Chaotic Extension of an Operator on 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 2 x , T 3 x , . . . } 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 2 x , T 3 x , . . . } 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 n x = 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 2 x , T 3 x , . . . } 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 n x = 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 α I + 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 α I + K, where α ∈ C and K compact. Take X = C ⊕ N , and A = 2 I : C → C . Suppose A has a hypercyclic extension T , which must take the form � 2 I � ⋆ 0 α I + 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 α I + K, where α ∈ C and K compact. Take X = C ⊕ N , and A = 2 I : C → C . Suppose A has a hypercyclic extension T , which must take the form � 2 I � ⋆ 0 α I + 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 α I + K, where α ∈ C and K compact. Take X = C ⊕ N , and A = 2 I : C → C . Suppose A has a hypercyclic extension T , which must take the form � 2 I � ⋆ 0 α I + 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 operator 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 operator 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 operator 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 2 h , . . . } = { [ h ] , [ Th ] , [ T 2 h ] , . . . } 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 operator 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 2 h , . . . } = { [ h ] , [ Th ] , [ T 2 h ] , . . . } 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 S n → 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 S n → 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 S n → 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 M 0 and write H = � ∞ j =0 M j , where each M j 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 M 0 and write H = � ∞ j =0 M j , where each M j is isomorphic to M . Let α > max { 1 , � A �} . Define T : H → H by suppressing symbols for isomorphisms: T ( h 0 , h 1 , h 2 , . . . ) = ( Ah 0 + α h 1 , α h 2 , α h 3 , . . . ) . 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 M 0 and write H = � ∞ j =0 M j , where each M j is isomorphic to M . Let α > max { 1 , � A �} . Define T : H → H by suppressing symbols for isomorphisms: T ( h 0 , h 1 , h 2 , . . . ) = ( Ah 0 + α h 1 , α h 2 , α h 3 , . . . ) . Thus if x = ( d 0 , d 1 , . . . , d k , 0 , 0 , 0 , . . . ) and if n ≥ k then T n x = ( A n − k ( A k d 0 + α A k − 1 d 1 + · · · + α k d k ) , 0 , 0 , 0 , . . . ) . Let S ( h 0 , h 1 , h 2 , . . . ) = 1 α (0 , h 0 , h 1 , h 2 , . . . ) . So TS = I . Kit Chan Chaotic Extension of an Operator on a Hilbert Subspace
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