We show that there are linear operators on Hilbert space that have n-dimensional subspaces with dense orbit, but no (n-1)-dimensional subspaces with dense orbit. This leads to a new class of operators, called the n-supercyclic operators. We show that many cohyponormal operators are n-supercyclic. Furthermore, we prove that for an n-supercyclic operator, there are n circles centered at the origin such that every component of the spectrum must intersect one of these circles.

We show that every class A operator has a scalar extension. In particular, such operators with rich spectra have nontrivial invariant subspaces. Also we give some spectral properties of the scalar extension of a class A operator. Finally, we show that every class A operator is nonhypertransitive.

The Cauchy dual operator T’, given by $T{(T*T)}^{-1}$, provides a bounded unitary invariant for a closed left-invertible T. Hence, in some special cases, problems in the theory of unbounded Hilbert space operators can be related to similar problems in the theory of bounded Hilbert space operators. In particular, for a closed expansive T with finite-dimensional cokernel, it is shown that T admits the Cowen-Douglas decomposition if and only if T’ admits the Wold-type decomposition (see Definitions 1.1 and 1.2 below)....

We introduce and discuss a class of operators, to be referred to as operators close to isometries. The Bergman-type operators, 2-hyperexpansions, expansive p-isometries, and certain alternating hyperexpansions are main examples of such operators. We establish a few decomposition theorems for operators close to isometries. Applications are given to the theory of p-isometries and of hyperexpansive operators.

2000 Mathematics Subject Classification: Primary 47B47, 47B10; Secondary 47A30.Let H be a separable infinite dimensional complex Hilbert space and let L(H) denote the algebra of all bounded linear operators on H into itself. Given A ∈ L(H), the derivation δA : L(H)→ L(H) is defined by δA(X) = AX-XA. In this paper we prove that if A is an n-multicyclic hyponormal operator and T is hyponormal such that AT = TA, then || δA(X)+T|| ≥ ||T|| for all X ∈ L(H). We establish the same inequality if A is...

We calculate the spectral multiplicity of the direct sum T⊕ A of a weighted shift operator T on a Banach space Y which is continuously embedded in $l^p$ and a suitable bounded linear operator A on a Banach space X.

This paper considers weak supercyclicity for bounded linear operators on a normed space. On the one hand, weak supercyclicity is investigated for classes of Hilbert-space operators: (i) self-adjoint operators are not weakly supercyclic, (ii) diagonalizable operators are not weakly $l$-sequentially supercyclic, and (iii) weak $l$-sequential supercyclicity is preserved between a unitary operator and its adjoint. On the other hand, weak supercyclicity is investigated for classes of normed-space operators:...

An operator T on a Banach space ℬ is said to be hypercyclic if there exists a vector x such that the orbit ${Tⁿx}_{n\ge 1}$ is dense in ℬ. Hypercyclicity is a strong kind of cyclicity which requires that the linear span of the orbit is dense in ℬ. If the arithmetic means of the orbit of x are dense in ℬ then the operator T is said to be Cesàro-hypercyclic. Apparently Cesàro-hypercyclicity is a strong version of hypercyclicity. We prove that an operator is Cesàro-hypercyclic if and only if there exists a vector...

Let T be a bounded linear operator on a (real or complex) Banach space X. If (aₙ) is a sequence of non-negative numbers tending to 0, then the set of x ∈ X such that ||Tⁿx|| ≥ aₙ||Tⁿ|| for infinitely many n’s has a complement which is both σ-porous and Haar-null. We also compute (for some classical Banach space) optimal exponents q > 0 such that for every non-nilpotent operator T, there exists x ∈ X such that $\left(\right|\left|Tⁿx\right||/\left|\right|Tⁿ\left|\right|)\notin {\ell }^q\left(\mathbb{N}\right)$, using techniques which involve the modulus of asymptotic uniform smoothness of X.

We study the orbits of isometries of L¹[0,1]. For a certain class of isometries we show that the set of functions f in L¹[0,1] for which the orbit of f under the isometry T is equivalent to the usual canonical basis e₁,e₂,e₃,... of l¹ is an open dense set. In the proof we develop a new method to get copies of l¹ inside L¹[0,1] using geometric progressions. This method does not use disjoint or relatively disjoint supports, which seems to be the most common way to get such copies. We also use this...

A bounded linear operator T defined on a Banach space X is said to be supercyclic if there exists a vector x ∈ X such that the projective orbit {λTⁿx : λ ∈ ℂ, n ∈ ℕ} is dense in X. The aim of this survey is to show the relationship between positivity and supercyclicity. This relationship comes from the so called Positive Supercyclicity Theorem. Throughout this exposition, interesting new directions and open problems will appear.

In these notes we report on recent progress in the theory of hypercyclic and chaotic operators. Our discussion will be guided by the following fundamental problems: How do we recognize hypercyclic operators? How many vectors are hypercyclic? How many operators are hypercyclic? How big can non-dense orbits be?

Let $X$ be a Banach space, $ℬ\left(X\right)$ the algebra of bounded linear operators on $X$ and $(J,\parallel ·{\parallel }_J)$ an admissible Banach ideal of $ℬ\left(X\right)$. For $T\in ℬ\left(X\right)$, let $L_{J,T}$ and $R_{J,T}\in ℬ\left(J\right)$ denote the left and right multiplication defined by $L_{J,T}\left(A\right)=TA$ and $R_{J,T}\left(A\right)=AT$, respectively. In this paper, we study the transmission of some concepts related to recurrent operators between $T\in ℬ\left(X\right)$, and their elementary operators $L_{J,T}$ and $R_{J,T}$. In particular, we give necessary and sufficient conditions for $L_{J,T}$ and $R_{J,T}$ to be sequentially recurrent. Furthermore, we prove that $L_{J,T}$ is recurrent if and only...

An eigenvalue criterion for hypercyclicity due to the first author is improved. As a consequence, some new sufficient conditions for a sequence of infinite order linear differential operators to be hypercyclic on the space of holomorphic functions on certain domains of ${ℂ}^N$ are shown. Moreover, several necessary conditions are furnished. The equicontinuity of a family of operators as above is also studied, and it is characterized if the domain is ${ℂ}^N$. The results obtained extend or improve earlier work...

We study the ``smallness'' of the set of non-hypercyclic vectors for some classical hypercyclic operators.

Let T be a continuous linear operator on a Hausdorff topological vector space 𝓧 over the field ℂ. We show that if T is N-supercyclic, i.e., if 𝓧 has an N-dimensional subspace whose orbit under T is dense in 𝓧, then T* has at most N eigenvalues (counting geometric multiplicity). We then show that N-supercyclicity cannot occur nontrivially in the finite-dimensional setting: the orbit of an N-dimensional subspace cannot be dense in an (N+1)-dimensional space. Finally, we show that a subnormal operator...