Hypercyclic N-linear operators Preliminary work, with Alberto Conejero (Univ. Polit´ ecnica de Valencia) Juan B` es Bowling Green State University Hypercyclic N-linear operators
Greetings from Alberto Conejero Hypercyclic N-linear operators
Theorem (N. Bernardes, 1998) No homogeneous polynomial of degree d ≥ 2 on a Banach space can be hypercyclic. Recall that a homogenous polynomial P of degree 2 is given by P ( x ) = L ( x , x ) , x ∈ X , where L : X × X → X is a continuous bilinear map. And a polynomial P is said to be hypercyclic if it has a dense orbit { P n ( x ) : n ≥ 0 } . Hypercyclic N-linear operators
If we drop the homogeneity requirement, F. Mart´ ınez-Gim´ enez, A. Peris (2009): Every separable, infinite dimensional Fr´ echet space supports non-homogeneous mixing polynomials of any positive degree. N. Bernardes, A. Peris (2014): If the Fr´ echet space has an unconditional basis, then it has non-homogeneous polynomials that are frequently hypercyclic and chaotic. Hypercyclic N-linear operators
We are interested in the new direction started by K.-G. Grosse-Erdmann and S. G. Kim (2013): Instead of looking at the homogeneous polynomial P , they considered the underlying bilinear mapping ( x , y ) �→ L ( x , y ) Question: Can a bilinear mapping be ’hypercyclic’, that is, have a dense ’orbit’? What is the ’orbit’ of a bilinear mapping? Hypercyclic N-linear operators
We are interested in the new direction started by K.-G. Grosse-Erdmann and S. G. Kim (2013): Instead of looking at the homogeneous polynomial P , they considered the underlying bilinear mapping ( x , y ) �→ L ( x , y ) Question: Can a bilinear mapping be ’hypercyclic’, that is, have a dense ’orbit’? What is the ’orbit’ of a bilinear mapping? Hypercyclic N-linear operators
For a linear map T : X → X and x ∈ X , Orb( x , T ) = { x , Tx , T 2 x , . . . } = { x } ∪ { x , Tx } ∪ { x , Tx , T 2 x } ∪ . . . Definition. (K.-G. Grosse-Erdmann, S. G. Kim, ’13) For a bilinear map L : X × X → X and ( x , y ) ∈ X × X , let Orb(( x , y ) , L ) = { x , y } ∪ { x , y , L ( x , x ) , L ( x , y ) , L ( y , x ) , L ( y , y ) } ∪ . . . = ∪ ∞ n =0 U n , where U 0 = { x , y } and U n := U n − 1 ∪ { L ( u , v ) : u , v ∈ U n − 1 } , n ≥ 1. L is bihypercyclic provided Orb(( x , y ) , L ) = X for some ( x , y ) ∈ X × X . Such ( x , y ) is called a bihypercyclic pair for L . Hypercyclic N-linear operators
Example. Is L : C × C → C , L ( x , y ) = xy , bihypercyclic? For x , y ∈ C Orb(( x , y ) , L ) = { x , y , x 2 , xy , y 2 , x 3 , x 2 y , xy 2 , y 3 , . . . , y 4 , . . . } = { x k y m } k , m ≥ 1 So the hypercyclicy follows by (N. Feldman, 2008) For each n ≥ 1, there are complex numbers a 1 , . . . , a n , b 1 , . . . , b n such that the set of vectors a k 1 b k 1 1 a k 2 b k 2 2 is dense in C n . . . . a k n b k n n k , k 1 , k 2 ,..., k n ≥ 1 In general, Grosse-Erdmann & Kim, 2013: Each K n has bihypercyclic operators. Hypercyclic N-linear operators
Example. Is L : C × C → C , L ( x , y ) = xy , bihypercyclic? For x , y ∈ C Orb(( x , y ) , L ) = { x , y , x 2 , xy , y 2 , x 3 , x 2 y , xy 2 , y 3 , . . . , y 4 , . . . } = { x k y m } k , m ≥ 1 So the hypercyclicy follows by (N. Feldman, 2008) For each n ≥ 1, there are complex numbers a 1 , . . . , a n , b 1 , . . . , b n such that the set of vectors a k 1 b k 1 1 a k 2 b k 2 2 is dense in C n . . . . a k n b k n n k , k 1 , k 2 ,..., k n ≥ 1 In general, Grosse-Erdmann & Kim, 2013: Each K n has bihypercyclic operators. Hypercyclic N-linear operators
Example. Is L : C × C → C , L ( x , y ) = xy , bihypercyclic? For x , y ∈ C Orb(( x , y ) , L ) = { x , y , x 2 , xy , y 2 , x 3 , x 2 y , xy 2 , y 3 , . . . , y 4 , . . . } = { x k y m } k , m ≥ 1 So the hypercyclicy follows by (N. Feldman, 2008) For each n ≥ 1, there are complex numbers a 1 , . . . , a n , b 1 , . . . , b n such that the set of vectors a k 1 b k 1 1 a k 2 b k 2 2 is dense in C n . . . . a k n b k n n k , k 1 , k 2 ,..., k n ≥ 1 In general, Grosse-Erdmann & Kim, 2013: Each K n has bihypercyclic operators. Hypercyclic N-linear operators
Example. Is L : C × C → C , L ( x , y ) = xy , bihypercyclic? For x , y ∈ C Orb(( x , y ) , L ) = { x , y , x 2 , xy , y 2 , x 3 , x 2 y , xy 2 , y 3 , . . . , y 4 , . . . } = { x k y m } k , m ≥ 1 So the hypercyclicy follows by (N. Feldman, 2008) For each n ≥ 1, there are complex numbers a 1 , . . . , a n , b 1 , . . . , b n such that the set of vectors a k 1 b k 1 1 a k 2 b k 2 2 is dense in C n . . . . a k n b k n n k , k 1 , k 2 ,..., k n ≥ 1 In general, Grosse-Erdmann & Kim, 2013: Each K n has bihypercyclic operators. Hypercyclic N-linear operators
Example. Is L : C × C → C , L ( x , y ) = xy , bihypercyclic? For x , y ∈ C Orb(( x , y ) , L ) = { x , y , x 2 , xy , y 2 , x 3 , x 2 y , xy 2 , y 3 , . . . , y 4 , . . . } = { x k y m } k , m ≥ 1 So the hypercyclicy follows by (N. Feldman, 2008) For each n ≥ 1, there are complex numbers a 1 , . . . , a n , b 1 , . . . , b n such that the set of vectors a k 1 b k 1 1 a k 2 b k 2 2 is dense in C n . . . . a k n b k n n k , k 1 , k 2 ,..., k n ≥ 1 In general, Grosse-Erdmann & Kim, 2013: Each K n has bihypercyclic operators. Hypercyclic N-linear operators
Example. Is L : C × C → C , L ( x , y ) = xy , bihypercyclic? For x , y ∈ C Orb(( x , y ) , L ) = { x , y , x 2 , xy , y 2 , x 3 , x 2 y , xy 2 , y 3 , . . . , y 4 , . . . } = { x k y m } k , m ≥ 1 So the hypercyclicy follows by (N. Feldman, 2008) For each n ≥ 1, there are complex numbers a 1 , . . . , a n , b 1 , . . . , b n such that the set of vectors a k 1 b k 1 1 a k 2 b k 2 2 is dense in C n . . . . a k n b k n n k , k 1 , k 2 ,..., k n ≥ 1 In general, Grosse-Erdmann & Kim, 2013: Each K n has bihypercyclic operators. Hypercyclic N-linear operators
Example. Is L : C × C → C , L ( x , y ) = xy , bihypercyclic? For x , y ∈ C Orb(( x , y ) , L ) = { x , y , x 2 , xy , y 2 , x 3 , x 2 y , xy 2 , y 3 , . . . , y 4 , . . . } = { x k y m } k , m ≥ 1 So the hypercyclicy follows by (N. Feldman, 2008) For each n ≥ 1, there are complex numbers a 1 , . . . , a n , b 1 , . . . , b n such that the set of vectors a k 1 b k 1 1 a k 2 b k 2 2 is dense in C n . . . . a k n b k n n k , k 1 , k 2 ,..., k n ≥ 1 In general, Grosse-Erdmann & Kim, 2013: Each K n has bihypercyclic operators. Hypercyclic N-linear operators
Example. Is L : C × C → C , L ( x , y ) = xy , bihypercyclic? For x , y ∈ C Orb(( x , y ) , L ) = { x , y , x 2 , xy , y 2 , x 3 , x 2 y , xy 2 , y 3 , . . . , y 4 , . . . } = { x k y m } k , m ≥ 1 So the hypercyclicy follows by (N. Feldman, 2008) For each n ≥ 1, there are complex numbers a 1 , . . . , a n , b 1 , . . . , b n such that the set of vectors a k 1 b k 1 1 a k 2 b k 2 2 is dense in C n . . . . a k n b k n n k , k 1 , k 2 ,..., k n ≥ 1 In general, Grosse-Erdmann & Kim, 2013: Each K n has bihypercyclic operators. Hypercyclic N-linear operators
Example. Is L : C × C → C , L ( x , y ) = xy , bihypercyclic? For x , y ∈ C Orb(( x , y ) , L ) = { x , y , x 2 , xy , y 2 , x 3 , x 2 y , xy 2 , y 3 , . . . , y 4 , . . . } = { x k y m } k , m ≥ 1 So the hypercyclicy follows by (N. Feldman, 2008) For each n ≥ 1, there are complex numbers a 1 , . . . , a n , b 1 , . . . , b n such that the set of vectors a k 1 b k 1 1 a k 2 b k 2 2 is dense in C n . . . . a k n b k n n k , k 1 , k 2 ,..., k n ≥ 1 In general, Grosse-Erdmann & Kim, 2013: Each K n has bihypercyclic operators. Hypercyclic N-linear operators
Example (Grosse-Erdmann and Kim) Suppose there exists x ∈ X so that T := L ( x , · ) : X → X is hypercyclic, with a hypercyclic vector y . Then ( x , y ) is a bihypercyclic pair for L . In fact, Ty = L ( x , y ) , T 2 y = L ( x , Ty ) = L ( x , L ( x , y )) , T 3 y = L ( x , T 2 y ) = L ( x , L ( x , L ( x , y ))) , . . . So Orb( T , y ) ⊂ Orb(( x , y ) , L ) � Theorem (Grosse-Erdmann and Kim) Every separable Banach space suports a bihypercyclic bilinear operator. If dim( X ) = ∞ , get T a hypercyclic operator on X , and any 0 � = x ∗ ∈ X ∗ . Then L := x ∗ ⊗ T , ( x , y ) �→ x ∗ ( x ) T ( y ) is bihypercyclic. Hypercyclic N-linear operators
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