Chaotic Extension of an Operator on a Hilbert Subspace Kit Chan - - PowerPoint PPT Presentation

Chaotic Extension of an Operator

• n a Hilbert Subspace

Kit Chan

Bowling Green State University

April 12, 2014

Kit Chan Chaotic Extension of an Operator on a Hilbert Subspace

Definition: Chaotic Extension

Let H be a separable, infinite-dimensional Hilbert space, and B(H) = {T : H → H|T is bounded and linear}. A bounded linear operator T in B(H) is hypercyclic if there is a vector x whose orbit orb(T, x) = {x, Tx, T 2x, T 3x, . . .} is dense in

• H. Such a vector x is called a hypercyclic vector.

Kit Chan Chaotic Extension of an Operator on a Hilbert Subspace

Definition: Chaotic Extension

Let H be a separable, infinite-dimensional Hilbert space, and B(H) = {T : H → H|T is bounded and linear}. A bounded linear operator T in B(H) is hypercyclic if there is a vector x whose orbit orb(T, x) = {x, Tx, T 2x, T 3x, . . .} is dense in

• H. Such a vector x is called a hypercyclic vector.

A vector x is a periodic point if there is a positive integer n such that T nx = x. An operator T in B(H) is chaotic if T is hypercyclic and has a dense set of periodic points.

Kit Chan Chaotic Extension of an Operator on a Hilbert Subspace

Definition: Chaotic Extension

Let H be a separable, infinite-dimensional Hilbert space, and B(H) = {T : H → H|T is bounded and linear}. A bounded linear operator T in B(H) is hypercyclic if there is a vector x whose orbit orb(T, x) = {x, Tx, T 2x, T 3x, . . .} is dense in

• H. Such a vector x is called a hypercyclic vector.

A vector x is a periodic point if there is a positive integer n such that T nx = x. An operator T in B(H) is chaotic if T is hypercyclic and has a dense set of periodic points. Theorem (Grivaux, 2005) If dim H/M = ∞, then every operator A ∈ B(M) has a chaotic extension T ∈ B(H); that is, a chaotic operator T : H → H whose restriction T|M = A.

Kit Chan Chaotic Extension of an Operator on a Hilbert Subspace

Problem with a General Banach Space

Kit Chan Chaotic Extension of an Operator on a Hilbert Subspace

Problem with a General Banach Space

Theorem (Argyros & Haydon, 2009) There exists a separable infinite dimensional Banach space N on which every bounded linear operator of the form αI + K, where α ∈ C and K compact.

Kit Chan Chaotic Extension of an Operator on a Hilbert Subspace

Problem with a General Banach Space

Theorem (Argyros & Haydon, 2009) There exists a separable infinite dimensional Banach space N on which every bounded linear operator of the form αI + K, where α ∈ C and K compact. Take X = C ⊕ N, and A = 2I : C → C. Suppose A has a hypercyclic extension T, which must take the form 2I ⋆ αI + K

• Kit Chan

Chaotic Extension of an Operator on a Hilbert Subspace

Problem with a General Banach Space

Theorem (Argyros & Haydon, 2009) There exists a separable infinite dimensional Banach space N on which every bounded linear operator of the form αI + K, where α ∈ C and K compact. Take X = C ⊕ N, and A = 2I : C → C. Suppose A has a hypercyclic extension T, which must take the form 2I ⋆ αI + K

• Thus the spectrum σ(T) = {2} ∪ (σ(K) + α).

Kit Chan Chaotic Extension of an Operator on a Hilbert Subspace

Problem with a General Banach Space

Theorem (Argyros & Haydon, 2009) There exists a separable infinite dimensional Banach space N on which every bounded linear operator of the form αI + K, where α ∈ C and K compact. Take X = C ⊕ N, and A = 2I : C → C. Suppose A has a hypercyclic extension T, which must take the form 2I ⋆ αI + K

• Thus the spectrum σ(T) = {2} ∪ (σ(K) + α).

K compact = ⇒ σ(K) has at most countable number of points. Contradiction – because Kitai proved in 1982 that every component of σ(T) must intersect the unit circle. ✷

Kit Chan Chaotic Extension of an Operator on a Hilbert Subspace

Finite Codimensional Subspaces

Fact If M is a closed subspace of H with dim H/M < ∞, then no

• perator A in B(M) can have a hypercyclic extension T in B(H).

Kit Chan Chaotic Extension of an Operator on a Hilbert Subspace

Finite Codimensional Subspaces

Fact If M is a closed subspace of H with dim H/M < ∞, then no

• perator A in B(M) can have a hypercyclic extension T in B(H).
• Proof. Suppose T ∈ B(H) is a hypercyclic extension of A.

Let π : H → H/M be the quotient map; that is, π(f ) = [f ] = f + M.

Kit Chan Chaotic Extension of an Operator on a Hilbert Subspace

Finite Codimensional Subspaces

Fact If M is a closed subspace of H with dim H/M < ∞, then no

• perator A in B(M) can have a hypercyclic extension T in B(H).
• Proof. Suppose T ∈ B(H) is a hypercyclic extension of A.

Let π : H → H/M be the quotient map; that is, π(f ) = [f ] = f + M. If h is a hypercyclic vector for T, then the set π{h, Th, T 2h, . . .} = {[h], [Th], [T 2h], . . .} is dense in H/M.

Kit Chan Chaotic Extension of an Operator on a Hilbert Subspace

Finite Codimensional Subspaces

Fact If M is a closed subspace of H with dim H/M < ∞, then no

• perator A in B(M) can have a hypercyclic extension T in B(H).
• Proof. Suppose T ∈ B(H) is a hypercyclic extension of A.

Let π : H → H/M be the quotient map; that is, π(f ) = [f ] = f + M. If h is a hypercyclic vector for T, then the set π{h, Th, T 2h, . . .} = {[h], [Th], [T 2h], . . .} is dense in H/M. If S : H/M → H/M is the linear map defined by S[x] = [Tx], then S is hypercyclic operator on a finite dimensional space H/M, which is impossible. ✷

Kit Chan Chaotic Extension of an Operator on a Hilbert Subspace

Extensions Satisfying the Hypercyclcity Criterion

Theorem (Kitai, 1982; Gethner & Shapiro, 1987) Let X be a Fr´ echet space. A continuous linear operator T : X → X is hypercyclic if there is a dense subset of vectors on which T n → 0 pointwise and if there are a (possibly different) dense subset Y of X and a (not necessarily linear and continuous) map S : Y → Y such that TS = identity on Y and Sn → 0 pointwise on Y .

Kit Chan Chaotic Extension of an Operator on a Hilbert Subspace

Extensions Satisfying the Hypercyclcity Criterion

Theorem (Kitai, 1982; Gethner & Shapiro, 1987) Let X be a Fr´ echet space. A continuous linear operator T : X → X is hypercyclic if there is a dense subset of vectors on which T n → 0 pointwise and if there are a (possibly different) dense subset Y of X and a (not necessarily linear and continuous) map S : Y → Y such that TS = identity on Y and Sn → 0 pointwise on Y . Theorem (with Turcu, 2010) If M is a closed subspace of a separable infinite dimensional Hilbert space H with dim H/M = ∞, then every bounded linear operator A : M → M has a chaotic extension T : H → H that satisfies the Hyercyclciity Criterion in the strongest sense.

Kit Chan Chaotic Extension of an Operator on a Hilbert Subspace

Extensions Satisfying the Hypercyclcity Criterion

Theorem (Kitai, 1982; Gethner & Shapiro, 1987) Let X be a Fr´ echet space. A continuous linear operator T : X → X is hypercyclic if there is a dense subset of vectors on which T n → 0 pointwise and if there are a (possibly different) dense subset Y of X and a (not necessarily linear and continuous) map S : Y → Y such that TS = identity on Y and Sn → 0 pointwise on Y . Theorem (with Turcu, 2010) If M is a closed subspace of a separable infinite dimensional Hilbert space H with dim H/M = ∞, then every bounded linear operator A : M → M has a chaotic extension T : H → H that satisfies the Hyercyclciity Criterion in the strongest sense. Extension T is the same as the one obtained by Grivaux. The norm of the extension T ≤ 2 max{1, A}.

Kit Chan Chaotic Extension of an Operator on a Hilbert Subspace

What Does an Extension Look Like?

Since dim H/M = ∞, we rename M as M0 and write H = ∞

j=0 Mj, where each Mj is isomorphic to M.

Kit Chan Chaotic Extension of an Operator on a Hilbert Subspace

What Does an Extension Look Like?

Since dim H/M = ∞, we rename M as M0 and write H = ∞

j=0 Mj, where each Mj is isomorphic to M.

Let α > max{1, A}. Define T : H → H by suppressing symbols for isomorphisms: T(h0, h1, h2, . . .) = (Ah0 + αh1, αh2, αh3, . . .).

Kit Chan Chaotic Extension of an Operator on a Hilbert Subspace

What Does an Extension Look Like?

Since dim H/M = ∞, we rename M as M0 and write H = ∞

j=0 Mj, where each Mj is isomorphic to M.

Let α > max{1, A}. Define T : H → H by suppressing symbols for isomorphisms: T(h0, h1, h2, . . .) = (Ah0 + αh1, αh2, αh3, . . .). Thus if x = (d0, d1, . . . , dk, 0, 0, 0, . . .) and if n ≥ k then T nx = (An−k(Akd0 + αAk−1d1 + · · · + αkdk), 0, 0, 0, . . .). Let S(h0, h1, h2, . . .) = 1

α(0, h0, h1, h2, . . .). So TS = I.

Kit Chan Chaotic Extension of an Operator on a Hilbert Subspace

What Does an Extension Look Like?

Since dim H/M = ∞, we rename M as M0 and write H = ∞

j=0 Mj, where each Mj is isomorphic to M.

Let α > max{1, A}. Define T : H → H by suppressing symbols for isomorphisms: T(h0, h1, h2, . . .) = (Ah0 + αh1, αh2, αh3, . . .). Thus if x = (d0, d1, . . . , dk, 0, 0, 0, . . .) and if n ≥ k then T nx = (An−k(Akd0 + αAk−1d1 + · · · + αkdk), 0, 0, 0, . . .). Let S(h0, h1, h2, . . .) = 1

α(0, h0, h1, h2, . . .). So TS = I.

Hence T has a right inverse S and satisfies the Hypercyclicity Criterion in the strongest sense.

Kit Chan Chaotic Extension of an Operator on a Hilbert Subspace

What Does an Extension Look Like?

Since dim H/M = ∞, we rename M as M0 and write H = ∞

j=0 Mj, where each Mj is isomorphic to M.

Let α > max{1, A}. Define T : H → H by suppressing symbols for isomorphisms: T(h0, h1, h2, . . .) = (Ah0 + αh1, αh2, αh3, . . .). Thus if x = (d0, d1, . . . , dk, 0, 0, 0, . . .) and if n ≥ k then T nx = (An−k(Akd0 + αAk−1d1 + · · · + αkdk), 0, 0, 0, . . .). Let S(h0, h1, h2, . . .) = 1

α(0, h0, h1, h2, . . .). So TS = I.

Hence T has a right inverse S and satisfies the Hypercyclicity Criterion in the strongest sense. Remark: ker T = {0} and so T is not left invertible.

Kit Chan Chaotic Extension of an Operator on a Hilbert Subspace

What Does an Extension Look Like?

Since dim H/M = ∞, we rename M as M0 and write H = ∞

j=0 Mj, where each Mj is isomorphic to M.

Let α > max{1, A}. Define T : H → H by suppressing symbols for isomorphisms: T(h0, h1, h2, . . .) = (Ah0 + αh1, αh2, αh3, . . .). Thus if x = (d0, d1, . . . , dk, 0, 0, 0, . . .) and if n ≥ k then T nx = (An−k(Akd0 + αAk−1d1 + · · · + αkdk), 0, 0, 0, . . .). Let S(h0, h1, h2, . . .) = 1

α(0, h0, h1, h2, . . .). So TS = I.

Hence T has a right inverse S and satisfies the Hypercyclicity Criterion in the strongest sense. Remark: ker T = {0} and so T is not left invertible. For periodic points, note: SnT nx ≤ An−k

αn

Akd0 + αAk−1d1 + · · · αkdk → 0.

Kit Chan Chaotic Extension of an Operator on a Hilbert Subspace

A Dense Set of Periodic Points

Let x = (d0, d1, . . . , dk, 0, 0, 0, . . .) ∈ H.

Kit Chan Chaotic Extension of an Operator on a Hilbert Subspace

A Dense Set of Periodic Points

Let x = (d0, d1, . . . , dk, 0, 0, 0, . . .) ∈ H. Let y = x − SnT nx. Since SnT nx → 0, such vectors y are dense.

Kit Chan Chaotic Extension of an Operator on a Hilbert Subspace

A Dense Set of Periodic Points

Let x = (d0, d1, . . . , dk, 0, 0, 0, . . .) ∈ H. Let y = x − SnT nx. Since SnT nx → 0, such vectors y are dense. Note T ny = T nx − (T nSn)T nx = T nx − T nx = 0.

Kit Chan Chaotic Extension of an Operator on a Hilbert Subspace

A Dense Set of Periodic Points

Let x = (d0, d1, . . . , dk, 0, 0, 0, . . .) ∈ H. Let y = x − SnT nx. Since SnT nx → 0, such vectors y are dense. Note T ny = T nx − (T nSn)T nx = T nx − T nx = 0. Let z =

∞

• j=0

Sjny = y +

∞

• j=1

Sjny.

Kit Chan Chaotic Extension of an Operator on a Hilbert Subspace

A Dense Set of Periodic Points

Let x = (d0, d1, . . . , dk, 0, 0, 0, . . .) ∈ H. Let y = x − SnT nx. Since SnT nx → 0, such vectors y are dense. Note T ny = T nx − (T nSn)T nx = T nx − T nx = 0. Let z =

∞

• j=0

Sjny = y +

∞

• j=1
• Sjny. Then

T nz = T ny +

∞

• j=1

T nSjny =

∞

• j=1

S(j−1)ny = z. Thus z is a periodic point.

Kit Chan Chaotic Extension of an Operator on a Hilbert Subspace

A Dense Set of Periodic Points

Let x = (d0, d1, . . . , dk, 0, 0, 0, . . .) ∈ H. Let y = x − SnT nx. Since SnT nx → 0, such vectors y are dense. Note T ny = T nx − (T nSn)T nx = T nx − T nx = 0. Let z =

∞

• j=0

Sjny = y +

∞

• j=1
• Sjny. Then

T nz = T ny +

∞

• j=1

T nSjny =

∞

• j=1

S(j−1)ny = z. Thus z is a periodic point. Such vectors z are dense because z = y +

∞

• j=1

Sjny = (x − SnT nx) +

∞

• j=1

Sjn(x − T nSnx) → x. ✷

Kit Chan Chaotic Extension of an Operator on a Hilbert Subspace

More about the Hypercyclicity Criterion

Theorem (De La Rosa & Read, 2009) There is a hypercyclic operator on a Banach space that does not satisfy the Criterion.

Kit Chan Chaotic Extension of an Operator on a Hilbert Subspace

More about the Hypercyclicity Criterion

Theorem (De La Rosa & Read, 2009) There is a hypercyclic operator on a Banach space that does not satisfy the Criterion. Theorem (Bayart & Matheron, 2007) There is a hypercyclic operator on a Hilbert space H that does not satisfy the Criterion.

Kit Chan Chaotic Extension of an Operator on a Hilbert Subspace

More about the Hypercyclicity Criterion

Theorem (De La Rosa & Read, 2009) There is a hypercyclic operator on a Banach space that does not satisfy the Criterion. Theorem (Bayart & Matheron, 2007) There is a hypercyclic operator on a Hilbert space H that does not satisfy the Criterion. Corollary If A in B(H) is a hypercyclic operator that does not satisfy the Criterion, then A has an extension to a “larger” Hilbert space that satisfies the Criterion.

Kit Chan Chaotic Extension of an Operator on a Hilbert Subspace

Can We Have ker T = ker A?

Kit Chan Chaotic Extension of an Operator on a Hilbert Subspace

Can We Have ker T = ker A?

Theorem (with Kadel, 2014) If dim (H/M) = ∞, and A in B(M) has a closed range, then A has a right invertible chaotic extension T in B(H) with ker A = ker T that satisfies the Criterion in the strongest sense.

Kit Chan Chaotic Extension of an Operator on a Hilbert Subspace

Can We Have ker T = ker A?

Theorem (with Kadel, 2014) If dim (H/M) = ∞, and A in B(M) has a closed range, then A has a right invertible chaotic extension T in B(H) with ker A = ker T that satisfies the Criterion in the strongest sense. What does an extension look like?

Kit Chan Chaotic Extension of an Operator on a Hilbert Subspace

Can We Have ker T = ker A?

Theorem (with Kadel, 2014) If dim (H/M) = ∞, and A in B(M) has a closed range, then A has a right invertible chaotic extension T in B(H) with ker A = ker T that satisfies the Criterion in the strongest sense. What does an extension look like? Write M = ran A ⊕ ran A⊥.

Kit Chan Chaotic Extension of an Operator on a Hilbert Subspace

Can We Have ker T = ker A?

Theorem (with Kadel, 2014) If dim (H/M) = ∞, and A in B(M) has a closed range, then A has a right invertible chaotic extension T in B(H) with ker A = ker T that satisfies the Criterion in the strongest sense. What does an extension look like? Write M = ran A ⊕ ran A⊥. Let M0, M1, M2, . . . be orthogonal subspaces of M⊥, each isomorphic to M.

Kit Chan Chaotic Extension of an Operator on a Hilbert Subspace

Can We Have ker T = ker A?

Theorem (with Kadel, 2014) If dim (H/M) = ∞, and A in B(M) has a closed range, then A has a right invertible chaotic extension T in B(H) with ker A = ker T that satisfies the Criterion in the strongest sense. What does an extension look like? Write M = ran A ⊕ ran A⊥. Let M0, M1, M2, . . . be orthogonal subspaces of M⊥, each isomorphic to M. Identify M0 with the original M.

Kit Chan Chaotic Extension of an Operator on a Hilbert Subspace

Can We Have ker T = ker A?

Theorem (with Kadel, 2014) If dim (H/M) = ∞, and A in B(M) has a closed range, then A has a right invertible chaotic extension T in B(H) with ker A = ker T that satisfies the Criterion in the strongest sense. What does an extension look like? Write M = ran A ⊕ ran A⊥. Let M0, M1, M2, . . . be orthogonal subspaces of M⊥, each isomorphic to M. Identify M0 with the original M. Let M−1, M−2, . . . be orthogonal subspaces of M⊥, each isomorphic to ran A

Kit Chan Chaotic Extension of an Operator on a Hilbert Subspace

Can We Have ker T = ker A?

Theorem (with Kadel, 2014) If dim (H/M) = ∞, and A in B(M) has a closed range, then A has a right invertible chaotic extension T in B(H) with ker A = ker T that satisfies the Criterion in the strongest sense. What does an extension look like? Write M = ran A ⊕ ran A⊥. Let M0, M1, M2, . . . be orthogonal subspaces of M⊥, each isomorphic to M. Identify M0 with the original M. Let M−1, M−2, . . . be orthogonal subspaces of M⊥, each isomorphic to ran A so that H = · · · ⊕ M−2 ⊕ M−1 ⊕ M0 ⊕ M1 ⊕ M2 ⊕ · · · .

Kit Chan Chaotic Extension of an Operator on a Hilbert Subspace

Can We Have ker T = ker A?

Theorem (with Kadel, 2014) If dim (H/M) = ∞, and A in B(M) has a closed range, then A has a right invertible chaotic extension T in B(H) with ker A = ker T that satisfies the Criterion in the strongest sense. What does an extension look like? Write M = ran A ⊕ ran A⊥. Let M0, M1, M2, . . . be orthogonal subspaces of M⊥, each isomorphic to M. Identify M0 with the original M. Let M−1, M−2, . . . be orthogonal subspaces of M⊥, each isomorphic to ran A so that H = · · · ⊕ M−2 ⊕ M−1 ⊕ M0 ⊕ M1 ⊕ M2 ⊕ · · · . A vector h in H is written as h = (· · · , h−2, h−1,

• h0 , h1, h2, · · · ),

where each hj ∈ Mj, and indicates the zeroth position.

Kit Chan Chaotic Extension of an Operator on a Hilbert Subspace

Can We Have ker T = ker A?

Theorem (with Kadel, 2014) If dim (H/M) = ∞, and A in B(M) has a closed range, then A has a right invertible chaotic extension T in B(H) with ker A = ker T that satisfies the Criterion in the strongest sense. What does an extension look like? Write M = ran A ⊕ ran A⊥. Let M0, M1, M2, . . . be orthogonal subspaces of M⊥, each isomorphic to M. Identify M0 with the original M. Let M−1, M−2, . . . be orthogonal subspaces of M⊥, each isomorphic to ran A so that H = · · · ⊕ M−2 ⊕ M−1 ⊕ M0 ⊕ M1 ⊕ M2 ⊕ · · · . A vector h in H is written as h = (· · · , h−2, h−1,

• h0 , h1, h2, · · · ),

where each hj ∈ Mj, and indicates the zeroth position. Suppress the symbols of isomorphisms. Assume h−1, h−2 . . . are in ran A, and h0, h1, h2, . . . are in M = ran A ⊕ ran M⊥.

Kit Chan Chaotic Extension of an Operator on a Hilbert Subspace

An Extension with the Same Kernel

Since the restriction A|ker A⊥ : ker A⊥ → ran A is invertible, there is a bounded linear operator B : ran A → ker A⊥ such that AB = I

• n ran A.

Kit Chan Chaotic Extension of an Operator on a Hilbert Subspace

An Extension with the Same Kernel

Since the restriction A|ker A⊥ : ker A⊥ → ran A is invertible, there is a bounded linear operator B : ran A → ker A⊥ such that AB = I

• n ran A.

An chaotic extension T is given by Th =

• · · · , 1

αh−2, 1 αh−1, αh′

1,

• Ah0 + αh1, αh2, αh3, · · ·
• ,

where h′

1 is the orthogonal component of h1 in ran A.

Kit Chan Chaotic Extension of an Operator on a Hilbert Subspace

An Extension with the Same Kernel

Since the restriction A|ker A⊥ : ker A⊥ → ran A is invertible, there is a bounded linear operator B : ran A → ker A⊥ such that AB = I

• n ran A.

An chaotic extension T is given by Th =

• · · · , 1

αh−2, 1 αh−1, αh′

1,

• Ah0 + αh1, αh2, αh3, · · ·
• ,

where h′

1 is the orthogonal component of h1 in ran A.

Then a right inverse S of T is given by Sh =

• · · · , αh−3, αh−2,
• B(h′

0 − h−1), 1

α(h−1 ⊕ h′′

0), 1

αh1, 1 αh2, · · ·

• ,

where h′

0 and h′′ 0 are orthogonal components of h0 in ran A and

ran A⊥ respectively.

Kit Chan Chaotic Extension of an Operator on a Hilbert Subspace

Verifying kerT = kerA

Recall Th =

• · · · , 1

αh−2, 1 αh−1, αh′

1,

• Ah0 + αh1, αh2, αh3, · · ·
• .

Kit Chan Chaotic Extension of an Operator on a Hilbert Subspace

Verifying kerT = kerA

Recall Th =

• · · · , 1

αh−2, 1 αh−1, αh′

1,

• Ah0 + αh1, αh2, αh3, · · ·
• .

Clearly if h ∈ M0 and Ah = 0, then Th = 0.

Kit Chan Chaotic Extension of an Operator on a Hilbert Subspace

Verifying kerT = kerA

Recall Th =

• · · · , 1

αh−2, 1 αh−1, αh′

1,

• Ah0 + αh1, αh2, αh3, · · ·
• .

Clearly if h ∈ M0 and Ah = 0, then Th = 0. Conversely, if Th = 0 then clearly h−1, h−2, · · · = 0 = h2 = h3 = · · · , and also h′

1 = Ah0 + αh1 = 0.

Kit Chan Chaotic Extension of an Operator on a Hilbert Subspace

Verifying kerT = kerA

Recall Th =

• · · · , 1

αh−2, 1 αh−1, αh′

1,

• Ah0 + αh1, αh2, αh3, · · ·
• .

Clearly if h ∈ M0 and Ah = 0, then Th = 0. Conversely, if Th = 0 then clearly h−1, h−2, · · · = 0 = h2 = h3 = · · · , and also h′

1 = Ah0 + αh1 = 0.

Since h1 = h′

1 + h

′′

1, where h′ 1 ∈ ran A, and h

′′ ∈ ran A⊥, we have Kit Chan Chaotic Extension of an Operator on a Hilbert Subspace

Verifying kerT = kerA

Recall Th =

• · · · , 1

αh−2, 1 αh−1, αh′

1,

• Ah0 + αh1, αh2, αh3, · · ·
• .

Clearly if h ∈ M0 and Ah = 0, then Th = 0. Conversely, if Th = 0 then clearly h−1, h−2, · · · = 0 = h2 = h3 = · · · , and also h′

1 = Ah0 + αh1 = 0.

Since h1 = h′

1 + h

′′

1, where h′ 1 ∈ ran A, and h

′′ ∈ ran A⊥, we have

Ah0 + αh

′′

1 = 0. But, Ah0 ∈ ran A, and so

Ah0 = h

′′

1 = 0.

✷

Kit Chan Chaotic Extension of an Operator on a Hilbert Subspace

Corollaries on Invertibility

Suppose dim H/M = ∞.

Kit Chan Chaotic Extension of an Operator on a Hilbert Subspace

Corollaries on Invertibility

Suppose dim H/M = ∞. Corollary An operator A ∈ B(M) has an invertible chaotic extension T ∈ B(H) if and only if A is bounded below.

Kit Chan Chaotic Extension of an Operator on a Hilbert Subspace

Corollaries on Invertibility

Suppose dim H/M = ∞. Corollary An operator A ∈ B(M) has an invertible chaotic extension T ∈ B(H) if and only if A is bounded below. Corollary An operator A ∈ B(M) has a chaotic Fredholm extension T ∈ B(H) if and only if A is left semi-Fredholm. Moreover, ind T ≥ ind A.

Kit Chan Chaotic Extension of an Operator on a Hilbert Subspace

Dual Hypercyclic Operators

On a separable, infinite dimensional Hilbert space H, a bounded linear operator T : H → H is said to be dual hypercyclic, if both T and T ∗ are hypercyclic.

Kit Chan Chaotic Extension of an Operator on a Hilbert Subspace

Dual Hypercyclic Operators

On a separable, infinite dimensional Hilbert space H, a bounded linear operator T : H → H is said to be dual hypercyclic, if both T and T ∗ are hypercyclic. Herrero (1991): Does there exist a dual hypercyclic operator? Salas (1991): Yes.

Kit Chan Chaotic Extension of an Operator on a Hilbert Subspace

Dual Hypercyclic Operators

On a separable, infinite dimensional Hilbert space H, a bounded linear operator T : H → H is said to be dual hypercyclic, if both T and T ∗ are hypercyclic. Herrero (1991): Does there exist a dual hypercyclic operator? Salas (1991): Yes. Let M be closed subspace of H with dim H/M = ∞, and P : H → H be the orthogonal projection onto M. Theorem (2012) For any operator A ∈ B(M), there exists an operator T ∈ B(H) such that (1) T is dual hypercyclic, (2) PTP|M = A, (3) PT ∗P|M = A∗.

Kit Chan Chaotic Extension of an Operator on a Hilbert Subspace

Dual Hypercyclic Extension

Theorem (with Kadel, preprint, 2013) Suppose dim H/M = ∞. An operator A ∈ B(M) has a dual hypercyclic extension T ∈ B(H) if and only if A∗ is hypercyclic.

Kit Chan Chaotic Extension of an Operator on a Hilbert Subspace

Dual Hypercyclic Extension

Theorem (with Kadel, preprint, 2013) Suppose dim H/M = ∞. An operator A ∈ B(M) has a dual hypercyclic extension T ∈ B(H) if and only if A∗ is hypercyclic. Proof for “only if”. Suppose h is a hypercyclic vector of T ∗. Write h = f + g, where f ∈ M and g ∈ M⊥.

Kit Chan Chaotic Extension of an Operator on a Hilbert Subspace

Dual Hypercyclic Extension

Theorem (with Kadel, preprint, 2013) Suppose dim H/M = ∞. An operator A ∈ B(M) has a dual hypercyclic extension T ∈ B(H) if and only if A∗ is hypercyclic. Proof for “only if”. Suppose h is a hypercyclic vector of T ∗. Write h = f + g, where f ∈ M and g ∈ M⊥. Since TM ⊂ M, we have T ∗M⊥ ⊂ M⊥. Also A∗n = PT ∗n|M, where P : H → H is the orthogonal projection onto M.

Kit Chan Chaotic Extension of an Operator on a Hilbert Subspace

Dual Hypercyclic Extension

Theorem (with Kadel, preprint, 2013) Suppose dim H/M = ∞. An operator A ∈ B(M) has a dual hypercyclic extension T ∈ B(H) if and only if A∗ is hypercyclic. Proof for “only if”. Suppose h is a hypercyclic vector of T ∗. Write h = f + g, where f ∈ M and g ∈ M⊥. Since TM ⊂ M, we have T ∗M⊥ ⊂ M⊥. Also A∗n = PT ∗n|M, where P : H → H is the orthogonal projection onto M. Thus T ∗nh = T ∗nf + T ∗ng = A∗nf + gn, where gn ∈ M⊥.

Kit Chan Chaotic Extension of an Operator on a Hilbert Subspace

Dual Hypercyclic Extension

Theorem (with Kadel, preprint, 2013) Suppose dim H/M = ∞. An operator A ∈ B(M) has a dual hypercyclic extension T ∈ B(H) if and only if A∗ is hypercyclic. Proof for “only if”. Suppose h is a hypercyclic vector of T ∗. Write h = f + g, where f ∈ M and g ∈ M⊥. Since TM ⊂ M, we have T ∗M⊥ ⊂ M⊥. Also A∗n = PT ∗n|M, where P : H → H is the orthogonal projection onto M. Thus T ∗nh = T ∗nf + T ∗ng = A∗nf + gn, where gn ∈ M⊥. Hence f ∈ M is a hypercyclic vector for A∗. ✷

Kit Chan Chaotic Extension of an Operator on a Hilbert Subspace

The General Case A ∈ B(M, H)

We seen results for extensions of operators A ∈ B(M). What about operators A ∈ B(M, H)? Does there exist a chaotic operator T : H → H such that T|M = A?

Kit Chan Chaotic Extension of an Operator on a Hilbert Subspace

The General Case A ∈ B(M, H)

We seen results for extensions of operators A ∈ B(M). What about operators A ∈ B(M, H)? Does there exist a chaotic operator T : H → H such that T|M = A? Major Problem: The proofs for our above results do not generalize to the case when ran A ∩ M⊥ = {0}, particularly when ran A = H.

Kit Chan Chaotic Extension of an Operator on a Hilbert Subspace

The General Case A ∈ B(M, H)

We seen results for extensions of operators A ∈ B(M). What about operators A ∈ B(M, H)? Does there exist a chaotic operator T : H → H such that T|M = A? Major Problem: The proofs for our above results do not generalize to the case when ran A ∩ M⊥ = {0}, particularly when ran A = H. In fact, if T|M = A and P : H → H is the orthogonal projection

• nto M, then for any vector h ∈ M, the component (PA)nh in M

may be nonzero, and so is the component (I − P)(PA)nh in M⊥. These cause a lot of difficulties in finding a chaotic extension.

Kit Chan Chaotic Extension of an Operator on a Hilbert Subspace

The General Case A ∈ B(M, H)

We seen results for extensions of operators A ∈ B(M). What about operators A ∈ B(M, H)? Does there exist a chaotic operator T : H → H such that T|M = A? Major Problem: The proofs for our above results do not generalize to the case when ran A ∩ M⊥ = {0}, particularly when ran A = H. In fact, if T|M = A and P : H → H is the orthogonal projection

• nto M, then for any vector h ∈ M, the component (PA)nh in M

may be nonzero, and so is the component (I − P)(PA)nh in M⊥. These cause a lot of difficulties in finding a chaotic extension. Theorem (with Pinheiro, preprint, 2014) Suppose dim (H/M) = ∞. Every operator A ∈ B(M, H) has a chaotic extension T ∈ B(H) that satisfies the Criterion in the strongest possible sense.

Kit Chan Chaotic Extension of an Operator on a Hilbert Subspace

What about operators A : H → K?

Corollary Suppose H, K are two separable, infinite dimensional Hilbert spaces, and A ∈ B(H, K). Then There is a chaotic operator T ∈ B(H ⊕ K) such that T|H = A.

Kit Chan Chaotic Extension of an Operator on a Hilbert Subspace

What about operators A : H → K?

Corollary Suppose H, K are two separable, infinite dimensional Hilbert spaces, and A ∈ B(H, K). Then There is a chaotic operator T ∈ B(H ⊕ K) such that T|H = A. Corollary (extending a linear functional) Suppose dim (H/M) = ∞ and g ∈ H. There exists a chaotic

• perator T ∈ B(H) such that Tx = | < x, g > | for all x ∈ M.

Kit Chan Chaotic Extension of an Operator on a Hilbert Subspace

What about operators A : H → K?

Corollary Suppose H, K are two separable, infinite dimensional Hilbert spaces, and A ∈ B(H, K). Then There is a chaotic operator T ∈ B(H ⊕ K) such that T|H = A. Corollary (extending a linear functional) Suppose dim (H/M) = ∞ and g ∈ H. There exists a chaotic

• perator T ∈ B(H) such that Tx = | < x, g > | for all x ∈ M.
• Proof. Define A : M → H by Ax = < x, g > e, where e is a unit
• vector. ✷

Kit Chan Chaotic Extension of an Operator on a Hilbert Subspace

Simultaneous Extension

Suppose dim (H/M) = ∞. For any sequence of operators (An) in B(M, H), we can take the point of view that An : H → H with An = 0 on M⊥.

Kit Chan Chaotic Extension of an Operator on a Hilbert Subspace

Simultaneous Extension

Suppose dim (H/M) = ∞. For any sequence of operators (An) in B(M, H), we can take the point of view that An : H → H with An = 0 on M⊥. Question: Does there exist one single operator V : M⊥ → H such that each operator An + V : H → H is chaotic? Here we take the point of view that V = 0 on M.

Kit Chan Chaotic Extension of an Operator on a Hilbert Subspace

Simultaneous Extension

Suppose dim (H/M) = ∞. For any sequence of operators (An) in B(M, H), we can take the point of view that An : H → H with An = 0 on M⊥. Question: Does there exist one single operator V : M⊥ → H such that each operator An + V : H → H is chaotic? Here we take the point of view that V = 0 on M. Theorem (with Pinheiro, 2014) If (An) is uniformly bounded, then there is a bounded linear

• perator V : M⊥ → H such that each operator An + V is chaotic

and satisfies the Criterion in the strongest sense.

Kit Chan Chaotic Extension of an Operator on a Hilbert Subspace

Hypercyclic Subspaces

A hypercyclic subspace of a hypercyclic operator T in B(H) is a closed infinite dimensional subspace of H that consists entirely, except for the zero vector, of hypercyclic vectors of T.

Kit Chan Chaotic Extension of an Operator on a Hilbert Subspace

Hypercyclic Subspaces

A hypercyclic subspace of a hypercyclic operator T in B(H) is a closed infinite dimensional subspace of H that consists entirely, except for the zero vector, of hypercyclic vectors of T. Theorem (Montes-Rodr´ ıguez, 1996) T has a hypercyclic subspace if T satisfies the Hypercyclicity Criterion, and H has a closed infinite dimensional subspace K such that T nf → 0, for every vector f in K.

Kit Chan Chaotic Extension of an Operator on a Hilbert Subspace

Hypercyclic Subspaces

A hypercyclic subspace of a hypercyclic operator T in B(H) is a closed infinite dimensional subspace of H that consists entirely, except for the zero vector, of hypercyclic vectors of T. Theorem (Montes-Rodr´ ıguez, 1996) T has a hypercyclic subspace if T satisfies the Hypercyclicity Criterion, and H has a closed infinite dimensional subspace K such that T nf → 0, for every vector f in K. Corollary All chaotic extensions in the above can have a hypercyclic subspace.

Kit Chan Chaotic Extension of an Operator on a Hilbert Subspace

Hypercyclic Subspaces

A hypercyclic subspace of a hypercyclic operator T in B(H) is a closed infinite dimensional subspace of H that consists entirely, except for the zero vector, of hypercyclic vectors of T. Theorem (Montes-Rodr´ ıguez, 1996) T has a hypercyclic subspace if T satisfies the Hypercyclicity Criterion, and H has a closed infinite dimensional subspace K such that T nf → 0, for every vector f in K. Corollary All chaotic extensions in the above can have a hypercyclic subspace. Proof: Take a closed subspace N in M⊥ with dim H/(M ⊕ N) = ∞, and extend A to be zero on N. Apply the Theorems on A on M ⊕ N. ✷

Kit Chan Chaotic Extension of an Operator on a Hilbert Subspace

Universal Extension

Let X, Y be two separable topological spaces. A sequence of continuous linear operators Tn : X → Y is universal if there is a vector x such that {Tnx} is dense in H. The sequence is densely universal if it has a dense set of universal vectors.

Kit Chan Chaotic Extension of an Operator on a Hilbert Subspace

Universal Extension

Let X, Y be two separable topological spaces. A sequence of continuous linear operators Tn : X → Y is universal if there is a vector x such that {Tnx} is dense in H. The sequence is densely universal if it has a dense set of universal vectors. Theorem (with Pinheiro, 2014) Suppose M is a closed complemented subspace of X with dim X/M = ∞. If An : M → Y is a sequence of continuous linear

• perators, then there is a densely universal sequence (Tn) in

B(X, Y ) so that Tn|M = An.

Kit Chan Chaotic Extension of an Operator on a Hilbert Subspace

One Last Proof

• Proof. Let X0 = {x1, x2, . . .} be a linearly independent dense

subset of X with X0 ∩ M = ∅. Let Y0 = {y1, y2, . . .} be a dense subset of Y .

Kit Chan Chaotic Extension of an Operator on a Hilbert Subspace

One Last Proof

• Proof. Let X0 = {x1, x2, . . .} be a linearly independent dense

subset of X with X0 ∩ M = ∅. Let Y0 = {y1, y2, . . .} be a dense subset of Y . Let X = M ⊕ N where N is a closed subspace, and write each xi = ui + vi, where ui ∈ M and vi ∈ N. Let Ek = span{v1, . . . , vk}. Thus Ek is complemented in N.

Kit Chan Chaotic Extension of an Operator on a Hilbert Subspace

One Last Proof

• Proof. Let X0 = {x1, x2, . . .} be a linearly independent dense

subset of X with X0 ∩ M = ∅. Let Y0 = {y1, y2, . . .} be a dense subset of Y . Let X = M ⊕ N where N is a closed subspace, and write each xi = ui + vi, where ui ∈ M and vi ∈ N. Let Ek = span{v1, . . . , vk}. Thus Ek is complemented in N. Write X = M ⊕ Ek ⊕ Nk, for a closed subspace Nk.

Kit Chan Chaotic Extension of an Operator on a Hilbert Subspace

One Last Proof

• Proof. Let X0 = {x1, x2, . . .} be a linearly independent dense

subset of X with X0 ∩ M = ∅. Let Y0 = {y1, y2, . . .} be a dense subset of Y . Let X = M ⊕ N where N is a closed subspace, and write each xi = ui + vi, where ui ∈ M and vi ∈ N. Let Ek = span{v1, . . . , vk}. Thus Ek is complemented in N. Write X = M ⊕ Ek ⊕ Nk, for a closed subspace Nk. Define Lk : Ek ⊕ Nk → Y by taking Lkvi = yk − Akui, for all i with 1 ≤ i ≤ k, and Lk = 0 on Nk.

Kit Chan Chaotic Extension of an Operator on a Hilbert Subspace

One Last Proof

• Proof. Let X0 = {x1, x2, . . .} be a linearly independent dense

subset of X with X0 ∩ M = ∅. Let Y0 = {y1, y2, . . .} be a dense subset of Y . Let X = M ⊕ N where N is a closed subspace, and write each xi = ui + vi, where ui ∈ M and vi ∈ N. Let Ek = span{v1, . . . , vk}. Thus Ek is complemented in N. Write X = M ⊕ Ek ⊕ Nk, for a closed subspace Nk. Define Lk : Ek ⊕ Nk → Y by taking Lkvi = yk − Akui, for all i with 1 ≤ i ≤ k, and Lk = 0 on Nk. Define Tk : M ⊕ Ek ⊕ Nk → Y by Tk = Ak ⊕ Lk.

Kit Chan Chaotic Extension of an Operator on a Hilbert Subspace

One Last Proof

• Proof. Let X0 = {x1, x2, . . .} be a linearly independent dense

subset of X with X0 ∩ M = ∅. Let Y0 = {y1, y2, . . .} be a dense subset of Y . Let X = M ⊕ N where N is a closed subspace, and write each xi = ui + vi, where ui ∈ M and vi ∈ N. Let Ek = span{v1, . . . , vk}. Thus Ek is complemented in N. Write X = M ⊕ Ek ⊕ Nk, for a closed subspace Nk. Define Lk : Ek ⊕ Nk → Y by taking Lkvi = yk − Akui, for all i with 1 ≤ i ≤ k, and Lk = 0 on Nk. Define Tk : M ⊕ Ek ⊕ Nk → Y by Tk = Ak ⊕ Lk. Thus, Tkxi = Tk(ui + vi) = Akui + Lkvi = Akui + (yk − Akui) = yk. Hence each xi is a universal vector for (Tk).

Kit Chan Chaotic Extension of an Operator on a Hilbert Subspace