ∈ A.

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Step 2. Construct the operator T1 ∈ B(X) for Proposition.

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Step 2. Construct the operator T1 ∈ B(X) for Proposition. Let (ek)∞

k=0 in X and (λk)∞ k=0 in X∗ such that

(a) span{ek : k ≥ 0} and ek = 1 (b) ∞

k=0 Ker(λk) = {0} and ∞ k=0 λk < ∞

(c) λk(ej) = 0 if j = k and λk(ek) = ck > 0.

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Step 2. Construct the operator T1 ∈ B(X) for Proposition. Let (ek)∞

k=0 in X and (λk)∞ k=0 in X∗ such that

(a) span{ek : k ≥ 0} and ek = 1 (b) ∞

k=0 Ker(λk) = {0} and ∞ k=0 λk < ∞

(c) λk(ej) = 0 if j = k and λk(ek) = ck > 0. Define T1 : X − → X by T1x = x +

∞

k=0

λk+1(x)ek.

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Step 3. Based on T1, construct the injective operator R ∈ B(X) and the vectors x1, . . . , xM, y1, . . . , yM for Proposition.

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Step 3. Based on T1, construct the injective operator R ∈ B(X) and the vectors x1, . . . , xM, y1, . . . , yM for Proposition. Essentially (x1, . . . , xM, y1, . . . , yM) ∈ HC 2M

i=1

T1

and

R = nuclear operator +

M

i=1
λi(·)fi

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Thank you for listening.

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References

1. L. Bernal-Gonz´

alez, Disjoint hypercyclic operators, Studia

Math. 182 (2) (2007), 113-130.
2. J. B`

es and A. Peris, Disjointness in hypercyclicity, J. Math.

Anal. Appl. 336 (2007), 297-315.
3. J. B`

es, ¨

O. Martin, and R. Sanders, Weighted shifts and

disjoint hypercyclicity, Journal of Operator Theory, to appear.

4. R. Sanders and S. Shkarin, Existence of disjoint weakly

mixing operators that fail to satisfy the Disjoint Hypercyclicity Criterion, J. Math. Anal. App., to appear.

Sanders Disjoint Hypercyclic Operators