Hypercyclicity for the Elements of the Commutant of an Operator

Given a bounded linear operator T acting on a complex Banach space, we obtain a spectral condition implying that each operator in the commutant of T different from λI has a hypercyclic multiple, and we show several examples of operators satisfying this condition. We emphasize that for some of these examples we do not have a description of the commutant of T.


Introduction
A bounded linear operator T on a Banach space is called hypercyclic if it has a dense orbit. A vector x for which the orbit {T n x} is dense is called a hypercyclic vector for T . We refer to the books [1] and [6] for information on hypercyclic operators.
When we say that an operator T has a hypercyclic commutant, we mean that each operator in the commutant different from λI (where I is the identity operator) has a hypercyclic multiple. The study of operators with hypercyclic commutant was originated by Godefroy and Shapiro in their seminal paper [4]. Their work suggests that operators behaving like backward shifts tend to transfer hypercyclicity to "appropriate" operators in their commutant. The word "appropriate" means that we must discard from the commutant some operators that can never be hypercyclic, like the multiples of the identity or the contractions.
In their work, Godefroy and Shapiro used some descriptions of the commutants of the involved operators acting on Hilbert spaces which are valid The first author was supported by the Kingdom of Spain, Grant MTM2010-20190. The second author was supported by Junta de Andalucía FQM-257 and Vicerrectorado de investigación UCA. IEOT neither in the Banach space setting, nor for bilateral weighted shifts and composition operators. Later, Herzog and Schmoeger [8] extended Godefroy and Shapiro's results to general operators T on Banach spaces. Using the holomorphic functional calculus, they transfer the hypercyclicity to operators ϕ(T ) under certain spectral conditions on T . This work was improved by Bermúdez and Miller [2]. Since the holomorphic functional calculus does not cover the whole commutant, the following general question arises: Given a hypercyclic operator T , when T has a hypercyclic commutant? Recently, Müller [10] improved the results of [2,8], and discovered that these results become stronger if we consider operators with closed range. His study allows to prove that any operator in the commutant of a generalized backward shift (acting on certain Banach spaces) different from λI is supercyclic, solving a longstanding question posed by Godefroy and Shapiro. In this paper we study Banach space operators T with hypercyclic commutant. To characterize the commutant of T is a really complicated problem even for well known operators in Hilbert spaces. However we show that, under certain spectral conditions and without a description of the commutant of T , it is possible to ensure that T has a hypercyclic conmutant.
The paper is structured as follows. In Sect. 2 we prove our main result (Theorem 2.1) that gives a sufficient spectral condition for an operator T to have a "hypercyclic commutant". Section 3 is devoted to give examples of operators satisfying that spectral condition: Bilateral weighted shifts, unilateral weighted backward shifts, the adjoint of the discrete Cesàro operator acting on the sequence spaces p , the continuous Cesàro operator acting on L p (0, 1), and the adjoint of the Cesàro operator acting on the Hardy spaces H p (D). We also show that some composition operators acting on the Hardy spaces, which are induced by linear fractional self-maps of the unit disk D, fail both the hypothesis and the thesis of Theorem 2.1. The results of the paper suggest some questions which will be formulated in the last section.

Main Theorem
In this section we give a general theorem for Banach space operators T asserting that, under certain spectral conditions, each operator in the commutant of T different from λI has a hypercyclic multiple. 1. Every λ ∈ U is a simple eigenvalue of T , 2. span{Ker(T − λI): λ ∈ U } is dense in X, 3. there exists a holomorphic functionx: λ ∈ U →x(λ) = x λ ∈ X such that 0 = x λ ∈ ker(T − λI).
Then every operator in the commutant of T different from λI has a hypercyclic multiple.
Thus by condition (1) we have Ax λ = a(λ)x λ for some complex number a(λ). So we get a function a: U → C. Boundedness of the operator A provides boundedness of a(λ) on U , and A = μI implies that a is non-constant. Let us see that a : U → C is a holomorphic function.
For each y ∈ X , y = 0, our function a satisfies This implies that our function is an open subset of the complex plane. Then for some m ∈ C the set m · a(U ) intersects the unit circle. To prove that mA is hypercyclic, let us see that the subsets span{ker(mA − λI): |λ| < 1} and span{ker(mA − λI): |λ| > 1} are dense in X, and the hypercyclicity of mA will follow from the Godefroy-Shapiro Criterion (see Corollary 1.10 in [1]). Indeed, is dense by the considerations at the beginning of the proof, and a similar argument is valid for the other set.

Some Applications
In this section we show that several concrete examples of operators satisfy the hypothesis of Theorem 2.1. IEOT

Bilateral Weighted Shifts
Let us consider a bilateral weighted shift W on the complex Hilbert space of sequences 2 (Z), that is defined by where n runs through the set of all integers Z, {e n : n ∈ Z} is the natural orthonormal basis of 2 (Z), and (w n ) is a bounded sequence of strictly positive numbers, the sequence of weights. We follow the notation used in the survey by Shields [14].
We consider the quantities It turns out (see [14] Theorem 9) that when r } is contained in the point spectrum of W , that every λ ∈ U is a simple eigenvalue of W and that a corresponding eigenvector is given by the following expression: Clearly the function f (λ) = x λ is analytic on U . Let us see that the set span{x λ : λ ∈ U} is dense in 2 (Z). Indeed, given (y i ) ∈ 2 (Z) we have Therefore if x λ , (y i ) = 0 for all λ ∈ U , then y i = 0 for all i ∈ Z. Thus W satisfies the conditions of Theorem 2.1 and we have the following result.

The Unilateral Weighted Backward Shift
Here we consider a unilateral weighted shift U on the complex Hilbert space of sequences 2 , that is defined by positive numbers, the sequence of weights. The relevant quantity to describe the spectrum of U is If r 2 (U ) > 0 then U p = {λ : |λ| < r 2 (U )} ⊂ σ p (U ) and consists of simple eigenvalues (see [14] Theorem 8).
A similar argument to the one given in Subsection 3.1 provides the following result.

The Adjoint of the Cesàro Operator on Spaces of Sequences
The Cesàro operator C: p → p (1 < p < ∞) is given by C((x n )) = x 1 + · · · + x n n and has norm equal to q, where 1/p + 1/q = 1. The adjoint of the Cesàro operator C : q → q is given by C e n = e 1 + · · · + e n n .
Note that x n (1/k) = 0 if and only if n ≤ k, and that the sequence (1/k) is contained in U q . Hence span{Ker(C − λI): λ ∈ U q } is dense in q . Moreover it is not difficult to show that the mapx: λ ∈ U q →x(λ) ∈ q is analytic.
Therefore the conditions of Theorem 2.1 are fully satisfied by C , and as in the previous examples we conclude that every operator in the commutant of C different from λI has a hypercyclic multiple.
Remarks 3.4. The Cesàro operator C is also bounded in the space c 0 of sequences converging to 0. In this case the adjoint C : 1 → 1 satisfies the same conditions that C on q with U 1 = {λ ∈ C: |λ − 1/2| < 1/2}. Thus it satisfies the hypothesis of Theorem 2.1.

The Cesàro Operator on Spaces of Integrable Functions
The Cesàro operator C: L p (0, 1) →: L p (0, 1) (1 < p < ∞) is given by M. González and F. León-Saavedra IEOT and has norm equal to q, where 1/p + 1/q = 1. Moreover, it was proved by Leibowitz (see [9]) that the point spectrum of C is the set that U p consists of simple eigenvalues of C, and that for every λ ∈ U p , the function g λ (x) = x −1+1/λ satisfies Cg λ = λg λ . Obviously, the function λ ∈ U p → g λ ∈ L p (0, 1) is analytic. Moreover, since f 1/n (x) = x n−1 for all n ≥ 1, the set span{f λ : λ ∈ U p } is dense in L p (0, 1). Therefore the conditions of Theorem 2.1 are fully satisfied by C, and we can conclude that every operator in the commutant of C different from λI has a hypercyclic multiple.

The Adjoint of the Cesàro Operator on Hardy Spaces of Analytic Functions
The Cesàro operator acting on the space H(D) of analytic functions on the unit disk D is defined formally by The operator C is bounded on several Banach spaces of analytic functions on D. Boundedness of C on the Hardy space H p (D) was proved by Hardy for p = 2, and by A. Siskakis for 1 ≤ p < ∞ ( [15]). It is not bounded acting on H ∞ (D). For further examples we refer to [11].
We will need the following properties provided by Persson [11,Theorem B]. Let us denote It follows from part (2) of Theorem 3.5 that each λ ∈ U p is a simple eigenvalue of the adjoint operator C . For each λ ∈ U p we consider the function f λ defined on the unit disk D by Proposition 3.6. Let 1 < p < ∞ and let q such that 1/p + 1/q = 1.

Hypercyclicity for the elements 271
Proof. Once we establish (1), part (2) is immediate. As usual, we identify H q (D) with the subspace of functions in L q (T) whose Fourier coefficients satisfyf (n) = 0 for n < 0.
To prove (1), it is enough to note that λ ∈ U p implies Re(1/λ) Recall that for 1 < p < ∞, H p (D) is a complemented subspace of L p (T). As a consequence, the dual space H p (D) is isomorphic to H q (D). In fact, for every G ∈ H p (D) there exists g ∈ H q (D) such that for each f ∈ H p (D), In this way, given the Cesàro operator C acting on H p (D) (1 < p < ∞), we can see C as an operator acting on H q (D). With this identification, each f ∈ H p (D) has the form f (θ) = ∞ n=0 a n e inθ , the action of C is given by and it easy to check that the action of the adjoint C on the elements e inθ of H q (D) is given by C e inθ = (n + 1) −1 n k=0 e ikθ . Therefore, if Proof. It is easy to check that for each λ ∈ U p \{1/n: n ∈ N}, the derivatives of f λ satisfy Proceeding as in the proof of Theorem 2 in [3], we get C f λ = λf λ . The case λ ∈ {1/n: n ∈ N} is easy. In fact, denoting F n the subspace generated by {e ikθ : 0 ≤ k ≤ n}, we have that C F n ⊂ F n for n ∈ N ∪ {0}, hence {f 1/n : n ∈ N} generates a dense subspace of H q (D).
As a consequence we obtain the following result.

Operators without a "Hypercyclic Commutant"
Here we analyze some examples of operators T which do not satisfy part of the hypothesis of Theorem 2.1, and also fail to have a hypercyclic commutant.
(a) In the absence of an open set contained in the point spectrum, an operator does not satisfy the hypothesis of Theorem 2.1. This is the case for the following example studied by Godefroy and Shapiro (see [4] Section 3.7).
Let B be a compact unilateral weighted backward shift on the sequence space p , 1 ≤ p < ∞. By a result of Salas (see [12]), I + B is hypercyclic. However λB is hypercyclic for no λ ∈ C because λB is quasinilpotent and the orbit of each vector is bounded.
(b) Next we consider some examples of operators for which the eigenvalues are not simple, and they do not have hypercyclic commutant.
Let 1 us consider the unilateral backward shift U defined on 2 (see Sect. 3.2) given by Ue 1 = 0, Ue n+1 = e n , and the diagonal operator A given by Ae n = (−1) n e n . The operator A has no hypercyclic multiple and belongs to the commutant of U 2 . Observe that U 2 satisfies conditions (2) and (3)  Other examples can be found among composition operators. Indeed, let ϕ denote a linear fractional selfmap of the unit disc which is a hyperbolic automorphism, and let C ϕ be the composition operator induced by ϕ acting on the Hardy space H 2 (D). It is proved in page 683 of [13] that C ϕ satisfies (2) and (3) of Theorem 2.1, but the eigenvalues are not simple, and there exists a multiplication operator M f in the commutant of C ϕ which has no hypercyclic multiple.

Concluding Remarks
To understand the influence of hypercyclic operators in linear dynamics it would be desirable to characterize operators with a hypercyclic commutant.
Let us denote by H an infinite dimensional complex Hilbert space. Using the ideas of Herrero in [7] (see also [10] Section 4) together with Theorem 2.1 we obtain the following result. Thus, although the set of operators with a hypercyclic commutant is invariant under similarity, it does not seem that this class can be characterized in spectral terms. To understand the nature of this phenomenon we suggest to analyze the following question.

Problem 4.2.
Let T be a unilateral (respectively bilateral) weighted backward shift with weight sequence {w n } defined on p , (respectively p (Z)) 1 ≤ p < ∞.
Is it be possible to characterize when T has a hypercyclic commutant in terms of the sequence {w n }?