On linear chaos in function spaces

John M. Jimenez; Marat V. Markin

doi:10.1515/dema-2022-0008

Article Open Access

On linear chaos in function spaces

John M. Jimenez and Marat V. Markin

Published/Copyright: May 5, 2022

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From the journal Demonstratio Mathematica Volume 55 Issue 1

Abstract

We show that, in L p ( 0 , ∞ ) ( 1 ≤ p < ∞ ), bounded weighted translations as well as their unbounded counterparts are chaotic linear operators. We also extend the unbounded case to C 0 [ 0 , ∞ ) and describe the spectra of the weighted translations provided the underlying spaces are complex.

Keywords: hypercyclic vector; periodic point; hypercyclic operator; chaotic operator; spectrum

MSC 2010: 47A16; 47B37; 47B38; 47A10

1 Introduction

Extending the classical Rolewicz’s result [1] and the results of [2] for the sequence spaces l p ( 1 ≤ p < ∞ ), we show that, in the space L p ( 0 , ∞ ) ( 1 ≤ p < ∞ ), the bounded weighted left translations

( T w , a x ) ( t ) = w x ( t + a ) ( ∣ w ∣ > 1 , a > 0 )

as well as their unbounded counterparts

( T w , a x ) ( t ) = w t x ( t + a ) ( w > 1 , a > 0 )

are chaotic linear operators (the latter forecasted in [2, Remark 3.1]).

The chaoticity of the bounded weighted left translations in C 0 [ 0 , ∞ ) established in [3], we stretch the unbounded case from the sequence space c 0 [2] to the space C 0 [ 0 , ∞ ) of real- or complex-valued functions continuous on [ 0 , ∞ ) and vanishing at infinity, which is Banach relative to the norm

C 0 [ 0 , ∞ ) ∋ x ↦ ‖ x ‖ ∞ ≔ sup t ≥ 0 ∣ x ( t ) ∣

(also forecasted in [2, Remark 3.1]) and describe the spectra of the weighted translations provided the underlying spaces are complex.

2 Preliminaries

2.1 Hypercyclicity and chaoticity

For a (bounded or unbounded) linear operator T in a (real or complex) Banach space X , a nonzero vector

x ∈ C ∞ ( T ) ≔ ⋂ n = 0 ∞ D ( T n )

( D ( ⋅ ) is the domain of an operator, T 0 ≔ I , I is the identity operator on X ) is called hypercyclic if its orbit under T

orb ( x , T ) ≔ { T n x } n ∈ Z +

( Z + ≔ { 0 , 1 , 2 , … } is the set of nonnegative integers) is dense in X .

Linear operators possessing hypercyclic vectors are said to be hypercyclic.

If there exist an N ∈ N ( N ≔ { 1 , 2 , … } is the set of natural numbers) and a vector

x ∈ D ( T N ) with T N x = x ,

such a vector is called a periodic point for the operator T of period N . If x ≠ 0 , we say that N is a period for T .

Hypercyclic linear operators with a dense in X set Per ( A ) of periodic points are said to be chaotic.

See [4,5,6].

Remarks 2.1

In the definition of hypercyclicity, the underlying space is necessarily infinite-dimensional and separable (see, e.g., [7]).
For a hypercyclic linear operator T , the set H C ( T ) of all its hypercyclic vectors is necessarily dense in X , and hence, the more so, is the subspace C ∞ ( T ) ⊇ H C ( T ) .
Observe that
Per ( A ) = ⋃ N = 1 ∞ Per N ( A ) ,
where
Per N ( A ) = ker ( A N − I ) , N ∈ N
is the subspace of N -periodic points of A .

Before [6,8], the notions of linear hypercyclicity and chaoticity had been studied exclusively for continuous linear operators on Fréchet spaces, in particular for bounded linear operators on Banach spaces (for a comprehensive survey, see [7,9]).

In [1], S. Rolewicz provides the first example of hypercyclic bounded linear operators on Banach spaces (see also [7]), which on the (real or complex) sequence space, l p ( 1 ≤ p < ∞ ) of p -summable sequences or c 0 of vanishing sequences, the latter equipped with the supremum norm

c 0 ∋ x ≔ ( x k ) k ∈ N ↦ ‖ x ‖ ∞ ≔ sup k ∈ N ∣ x k ∣ ,

are the weighted backward shifts

T w ( x k ) k ∈ N ≔ w ( x k + 1 ) k ∈ N

with w ∈ F ( F ≔ R or F ≔ C ) such that ∣ w ∣ > 1 . Furthermore, Rolewicz’s shifts are established to be chaotic [5].

In [2] (see also [10]), it is shown that the weighted backward shifts

T w x ≔ ( w k x k + 1 ) k ∈ N

with w ∈ F such that ∣ w ∣ > 1 and maximal domain in the (real or complex) sequence spaces l p ( 1 ≤ p < ∞ ) and c 0 are chaotic unbounded linear operators and, provided the underlying space is complex, each λ ∈ C is a simple eigenvalue for T w .

When establishing hypercyclicity, we obviate explicit construction of hypercyclic vectors by applying the subsequent version of the classical Birkhoff transitivity theorem [7, Theorem 1.16] or the following Sufficient condition for hypercyclicity [6, Theorem 2.1], which is an extension of Kitai’s ctriterion [11,12].

Theorem 2.1

(Birkhoff transitivity theorem) A bounded linear operator T on a (real or complex) infinite-dimensional separable Banach space X is hypercyclic iff it is topologically transitive, i.e., for any nonempty open subsets U and V of X , there exists an n ∈ Z + such that

T n ( U ) ∩ V ≠ ∅ .

Cf. [7, Theorem 2.19].

Theorem 2.2

(Sufficient condition for hypercyclicity) Let X be a (real or complex) infinite-dimensional separable Banach space and T be a densely defined linear operator in X such that each power T n ( n ∈ N ) is a closed operator. If there exists a set

Y ⊆ C ∞ ( T ) ≔ ⋂ n = 1 ∞ D ( T n )

dense in X and a mapping S : Y → Y such that

∀ x ∈ Y : T S x = x and
∀ x ∈ Y : T n x , S n x → 0 , n → ∞ ,

then the operator T is hypercyclic.

2.2 Resolvent set and spectrum

For a linear operator T in a complex Banach space X , the set

ρ ( A ) ≔ { λ ∈ C ∣ ∃ ( T − λ I ) − 1 ∈ L ( X ) } .

( L ( X ) is the space of bounded linear operators on X ) and its complement σ ( T ) ≔ C ⧹ ρ ( T ) are called the resolvent set and the spectrum of T , respectively.

The spectrum σ ( T ) of a closed linear operator T in a complex Banach space X is the union of the following pairwise disjoint sets:

σ p ( T ) ≔ { λ ∈ C ∣ T − λ I is not injective , i.e. , λ is an eigen value of T } , σ c ( T ) ≔ { λ ∈ C ∣ T − λ I is injective , not surjective , and R ( T − λ I ) ¯ = X } , σ r ( T ) ≔ { λ ∈ C ∣ T − λ I is injective and R ( T − λ I ) ¯ ≠ X } .

( R ( ⋅ ) is the range of an operator, and • ¯ is the closure of a set), called the point, continuous and residual spectrum of T , respectively (see, e.g., [13,14]).

3 Bounded weighted translations on L p ( 0 , ∞ )

Theorem 3.1

(Bounded weighted translations on L p ( 0 , ∞ ) ) On the (real or complex) space L p ( 0 , ∞ ) ( 1 ≤ p < ∞ ), the weighted left translation

( T w , a x ) ( t ) ≔ w x ( t + a ) , x ∈ L p ( 0 , ∞ ) , t ≥ 0 ,

with w ∈ F such that ∣ w ∣ > 1 and a > 0 is a chaotic bounded linear operator.

Furthermore, provided the underlying space is complex,

(3.1) σ ( T w , a ) = { λ ∈ C ∣ ∣ λ ∣ ≤ ∣ w ∣ }

with

(3.2) σ p ( T w , a ) = { λ ∈ C ∣ ∣ λ ∣ < ∣ w ∣ } and σ c ( T w , a ) = { λ ∈ C ∣ ∣ λ ∣ = ∣ w ∣ } .

Proof

Let 1 ≤ p < ∞ , w ∈ F such that ∣ w ∣ > 1 , and a > 0 be arbitrary and, for the simplicity of notation, let T ≔ T w , a .

The linearity of T is obvious. Its boundedness immediately follows from the fact that

T = w B ,

where

( B x ) ( t ) ≔ x ( t + a ) , x ∈ L p ( 0 , ∞ ) , t ≥ 0 ,

is a left translation operator with ‖ B ‖ = 1 , and hence,

(3.3) ‖ T ‖ = ∣ w ∣ ‖ B ‖ = ∣ w ∣

(here and wherever appropriate, ‖ ⋅ ‖ also stands for the operator norm).

Suppose that

U , V ⊆ L p ( 0 , ∞ )

are arbitrary nonempty open sets.

By the denseness in L p ( 0 , ∞ ) of the equivalence classes represented by p -integrable on ( 0 , ∞ ) eventually zero functions (see, e.g., [15]), there exist equivalence classes

x ∈ U and y ∈ V

represented by such functions x ( ⋅ ) and y ( ⋅ ) , respectively. Since the representative functions are eventually zero,

∃ N ∈ N ∀ t > N a : x ( t ) = 0 and y ( t ) = 0 .

For an arbitrary n ≥ N , the p -integrable on ( 0 , ∞ ) eventually zero function

z n ( t ) ≔ x ( t ) , t ∈ [ 0 , N a ) , w − n y ( t − a n ) , t ∈ [ n a , N a + n a ) , 0 , otherwise

represents an equivalence class z n ∈ L p ( 0 , ∞ ) .

Observe that, for all n ≥ N ,

( T n z n ) ( t ) = y ( t ) , t ≥ 0 ,

and

‖ z n − x ‖ p = ∣ w ∣ − n ‖ y ‖ p → 0 , n → ∞ .

Hence, for all sufficiently large n ∈ N ,

z n ∈ U and T n z n = y ∈ V

(see Figure 1).

By the Birkhoff transitivity theorem (Theorem 2.1), we infer that the operator T is hypercyclic.

To prove that T has a dense set of periodic points, let us first show that each N ∈ N is a period for T .

For an arbitrary N ∈ N , let

x ∈ ker T N ⧹ { 0 } ,

where

ker T N = { f ∈ L p ( 0 , ∞ ) ∣ f ( t ) = 0 , t > N a } .

Figure 1

Topological transitivity.

Then, the p -integrable on ( 0 , ∞ ) function

(3.4) x N ( t ) ≔ w − k N x ( t − k N a ) , t ∈ D k ≔ [ k N a , ( k + 1 ) N a ) , k ∈ Z + ,

represents an N -periodic point x N of T .

Indeed, in view of ∣ w ∣ > 1 ,

∫ 0 ∞ ∣ x N ( t ) ∣ p d t = ∑ k = 0 ∞ ∫ D k ∣ w − k N x ( t − k N a ) ∣ p d t = ∑ k = 0 ∞ ( ∣ w ∣ − p N ) k ∫ 0 N a ∣ x ( t ) ∣ p d t = ∑ k = 0 ∞ ( ∣ w ∣ − p N ) k ‖ x ‖ p p = 1 1 − ∣ w ∣ − p N ‖ x ‖ p p < ∞ ,

and hence, x N ∈ L p ( 0 , ∞ ) .

Furthermore, since

( T N x N ) ( t ) = w N x N ( t + N a ) = w N w − k N x ( t + N a − k N a ) = w − ( k − 1 ) N y ( t − ( k − 1 ) N a ) , t ∈ D k − 1 , k ∈ N ,

we infer that

T N x N = x N .

Suppose that x ∈ L p ( 0 , ∞ ) is an arbitrary equivalence class represented by a p -integrable on ( 0 , ∞ ) eventually zero function x ( ⋅ ) . Then,

∃ M ∈ N : x ( t ) = 0 , t > M a .

Let x N be the periodic point of the operator T of an arbitrary period N ≥ M defined based on x by (3.4). Then,

‖ x N − x ‖ p p = ∑ k = 0 ∞ ∫ D k ∣ x N ( t ) − x ( t ) ∣ p d t = ∑ k = 1 ∞ ∫ D k ∣ w − k N x ( t − k N a ) ∣ p d t = ∑ k = 1 ∞ ( ∣ w ∣ − p N ) k ∫ 0 N a ∣ x ( t ) ∣ p d t = ∑ k = 1 ∞ ( ∣ w ∣ − p N ) k ‖ x ‖ p p = ∣ w ∣ − p N 1 − ∣ w ∣ − p N ‖ x ‖ p p → 0 , N → ∞ .

By the denseness in L p ( 0 , ∞ ) ( 1 ≤ p < ∞ ) of the subspace

(3.5) Y ≔ ⋃ n = 1 ∞ ker T n ,

where

(3.6) ker T n = { f ∈ L p ( 0 , ∞ ) ∣ f ( t ) = 0 , t > n a } , n ∈ N ,

of the equivalence classes represented by p -integrable on ( 0 , ∞ ) eventually zero functions, we infer that the set Per ( T ) of periodic points of T is dense in L p ( 0 , ∞ ) as well, and hence, the operator T is chaotic.

Now, assuming that the space L p ( 0 , ∞ ) is complex, let us prove (3.1) and (3.2).

In view of (3.3), by Gelfand’s spectral radius theorem [14],

(3.7) σ ( T ) ⊆ { λ ∈ C ∣ ∣ λ ∣ ≤ ∣ w ∣ } .

For an arbitrary λ ∈ C with ∣ λ ∣ < ∣ w ∣ , let

x ∈ ker T ⧹ { 0 } ⊆ Y ⧹ { 0 } ,

where

ker T = { f ∈ L p ( 0 , ∞ ) ∣ f ( t ) = 0 , t > a }

(see (3.6)).

Then, the p -integrable on ( 0 , ∞ ) function

(3.8) x λ ( t ) ≔ λ w k x ( t − k a ) , t ∈ [ k a , ( k + 1 ) a ) , k ∈ Z + , ( 0 0 ≔ 1 )

is an eigenvector for T associated with λ .

Indeed, in view of ∣ λ ∣ < ∣ w ∣ ,

0 < ‖ x λ ‖ p p = ∫ 0 ∞ ∣ x λ ( t ) ∣ p d t = ∑ k = 0 ∞ ∫ k a ( k + 1 ) a λ w k x ( t − k a ) p d t = ∑ k = 0 ∞ λ w k p ∫ k a ( k + 1 ) a ∣ x ( t − k a ) ∣ p d t = ∑ k = 0 ∞ λ w p k ∫ 0 a ∣ x ( t ) ∣ p d t = ∑ k = 0 ∞ λ w p k ‖ x ‖ p p < ∞ ,

and hence, x λ ∈ L p ( 0 , ∞ ) ⧹ { 0 } .

Furthermore,

( T x λ ) ( t ) = w x λ ( t + a ) = w λ w k x ( t + a − k a ) = λ λ w k − 1 x ( t − ( k − 1 ) a ) , t ∈ [ ( k − 1 ) a , k a ) , k ∈ N ,

which implies that

(3.9) T x λ = λ x λ ,

and hence, λ ∈ σ p ( T ) .

Conversely, let λ ∈ σ p ( T ) be an arbitrary eigenvalue for T with an associated eigenvector x λ ∈ L p ( 0 , ∞ ) ⧹ { 0 } . Then, for

x k ( t ) ≔ x λ ( t ) , t ∈ [ k a , ( k + 1 ) a ) , k ∈ Z + ,

by (3.9), we have:

λ x k − 1 ( t ) = w x k ( t + a ) , on [ k a , ( k + 1 ) a ) ( mod λ 1 ) , k ∈ N ,

( λ 1 is the Lebesgue measure on R ).

Whence,

x k ( t ) = λ w k x λ ( t − k a ) , on [ k a , ( k + 1 ) a ) ( mod λ 1 ) , k ∈ Z + ,

which, in view of x λ ≠ 0 , implies that

0 < ∫ 0 a ∣ x λ ( t ) ∣ p d t ≤ ∫ 0 ∞ ∣ x λ ( t ) ∣ p d t = ‖ x λ ‖ p p < ∞

and

∞ > ‖ x λ ‖ p p = ∫ 0 ∞ ∣ x λ ( t ) ∣ p d t = ∑ k = 0 ∞ ∫ k a ( k + 1 ) a λ w k x λ ( t − k a ) p d t = ∑ k = 0 ∞ λ w k p ∫ k a ( k + 1 ) a ∣ x λ ( t − k a ) ∣ p d t = ∑ k = 0 ∞ λ w p k ∫ 0 a ∣ x λ ( t ) ∣ p d t .

The convergence of the latter series implies that

λ w p k → 0 , k → ∞ ,

which, in its turn, means that

∣ λ ∣ < ∣ w ∣ .

Thus, x λ can be represented by a p -integrable on ( 0 , ∞ ) function x λ ( ⋅ ) of the form given by (3.8), where the corresponding x ∈ ker T ⧹ { 0 } is represented by

x ( t ) ≔ χ [ 0 , a ] ( t ) x λ ( t ) , t ≥ 0

( χ δ ( ⋅ ) is the characteristic function of a set δ ).

The aforementioned explanation proves that

(3.10) σ p ( T ) = { λ ∈ C ∣ ∣ λ ∣ < ∣ w ∣ } .

Considering that σ ( T ) is a closed set in C (see, e.g., [13,14]), we infer from (3.7) and (3.10) that (3.1) holds.

Since, by [7, Lemma 2.53], the hypercyclicity of T implies the operator T − λ I has a dense range for all λ ∈ C , we infer that

σ r ( T ) = ∅

(cf. [16, Proposition 4.1], [17, Lemma 1]), and hence, in view of (3.1) and (3.10), we conclude that

σ c ( T ) = { λ ∈ C ∣ ∣ λ ∣ = ∣ w ∣ } .

Thus, (3.2) holds as well.□

4 Unbounded weighted translations in L p ( 0 , ∞ )

Lemma 4.1

(Closedness of powers) In the (real or complex) space L p ( 0 , ∞ ) ( 1 ≤ p < ∞ ), for the weighted left translation

( T w , a x ) ( t ) ≔ w t x ( t + a ) , t ≥ 0 ,

with w > 1 , a > 0 , and domain

D ( T w , a ) ≔ x ∈ L p ( 0 , ∞ ) ∣ ∫ 0 ∞ ∣ w t x ( t + a ) ∣ p d t < ∞ ,

each power T w , a n ( n ∈ N ) is a densely defined unbounded closed linear operator.

Proof

Let 1 ≤ p < ∞ , w > 1 , a > 0 , and n ∈ N be arbitrary and, for the simplicity of notation, let T ≔ T w , a .

The linearity of T is obvious and implies that for T n .

Inductively,

(4.1) ( T n x ) ( t ) = w t w t + a … w t + ( n − 1 ) a x ( t + n a ) = w n t + ( n − 1 ) n a 2 x ( t + n a ) , t ≥ 0

and

(4.2) D ( T n ) = x ∈ L p ( 0 , ∞ ) ∣ ∫ 0 ∞ w n t + ( n − 1 ) n a 2 x ( t + n a ) p d t < ∞ .

By the denseness in L p ( 0 , ∞ ) ( 1 ≤ p < ∞ ) of the subspace

(4.3) Y ≔ ⋃ m = 1 ∞ ker T m ,

where

(4.4) ker T m = { f ∈ L p ( 0 , ∞ ) ∣ f ( t ) = 0 , t > m a } , m ∈ N ,

of the equivalence classes represented by p -integrable on ( 0 , ∞ ) eventually zero functions and the inclusion

(4.5) Y ⊂ C ∞ ( T ) ≔ ⋂ m = 1 ∞ D ( T m ) ,

which follows from (4.2), we infer that the operator T n is densely defined.

The unboundedness of T n follows from the fact that, for the equivalence classes e m ∈ L p ( 0 , ∞ ) , m ∈ N , represented by

e m ( t ) ≔ χ [ m , m + 1 ] ( t ) , m ∈ N , t ≥ 0 ,

and we have

e m ∈ D ( T n ) , ‖ e m ‖ p = 1 , m ∈ N ,

and, for all m ∈ N sufficiently large so that m ≥ n a , in view of w > 1 ,

‖ T n e m ‖ p = ∫ 0 ∞ w n t + ( n − 1 ) n a 2 e m ( t + n a ) p d t 1 / p = ∫ m − n a m + 1 − n a w p n t + ( n − 1 ) n a 2 d t 1 / p ≥ w n ( m − n a ) + ( n − 1 ) n a 2 → ∞ , m → ∞ .

Let a sequence ( x m ) m ∈ N in L p ( 0 , ∞ ) be such that

D ( T n ) ∋ x m → x ∈ L p ( 0 , ∞ ) , m → ∞

and

T n x m → y ∈ L p ( 0 , ∞ ) , m → ∞ .

The sequences ( x m ( ⋅ ) ) m ∈ N and ( ( T n x m ) ( ⋅ ) ) m ∈ N of the p -integrable on ( 0 , ∞ ) representatives of the corresponding equivalence classes converging in p -norm on ( 0 , ∞ ) , also converge in the Lebesgue measure λ 1 on ( 0 , ∞ ) , and hence, by the Riesz theorem (see, e.g., [15]), there exist subsequences ( x m ( k ) ( ⋅ ) ) k ∈ N and ( ( T n x m ( k ) ) ( ⋅ ) ) k ∈ N convergent a.e. on ( 0 , ∞ ) relative to λ 1 , i.e.,

(4.6) x m ( k ) ( t ) → x ( t ) on ( 0 , ∞ ) ( mod λ 1 )

and

(4.7) ( T n x m ( k ) ) ( t ) → y ( t ) on ( 0 , ∞ ) ( mod λ 1 ) .

By (4.6),

( T n x m ( k ) ) ( t ) = w n t + ( n − 1 ) n a 2 x m ( k ) ( t + n a ) → w n t + ( n − 1 ) n a 2 x ( t + n a ) on ( 0 , ∞ ) ( mod λ 1 ) ,

which by (4.7), in view of the completeness of the Lebesgue measure (see, e.g., [15]), implies that

w n t + n ( n − 1 ) a 2 x ( t + n a ) = y ( t ) ( mod λ 1 ) ,

and hence,

x ∈ D ( T n ) and T n x = y .

By the Sequential Characterization of Closed Linear Operators (see, e.g., [14]) the operator T n is closed.□

Theorem 4.1

(Unbounded weighted translations in L p ( 0 , ∞ ) ) In the (real or complex) space L p ( 0 , ∞ ) ( 1 ≤ p < ∞ ), the weighted left translation

( T w , a x ) ( t ) ≔ w t x ( t + a ) , t ≥ 0 ,

with w > 1 , a > 0 , and domain

D ( T w , a ) ≔ x ∈ L p ( 0 , ∞ ) ∣ ∫ 0 ∞ ∣ w t x ( t + a ) ∣ p d t < ∞

is a chaotic unbounded linear operator.

Furthermore, provided the underlying space is complex,

(4.8) σ ( T w , a ) = σ p ( T w , a ) = C .

Proof

Let 1 ≤ p < ∞ , w > 1 , and a > 0 be arbitrary and, for the simplicity of notation, let T ≔ T w , a .

For the dense in L p ( 0 , ∞ ) subspace Y of the equivalence classes represented by p -integrable eventually zero functions (see (4.3) and (4.4)), we have inclusion (4.5).

The mapping

Y ∋ x ↦ S x ∈ Y ,

where the equivalence class S x is represented by

(4.9) ( S x ) ( t ) ≔ w − ( t − a ) x ( t − a ) , t > a , 0 , otherwise ,

is well defined since the function ( S x ) ( ⋅ ) is eventually zero and, in view of w > 1 ,

∫ 0 ∞ ∣ ( S x ) ( t ) ∣ p d t = ∫ a ∞ ∣ w − ( t − a ) p x ( t − a ) ∣ p d t = ∫ 0 ∞ w − t p ∣ x ( t ) ∣ p d t ≤ ∫ 0 ∞ ∣ x ( t ) ∣ p d t < ∞ .

As is easily seen,

(4.10) ∀ x ∈ Y : T S x = x .

Let x ∈ Y , represented by a p -integrable on ( 0 , ∞ ) eventually zero function x ( ⋅ ) , be arbitrary. Then,

∃ M ∈ N : supp x ≔ { t ∈ ( 0 , ∞ ) ∣ x ( t ) ≠ 0 } ¯ ⊆ [ 0 , M a ] .

By (4.1),

∀ n ≥ M : T n x = 0 ,

and hence,

T n x → 0 , n → ∞ .

Based on (4.9), inductively,

(4.11) ( S n x ) ( t ) = 0 , 0 ≤ t < n a , w − ( t − a ) w − ( t − 2 a ) … w − ( t − n a ) x ( t − n a ) , t ≥ n a , = 0 , 0 ≤ t < n a , w − n t + n ( n + 1 ) a 2 x ( t − n a ) , t ≥ n a , x ∈ Y , n ∈ N .

In view of w > 1 , we have

‖ S n x ‖ p = ∫ 0 ∞ ∣ ( S n x ) ( t ) ∣ p d t 1 / p = ∫ n a ∞ w − n t + n ( n + 1 ) a 2 x ( t − n a ) p d t 1 / p ≤ w − n ⋅ n a + n ( n + 1 ) a 2 ∫ n a ∞ ∣ x ( t − n a ) ∣ p d t 1 / p = w − n ( n − 1 ) a 2 ∫ 0 ∞ ∣ x ( t ) ∣ p d t 1 / p = w − n ( n − 1 ) a 2 ‖ x ‖ p , x ∈ Y , n ∈ N .

Whence, since w > 1 and a > 0 , we deduce that

∀ x ∈ Y : lim n → ∞ ‖ S n x ‖ p 1 / n = 0 ,

or equivalently,

(4.12) ∀ x ∈ Y , ∀ α ∈ ( 0 , 1 ) ∃ c = c ( x , α ) > 0 ∀ n ∈ N : ‖ S n x ‖ p ≤ c α n ‖ x ‖ p ,

which implies

∀ x ∈ Y : S n x → 0 , n → ∞ .

From the details presented earlier and the fact that, by the Closedness of powers lemma (Lemma 4.1), each power T n ( n ∈ N ) is a closed operator, by the Sufficient condition for hypercyclicity (Theorem 2.2), we infer that the operator T is hypercyclic.

To prove that T has a dense set of periodic points, let us first show that each N ∈ N is a period for T .

Let N ∈ N and

(4.13) x ∈ ker T N ⧹ { 0 } ⊆ Y ⧹ { 0 } ,

where

ker T N = { f ∈ L p ( 0 , ∞ ) ∣ f ( t ) = 0 , t > N a }

be arbitrary.

By estimate (4.12),

(4.14) x N ≔ ∑ k = 0 ∞ S k N x ∈ L p ( 0 , ∞ )

is well defined and, in view of (4.11), is represented by the p -integrable on ( 0 , ∞ ) function

x N ( t ) ≔ w − k N t + k N ( k N + 1 ) a 2 x ( t − k N a ) , t ∈ D k ≔ [ k N a , ( k + 1 ) N a ) , k ∈ Z + .

Since, in view of (4.13) and (4.10),

∑ k = 0 ∞ T N S k N x = ∑ k = 1 ∞ S ( k − 1 ) N x = x N ,

by the closedness of the operator T N , we infer that

x N ∈ D ( T N ) and T N x N = x N ,

(see, e.g., [14]), and hence, x N is an N -periodic point for T .

Suppose that x ∈ Y is an arbitrary equivalence class represented by a p -integrable on ( 0 , ∞ ) eventually zero function x ( ⋅ ) . Then,

∃ M ∈ N : x ( t ) = 0 , t > M a .

Then, for an arbitrary period N ≥ M , (4.13) holds, and there exists an N -periodic point x N for the operator T defined based on x by (4.14). By estimate (4.12),

∥ x N − x ∥ = ∑ k = 1 ∞ S k N x p ≤ ∑ k = 1 ∞ ∥ S k N x ∥ p ≤ c ∑ k = 1 ∞ ( α N ) k ‖ x ‖ p = c α N 1 − α N ‖ x ‖ p → 0 , N → ∞ .

Whence, in view of the denseness of Y in L p ( 0 , ∞ ) , we infer that the set Per ( T ) of periodic points of T is dense in L p ( 0 , ∞ ) as well, and hence, the operator T is chaotic.

Now, assuming that the space L p ( 0 , ∞ ) is complex, let us prove (4.8).

Let λ ∈ C and

(4.15) x ∈ ker T ⧹ { 0 } ⊆ Y ⧹ { 0 } ,

where

ker T = { f ∈ L p ( 0 , ∞ ) ∣ f ( t ) = 0 , t > a } ,

be arbitrary.

By estimate (4.12), for

α ≔ ( ∣ λ ∣ + 1 ) − 1 ∈ ( 0 , 1 ) ,

we have

(4.16) ∃ c = c ( x , α ) > 0 ∀ k ∈ N : ‖ λ k S k x ‖ p ≤ ∣ λ ∣ k c α k ‖ x ‖ p = c ( ∣ λ ∣ α ) k ‖ x ‖ p ,

where 0 ≤ ∣ λ ∣ α k = ∣ λ ∣ ( ∣ λ ∣ + 1 ) − k < 1 .

By estimate (4.16),

x λ ≔ ∑ k = 0 ∞ λ k S k x ∈ L p ( 0 , ∞ )

is well defined and, in view of (4.11), is represented by the p -integrable on ( 0 , ∞ ) function

x λ ( t ) ≔ λ k w − k t + k ( k + 1 ) a 2 x ( t − k a ) , t ∈ [ k a , ( k + 1 ) a ) , k ∈ Z + , ( 0 0 ≔ 1 ) .

Since

‖ x λ ‖ p p = ∫ 0 ∞ ∣ x λ ( t ) ∣ p d t ≥ ∫ 0 a ∣ x λ ( t ) ∣ p d t = ∫ 0 a ∣ x ( t ) ∣ p d t = ∫ 0 ∞ ∣ x ( t ) ∣ p d t = ‖ x ‖ p p > 0 ,

we infer that x λ ≠ 0 .

Furthermore, since, in view of (4.15) and (4.10),

∑ k = 0 ∞ T ( λ k S k x ) = λ ∑ k = 1 ∞ λ k − 1 S k − 1 = λ x λ ,

by the closedness of the operator T , we conclude that

x λ ∈ D ( T ) and T x λ = λ x λ .

See, e.g., [14].

Thus, λ ∈ σ p ( T ) and x λ is an eigenvector of T associated with λ , which proves (4.8).□

5 Bounded weighted translations on C 0 [ 0 , ∞ )

In [3, Theorem 2.3], it is shown that, on the (real or complex) space C 0 [ 0 , ∞ ) , the bounded linear weighted left translation operator

( T w , a x ) ( t ) ≔ w x ( t + a ) , t ≥ 0 ,

with ∣ w ∣ > 1 and a > 0 is chaotic and

{ λ ∈ C ∣ 0 < ∣ λ ∣ < ∣ w ∣ } ⊆ σ p ( T )

based on the simple fact that, for each λ ∈ C , Re λ < 0 , the equation

T w , a x = w e a λ x

is satisfied by the function

x ( t ) ≔ e λ t , t ≥ 0 .

It is also stated (without proof) that one can show that

σ p ( T w , a ) = { λ ∈ C ∣ ∣ λ ∣ < ∣ w ∣ } .

Here, we completely describe the spectrum of such operators.

Proposition 5.1

(Spectrum) On the complex space C 0 [ 0 , ∞ ) , for the bounded linear weighted left translation operator,

( T x ) ( t ) ≔ w x ( t + a ) , t ≥ 0 ,

where w ∈ C with ∣ w ∣ > 1 and a > 0 ,

(5.1) σ ( T ) = { λ ∈ C ∣ ∣ λ ∣ ≤ ∣ w ∣ }

with

(5.2) σ p ( T w , a ) = { λ ∈ C ∣ ∣ λ ∣ < ∣ w ∣ } and σ c ( T w , a ) = { λ ∈ C ∣ ∣ λ ∣ = ∣ w ∣ } .

Proof

Let w ∈ C with w > 1 and a > 0 be arbitrary and, for the simplicity of notation, let T ≔ T w , a .

Since

T = w B ,

where

( B x ) ( t ) ≔ x ( t + a ) , x ∈ C 0 [ 0 , ∞ ) , t ≥ 0 ,

is a left translation with ‖ B ‖ = 1 , and hence,

‖ T ‖ = ∣ w ∣ ‖ B ‖ = ∣ w ∣ ,

and by Gelfand’s spectral radius theorem [14],

(5.3) σ ( T ) ⊆ { λ ∈ C ∣ ∣ λ ∣ ≤ ∣ w ∣ } .

Let λ ∈ C with ∣ λ ∣ < ∣ w ∣ and a nonzero x ∈ C [ 0 , a ] , with

x ( a ) = λ w x ( 0 )

be arbitrary. For example, for 0 < ∣ λ ∣ < ∣ w ∣ ,

y ( t ) ≔ e c t , t ∈ [ 0 , a ] ,

with c ≔ 1 a ln λ w = 1 a ln λ w + i Im λ w ( i is the imaginary unit).

Then, as is readily verified,

(5.4) x λ ( t ) ≔ λ w k x ( t − k a ) , t ∈ [ k a , ( k + 1 ) a ) , k ∈ Z + , ( 0 0 ≔ 1 )

is a nonzero function continuous on [ 0 , ∞ ) .

Since, in view of ∣ λ / w ∣ < 1 , for any k ∈ Z + ,

max k a ≤ t ≤ ( k + 1 ) a ∣ x λ ( t ) ∣ = max k a ≤ t ≤ ( k + 1 ) a λ w k x ( t − k a ) = λ w k max 0 ≤ t ≤ a ∣ x ( t ) ∣ → 0 , k → ∞ ,

we infer that x λ ∈ C 0 [ 0 , ∞ ) ⧹ { 0 } .

Also,

( T x λ ) ( t ) = w x ( t + a ) = w λ w k x ( t + a − k a ) = λ λ w k − 1 x ( t − ( k − 1 ) a ) , t ∈ [ ( k − 1 ) a , k a ) , k ∈ N ,

which implies that

(5.5) T x λ = λ x λ .

Thus, λ ∈ σ p ( T ) .

Conversely, let λ ∈ σ p ( T ) be an arbitrary eigenvalue for T with an associated eigenvector x λ ∈ C 0 [ 0 , ∞ ) ⧹ { 0 } . Then, for

x k ( t ) ≔ x λ ( t ) , t ∈ [ k a , ( k + 1 ) a ) , k ∈ Z + ,

by (5.5), we have

λ x k − 1 ( t ) = w x k ( t + a ) , t ∈ [ k a , ( k + 1 ) a ) , k ∈ N .

Whence,

x k ( t ) = λ w k x λ ( t − k a ) , t ∈ [ k a , ( k + 1 ) a ) , k ∈ Z + ,

which, in view of x λ ≠ 0 , implies that

0 < max 0 ≤ t ≤ a ∣ x λ ( t ) ∣ ≤ sup t ≥ 0 ∣ x λ ( t ) ∣ < ∞ .

Further,

lim t → ∞ x λ ( t ) = 0

implies

λ w k max 0 ≤ t ≤ a ∣ x λ ( t ) ∣ → 0 , k → ∞ ,

which, in its turn, means that

∣ λ ∣ < ∣ w ∣ .

Thus, x λ is of the form given by (5.4), where x is the restriction to [ 0 , a ] of x λ .

The aforementioned equation proves that

(5.6) σ p ( T ) = { λ ∈ C ∣ ∣ λ ∣ < ∣ w ∣ } .

Considering that σ ( T ) is a closed set in C (see, e.g., [13,14]), we infer from (5.3) and (5.6) that (5.1) holds.

Since, by [7, Lemma 2.53], the hypercyclicity of T implies the operator T − λ I has a dense range for all λ ∈ C , we infer that

σ r ( T ) = ∅

(cf. [16, Proposition 4.1], [17, Lemma 1]), and hence, in view of (5.1) and (5.6), we conclude that

σ c ( T ) = { λ ∈ C ∣ ∣ λ ∣ = ∣ w ∣ } .

Thus, (5.2) holds as well.□

6 Unbounded weighted translations in C 0 [ 0 , ∞ )

Lemma 6.1

(Closedness of powers) In the (real or complex) space ( C 0 [ 0 , ∞ ) , ‖ ⋅ ‖ ∞ ) , for the weighted left translation

( T w , a x ) ( t ) ≔ w t x ( t + a ) , t ≥ 0 ,

with w > 1 , a > 0 , and domain

D ( T w , a ) ≔ { x ∈ C 0 [ 0 , ∞ ) ∣ lim t → ∞ w t x ( t + a ) = 0 } ,

and each power T w , a n ( n ∈ N ) is a densely defined unbounded closed linear operator.

Proof

Let w > 1 , a > 0 , and n ∈ N be arbitrary and, for the simplicity of notation, let T ≔ T w , a .

The linearity of T is obvious and implies that for T n .

Inductively,

(6.1) ( T n x ) ( t ) = w t w t + a … w t + ( n − 1 ) a x ( t + n a ) = w n t + ( n − 1 ) n a 2 x ( t + n a ) , t ≥ 0

and

(6.2) D ( T n ) = x ∈ C 0 [ 0 , ∞ ) ∣ lim t → ∞ w n t + ( n − 1 ) n a 2 x ( t + n a ) = 0

(cf. (4.1) and (4.2)).

By the denseness in C 0 [ 0 , ∞ ) ( 1 ≤ p < ∞ ) of the subspace

(6.3) Y ≔ ⋃ m = 1 ∞ ker T m ,

where

(6.4) ker T m = { f ∈ C 0 [ 0 , ∞ ) ∣ f ( t ) = 0 , t ≥ m a } , m ∈ N ,

of the equivalence classes represented by p -integrable on ( 0 , ∞ ) eventually zero functions and the inclusion

(6.5) Y ⊂ C ∞ ( T ) ≔ ⋂ m = 1 ∞ D ( T m ) ,

which follows from (6.2), we infer that the operator T n is densely defined.

The unboundedness of T n follows from the fact that, for

e m ( t ) ≔ 1 , 0 ≤ t < m a , w − ( t − m a ) 2 , t ≥ m a , m ∈ N

we have

e n ∈ D ( T ) , ‖ e n ‖ ∞ = 1 , m ∈ N ,

and, for all m ≥ n , in view of w > 1 ,

‖ T n e m ‖ ∞ = sup t ≥ 0 w n t + ( n − 1 ) n a 2 e m ( t + n a ) ≥ w n t + ( n − 1 ) n a 2 e m ( t + n a ) t = m a − n a ≥ w n ( m a − n a ) + ( n − 1 ) n a 2 → ∞ , m → ∞ .

Let a sequence ( x m ) m ∈ N in C 0 [ 0 , ∞ ) be such that

D ( T n ) ∋ x m → x ∈ C 0 [ 0 , ∞ ) , m → ∞

and

T n x m → y ∈ C 0 [ 0 , ∞ ) , m → ∞ .

Then, for each t ≥ 0 ,

(6.6) x m ( t ) → x ( t ) and ( T n x m ) ( t ) → y ( t ) , m → ∞ .

By (6.6), for each t ≥ 0 ,

( T n x m ) ( t ) = w n t + ( n − 1 ) n a 2 x m ( t + m a ) → w n t + ( n − 1 ) n a 2 x ( t + n a ) , m → ∞

and

w n t + ( n − 1 ) n a 2 x ( t + n a ) = y ( t ) , t ≥ 0 ,

which implies

x ∈ D ( T n ) and T n x = y .

Thus, by the sequential characterization of closed linear operators (see, e.g., [14]) the operator T n is closed.□

Theorem 6.1

(Unbounded weighted translations in C 0 [ 0 , ∞ ) ) In the (real or complex) space ( C 0 [ 0 , ∞ ) , ‖ ⋅ ‖ ∞ ) , the weighted left translation

( T w , a x ) ( t ) ≔ w t x ( t + a ) , t ≥ 0 ,

with w > 1 , a > 0 , and domain

D ( T w , a ) ≔ { x ∈ C 0 [ 0 , ∞ ) ∣ lim t → ∞ w t x ( t + a ) = 0 }

is a chaotic unbounded linear operator.

Furthermore, provided the underlying space is complex,

(6.7) σ ( T w , a ) = σ p ( T w , a ) = C .

Proof

Let w > 1 and a > 0 be arbitrary and, for the simplicity of notation, let T ≔ T w , a .

For the dense in C 0 [ 0 , ∞ ) subspace Y of eventually zero functions (see (6.3) and (6.4)), we have inclusion (6.5).

The mapping

Y ∋ x ↦ S x ∈ Y ,

where

(6.8) ( S x ) ( t ) ≔ x ( 0 ) a t , 0 ≤ t < a , w − ( t − a ) x ( t − a ) , t ≥ a ,

is well defined since the function S x is eventually zero and, as is easily seen,

(6.9) ∀ x ∈ Y : T S x = x .

Let x ∈ Y be arbitrary. Then,

∃ M ∈ N : supp x ≔ { t ∈ [ 0 , ∞ ) ∣ x ( t ) ≠ 0 } ¯ ⊆ [ 0 , M a ] .

By (6.1),

∀ n ≥ M : T n x = 0 ,

and hence,

T n x → 0 , n → ∞ .

Based on (6.8), inductively,

(6.10) ( S n x ) ( t ) = 0 , 0 ≤ t < ( n − 1 ) a , w − ( t − a ) … w − ( t − ( n − 1 ) a ) x ( 0 ) a ( t − ( n − 1 ) a ) , ( n − 1 ) a ≤ t < n a , w − ( t − a ) … w − ( t − n a ) x ( t − n a ) , t ≥ n a , = 0 , 0 ≤ t < ( n − 1 ) a , w − ( t − a ) x ( 0 ) a ( t − ( n − 1 ) a ) , ( n − 1 ) a ≤ t < n a , w − n t + n ( n + 1 ) a 2 x ( t − n a ) , t ≥ n a , x ∈ Y , n ∈ N .

In view of w > 1 ,

‖ S n x ‖ ∞ = sup t ≥ 0 ∣ ( S n x ) ( t ) ∣ ≤ w − n ⋅ n a + n ( n + 1 ) a 2 ‖ x ‖ ∞ = w − n ( n − 1 ) a 2 ‖ x ‖ ∞ , x ∈ Y , n ∈ N .

Whence, since w > 1 and a > 0 , we deduce that

∀ x ∈ Y : lim n → ∞ ‖ S n x ‖ ∞ 1 / n = 0 ,

or equivalently,

(6.11) ∀ x ∈ Y , ∀ α ∈ ( 0 , 1 ) ∃ c = c ( x , α ) > 0 ∀ n ∈ N : ‖ S n x ‖ ∞ ≤ c α n ‖ x ‖ ∞ ,

which implies

∀ x ∈ Y : S n x → 0 , n → ∞ .

From the aforementioned argument and the fact that, by the Closedness of powers lemma (Lemma 6.1), each power T n ( n ∈ N ) is a closed operator, by the Sufficient condition for hypercyclicity (Theorem 2.2), we infer that the operator T is hypercyclic.

Based on estimate (6.11), proving that T has a dense set of periodic points, and hence, is chaotic and that (6.7) holds is identical to proving the same parts in Theorem 4.1.□

7 Concluding remarks

The foregoing results are consistent with the recent findings of [16]. According to the latter, under the premises of Theorems 3.1, 4.1, [3, Theorem 2.3], or Theorem 6.1, not only is the operator T w , a chaotic but also its every power T w , a n ( n ∈ N ), and furthermore,

dim ker ( T w , a n − λ I ) = dim ker T w , a n = dim { f ∈ X ∣ f ( t ) = 0 , t > n a } ,

where X ≔ L p ( 0 , ∞ ) ( 1 ≤ p < ∞ ) or X ≔ C 0 [ 0 , ∞ ) , holds in Theorem 3.1 and Proposition 5.1 for all n ∈ N and λ ∈ C with ∣ λ ∣ < ∣ w ∣ n and in Theorems 4.1 and 6.1 for all n ∈ N and λ ∈ C , i.e., all eigenvalues of T w , a n are of the same geometric multiplicity.

Conflict of interest: The authors state no conflict of interest.

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Received: 2021-09-26

Revised: 2022-01-23

Accepted: 2022-03-29

Published Online: 2022-05-05

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Articles in the same Issue

https://doi.org/10.1515/dema-2022-0008

Keywords for this article

hypercyclic vector; periodic point; hypercyclic operator; chaotic operator; spectrum

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