On the non-hypercyclicity of scalar type spectral operators and collections of their exponentials

We give a straightforward proof of the non-hypercyclicity of an arbitrary (bounded or not) scalar type spectral operator $A$ in a complex Banach space as well as of the collection $\left\{e^{tA}\right\}_{t\ge 0}$ of its exponentials, which, under a certain condition on the spectrum of $A$, coincides with the $C_0$-semigroup generated by it. The spectrum of $A$ lying on the imaginary axis, it is shown that non-hypercyclic is also the strongly continuous group $\left\{e^{tA}\right\}_{t\in {\mathbb R}}$ of bounded linear operators generated by $A$. The important particular case of a normal operator $A$ in a complex Hilbert space follows immediately. From the general results, we infer that, in the complex Hilbert space $L_2({\mathbb R})$, the anti-self-adjoint differentiation operator $A:=\dfrac{d}{dx}$ with the domain $D(A):=W_2^1({\mathbb R})$ is not hypercyclic and neither is the left-translation operator group generated by it.


Introduction
In this paper, we give a straightforward proof of the non-hypercyclicity of an arbitrary scalar type spectral operator A (bounded or not) in a complex Banach space as well as of the collection e tA t≥0 of its exponentials (see, e.g., [4,5,8]), which, provided the spectrum of A is located in a left half-plane {λ ∈ C | Re λ ≤ ω} with some ω ∈ R, coincides with the C 0 -semigroup generated by A [14] (see also [3,20]). The spectrum of A lying on the imaginary axis iR (i is the imaginary unit ), it is shown that non-hypercyclic is also the generated by it strongly continuous group e tA t∈R of bounded linear operators. As an important particular case, we immediately obtain that of a normal operator A in a complex Hilbert space.
From the general results, we infer that, in the complex Hilbert space L 2 (R), the anti-self-adjoint differentiation operator A := d dx with the domain W 1 2 (R) := {f ∈ L 2 (R)|f (·) is absolutely continuous on R with f ′ ∈ L 2 (R)} is not hypercyclic and neither is the left-translation strongly continuous unitary group generated by it [9,23].
First, we are going to extend the definitions of hypercyclicity, traditionally given for bounded linear operators (see, e.g., [10,11]), to unbounded ones. The reason for such a shortcoming appears to stem out of the fact that, for an unbounded linear operator is the domain of an operator) in a normed vector space (X, · ), the subspace . . } is the set of nonnegative integers) is well defined, can be meager and even degenerate to {0}.
However, this not being the case for many important unbounded operators, including the scalar type spectral in a complex Banach space (X, · ), for which the subspace of all permissible orbit starters defined by (1.1) is dense in X (see Preliminaries), without further reservations, we naturally extend the known definition of hypercyclicity (see, e.g., [10,11]) as follows.

Definition 1.1 (Hypercyclicity). Let
A : X ⊇ D(A) → X be a linear operator in a (real or complex) normed vector space (X, · ). A vector Operators possessing hypercyclic vectors are called hypercyclic.
More generally, a collection {T (t)} t∈I (I is a nonempty indexing set ) of linear operators in X is called hypercyclic if it possesses hypercyclic vectors, i.e., such vectors f ∈ t∈I D(T (t)), whose orbit Remarks 1.1.
• It is quite obvious that, in the definition of hypercyclicity for an operator, the underlying space must necessarily be separable. Generally, for a collection of operators, this need not be so.
• According to [10,Observation 2.17], an operator has no nontrivial invariant closed subsets iff every nonzero vector is hypercyclic. Since nontrivial scalar type spectral operators are certain to have nontrivial invariant closed subspaces [4,5,8], our quest to prove their non-hypercyclicity appears to be amply justified.

Preliminaries
Here, for the reader's convenience, we outline certain essential preliminaries.
Henceforth, unless specified otherwise, A is supposed to be a scalar type spectral operator in a complex Banach space (X, · ) with strongly σ-additive spectral measure (the resolution of the identity) E A (·) assigning to each Borel set δ of the complex plane C a projection operator E A (δ) on X and having the operator's spectrum σ(A) as its support [4,5,8].
Observe that, in a complex finite-dimensional space, the scalar type spectral operators are all linear operators on the space, for which there is an eigenbasis (see, e.g., [4,5,8]) and, in a complex Hilbert space, the scalar type spectral operators are precisely all those that are similar to the normal ones [24].
Associated with a scalar type spectral operator in a complex Banach space is the Borel operational calculus analogous to that for a normal operator in a complex Hilbert space [7,21], which assigns to any Borel measurable function F : σ(A) → C a scalar type spectral operator (see [5,8]).
In particular, Provided σ(A) ⊆ {λ ∈ C | Re λ ≤ ω} , with some ω ∈ R, the collection of exponentials e tA t≥0 coincides with the C 0semigroup generated by A [14, Proposition 3.1] (cf. also [3,20]), and hence, if with some ω ≥ 0, the collection of exponentials e tA t∈R coincides with the strongly continuous group of bounded linear operators generated by A.
• By [13,Theorem 4.2], the orbits describe all weak/mild solutions of the abstract evolution equation whereas, by [19,Theorem 7], the orbits describe all weak/mild solutions of the abstract evolution equation which need not be differentiable in the strong sense and encompass the classical ones, strongly differentiable and satisfying the equations in the traditional plug-in sense, [2] (cf. [9, Ch. II, Definition 6.3], see also [18,Preliminaries]).
• The operator A generating a C 0 -semigroup or a strongly continuous group of bounded linear operators (see, e.g., [9,12]), the associated abstract Cauchy problem (ACP ) • Observe that all three subspaces are dense in X since they contain the dense in X subspace which coincides with the class E {0} (A) of entire vectors of A of exponential type [17,22].
The properties of the spectral measure and operational calculus, exhaustively delineated in [5,8], underlie the subsequent discourse. Here, we touch upon a few facts of particular importance.
Due to its strong countable additivity, the spectral measure E A (·) is bounded [6,8], i.e., there is such an M ≥ 1 that, for any Borel set δ ⊆ C, Observe that the notation · is used here to designate the norm in the space L(X) of all bounded linear operators on X. We adhere to this rather conventional economy of symbols in what follows also adopting the same notation for the norm in the dual space X * .
For any f ∈ X and g * ∈ X * , the total variation measure v(f, g * , ·) of the complexvalued Borel measure E A (·)f, g * ( ·, · is the pairing between the space X and its dual X * ) is a finite positive Borel measure with (see, e.g., [15,16]).

Main Results
Theorem 3.1. An arbitrary scalar type spectral operator A in a complex Banach space (X, · ) with spectral measure E A (·) is not hypercyclic and neither is the collection e tA t≥0 of its exponentials, which, provided the spectrum of A is located in a left half-plane {λ ∈ C | Re λ ≤ ω} with some ω ∈ R, coincides with the C 0 -semigroup generated by A.
There are two possibilities: either In the first case, as follows from the Hahn-Banach Theorem (see, e.g., [6]), there is a g * ∈ X * \ {0} such that and hence, for any n ∈ Z + , A n f by (2.12); |λ| n dv(f, g * , λ) which immediately implies that the orbit {A n f } n∈Z+ is not dense in X.
In the second case, = sup by the properties of the operational calculus; |λ| n dv(f, g * , λ) 1 dv(f, g * , λ) by (2.11) with F (λ) ≡ 1; which also implies that the orbit {A n f } n∈Z+ , being bounded, is not dense in X and completes the proof for the operator case.
Now, let us consider the case of the exponential collection e tA t≥0 assuming that f ∈ t≥0 D(e tA ) \ {0} is arbitrary.
There are two possibilities: either In the first case, as follows from the Hahn-Banach Theorem, there is a g * ∈ X * \{0} such that E A ({λ ∈ σ(A) | Re λ > 0}) f, g * = 0 and hence, for any t ≥ 0, e tA f by (2.12); e tλ dv(f, g * , λ) which immediately implies that the orbit e tA f t≥0 is not dense in X.
In the second case, and, for any t ≥ 0, = sup which also implies that the orbit e tA f t≥0 , being bounded, is not dense on X and completes the entire proof.
If further σ(A) ⊆ iR, by [8,Theorem XVIII.2.11 (c)], for any t ∈ R where the constant M ≥ 1 is from (2.9). Therefore, the generated by A strongly continuous group e tA t∈R is bounded (cf. [3]), which implies that every orbit e tA f t∈R , f ∈ X, is bounded, and hence, cannot be dense in X, and we arrive at the following Proposition 3.1. For a scalar type spectral operator A in a complex Banach space (X, · ) with σ(A) ⊆ iR, the generated by it strongly continuous group e tA t∈R is bounded, and hence, non-hypercyclic.

The Case of a Normal Operator
As an important particular case of Theorem 3.1, we obtain An arbitrary normal operator A in a complex Hilbert space is not hypercyclic and neither is the collection e tA t≥0 of its exponentials, which, provided the spectrum of A is located in a left half-plane {λ ∈ C | Re λ ≤ ω} with some ω ∈ R, coincides with the C 0 -semigroup generated by A.
As is known [23], for an anti-self-adjoint operator A in a complex Hilbert space, σ(A) ⊆ iR and the generated by it strongly continuous group e tA t∈R is unitary, which, in particular, implies that e tA = 1, t ∈ R.
Hence, in this case, Proposition 3.1 acquires the following form. For an anti-self-adjoint operator A in a complex Hilbert space, the generated by it strongly continuous group e tA t∈R is unitary, and hence, non-hypercyclic.

An Application
Since, in the complex Hilbert space L 2 (R), the differentiation operator A := d dx with the domain W 1 2 (R) := {f ∈ L 2 (R)|f (·) is absolutely continuous on R with f ′ ∈ L 2 (R)} is anti-self-adjoint (see, e.g., [1]), and hence, the generated by it left-translation group is strongly continuous and unitary [9,23], by the Corollaries 4.1 and 4.2, we obtain

Concluding Remark
The exponentials given by (2.4) describing all weak/mild solutions of equation (2.5) (see Remarks 2.1), Theorem 3.1, in particular, implies that the latter is void of chaos (cf. [10]). The same, by Proposition 3.1, is true also for equation (2.7) provided σ(A) ⊆ iR.

Acknowledgments
The author would like to express sincere appreciation to his colleague, Dr. Oscar Vega of the Department of Mathematics, California State University, Fresno, for his gift of the book [11], reading which inspired the above findings.