scholarly journals Szemerédi's theorem, frequent hypercyclicity and multiple recurrence

2012 ◽  
Vol 110 (2) ◽  
pp. 251 ◽  
Author(s):  
George Costakis ◽  
Ioannis Parissis
Keyword(s):  
Banach Space ◽  
Linear Operator ◽  
Multiplication Operator ◽  
Complex Banach Space ◽  
Multiplication Operators ◽  
Complex Numbers ◽  
Multiple Recurrence ◽  
Weighted Shifts ◽  
Bounded Linear ◽  
Previous Assumption

Let $T$ be a bounded linear operator acting on a complex Banach space $X$ and $(\lambda_n)_{n\in\mathsf{N}}$ a sequence of complex numbers. Our main result is that if $|\lambda_n|/|\lambda_{n+1}|\to 1$ and the sequence $(\lambda_n T^n)_{n\in\mathsf{N}}$ is frequently universal then $T$ is topologically multiply recurrent. To achieve such a result one has to carefully apply Szemerédi's theorem in arithmetic progressions. We show that the previous assumption on the sequence $( \lambda_n)_{n\in\mathsf{N}}$ is optimal among sequences such that $|\lambda_{n}|/|\lambda_{n+1}|$ converges in $[0,\infty]$. In the case of bilateral weighted shifts and adjoints of multiplication operators we provide characterizations of topological multiple recurrence in terms of the weight sequence and the symbol of the multiplication operator respectively.

On Certain Classes of Bounded Linear Operators

1970 ◽  
Vol 13 (4) ◽  
pp. 469-473
Author(s):  
C-S Lin
Keyword(s):  
Banach Space ◽  
Linear Operator ◽  
Banach Algebra ◽  
Fredholm Operator ◽  
Bounded Linear Operator ◽  
Complex Banach Space ◽  
Linear Operators ◽  
Complex Numbers ◽  
Bounded Linear Operators ◽  
Bounded Linear

Let T—c be a Fredholm operator, where T is a bounded linear operator on a complex Banach space and c is a scalar, the set of all such scalars is called the Φ-set of T [2] and was studied by many authors. In this connection, the purpose of the present paper is to investigate some classes Φ(V) of all such operators for any subset V of the complex plane.Let X be a Banach space over the field C of complex numbers with dim Z = ∞, unless otherwise stated, B(X) the Banach algebra of all bounded linear operators and K(X) the closed two-sided ideal of all compact operators on X.

A Note on the Caradus Class of Bounded Linear Operators on a Complex Banach Space

1969 ◽  
Vol 21 ◽  
pp. 592-594 ◽  
Author(s):  
A. F. Ruston
Keyword(s):  
Banach Space ◽  
Linear Operator ◽  
Finite Number ◽  
Bounded Linear Operator ◽  
Present Note ◽  
Complex Banach Space ◽  
Linear Operators ◽  
Bounded Linear Operators ◽  
Bounded Linear ◽  
The Subject

1. In a recent paper (1) on meromorphic operators, Caradus introduced the class of bounded linear operators on a complex Banach space X. A bounded linear operator T is put in the class if and only if its spectrum consists of a finite number of poles of the resolvent of T. Equivalently, T is in if and only if it has a rational resolvent (8, p. 314).Some ten years ago (in May, 1957), I discovered a property of the class g which may be of interest in connection with Caradus' work, and is the subject of the present note.2. THEOREM. Let X be a complex Banach space. If T belongs to the class, and the linear operator S commutes with every bounded linear operator which commutes with T, then there is a polynomial p such that S = p(T).

scholarly journals Expansion of analytic functions of an operator in series of Faber polynomials

1997 ◽  
Vol 56 (2) ◽  
pp. 303-318 ◽  
Author(s):  
Maurice Hasson
Keyword(s):  
Banach Space ◽  
Linear Operator ◽  
Analytic Functions ◽  
Bounded Linear Operator ◽  
Complex Banach Space ◽  
Faber Polynomials ◽  
Bounded Linear ◽  
In Series ◽  
Image Position

Let T: B → B be a bounded linear operator on the complex Banach space B and let f(z) be analytic on a domain D containing the spectrum Sp(T) of T. Then f(T) is defined bywhere C is a contour surrounding SP(T) and contained in D.

scholarly journals Perturbations of local C-cosine functions

2020 ◽  
Vol 65 (4) ◽  
pp. 585-597
Author(s):  
Chung-Cheng Kuo
Keyword(s):  
Banach Space ◽  
Linear Operator ◽  
Bounded Linear Operator ◽  
Complex Banach Space ◽  
Cosine Function ◽  
Bounded Linear ◽  
Cosine Functions

"We show that $\tA+\tB$ is a closed subgenerator of a local $\tC$-cosine function $\tT(\cdot)$ on a complex Banach space $\tX$ defined by $$\tT(t)x=\sum\limits_{n=0}^\infty \tB^n\int_0^tj_{n-1}(s)j_n(t-s)\tC(|t-2s|)xds$$ for all $x\in\tX$ and $0\leq t<T_0$, if $\tA$ is a closed subgenerator of a local $\tC$-cosine function $\tC(\cdot)$ on $\tX$ and one of the following cases holds: $(i)$ $\tC(\cdot)$ is exponentially bounded, and $\tB$ is a bounded linear operator on $\overline{\tD(\tA)}$ so that $\tB\tC=\tC\tB$ on $\overline{\tD(\tA)}$ and $\tB\tA\subset\tA\tB$; $(ii)$ $\tB$ is a bounded linear operator on $\overline{\tD(\tA)}$ which commutes with $\tC(\cdot)$ on $\overline{\tD(\tA)}$ and $\tB\tA\subset\tA\tB$; $(iii)$ $\tB$ is a bounded linear operator on $\tX$ which commutes with $\tC(\cdot)$ on $\tX$. Here $j_n(t)=\frac{t^n}{n!}$ for all $t\in\Bbb R$, and $$\int_0^tj_{-1}(s)j_0(t-s)\tC(|t-2s|)xds=\tC(t)x$$ for all $x\in\tX$ and $0\leq t<T_0$."

Representations and Divisibility of Operator Polynomials

1978 ◽  
Vol 30 (5) ◽  
pp. 1045-1069 ◽  
Author(s):  
I. Gohberg ◽  
P. Lancaster ◽  
L. Rodman
Keyword(s):  
Banach Space ◽  
Complex Banach Space ◽  
Linear Operators ◽  
Complex Numbers ◽  
Bounded Linear Operators ◽  
Bounded Linear ◽  
Image Position

Let be a complex Banach space and the algebra of bounded linear operators on . In this paper we study functions from the complex numbers to of the form

scholarly journals Non-recurrence sets for weakly mixing linear dynamical systems

2012 ◽  
Vol 34 (1) ◽  
pp. 132-152 ◽  
Author(s):  
SOPHIE GRIVAUX
Keyword(s):  
Banach Space ◽  
Dynamical Systems ◽  
Linear Operator ◽  
Gaussian Measure ◽  
Bounded Linear Operator ◽  
Probability Space ◽  
Complex Banach Space ◽  
Bounded Linear ◽  
Linear Dynamical Systems ◽  
Weakly Mixing

AbstractWe study non-recurrence sets for weakly mixing dynamical systems by using linear dynamical systems. These are systems consisting of a bounded linear operator acting on a separable complex Banach space$X$, which becomes a probability space when endowed with a non-degenerate Gaussian measure. We generalize some recent results of Bergelson, del Junco, Lemańczyk and Rosenblatt, and show in particular that sets$\{n_{k}\}$such that$n_{k+1}/n_{k}\to +\infty $, or such that$n_{k}$divides$n_{k+1}$for each$k\ge 0$, are non-recurrence sets for weakly mixing linear dynamical systems. We also give examples, for each$r\ge 1$, of$r$-Bohr sets which are non-recurrence sets for some weakly mixing systems.

An unbounded spectral mapping theorem

1968 ◽  
Vol 8 (1) ◽  
pp. 119-127 ◽  
Author(s):  
S. J. Bernau
Keyword(s):  
Banach Space ◽  
Linear Operator ◽  
Closed Subset ◽  
Complex Banach Space ◽  
Complex Numbers ◽  
Spectral Mapping Theorem ◽  
Complex Polynomials ◽  
Spectral Mapping ◽  
Image Position ◽  
Densely Defined

Recall that the spectrum, σ(T), of a linear operator T in a complex Banach space is the set of complex numbers λ such that T—λI does not have a densely defined bounded inverse. It is known [7, § 5.1] that σ(T) is a closed subset of the complex plane C. If T is not bounded, σ(T) may be empty or the whole of C. If σ(T) ≠ C and T is closed the spectral mapping theorem, is valid for complex polynomials p(z) [7, §5.7]. Also, if T is closed and λ ∉ σ(T), (T–λI)−1 is everywhere defined.

scholarly journals On generalization of decomposability

1981 ◽  
Vol 22 (1) ◽  
pp. 77-81 ◽  
Author(s):  
Ridgley Lange
Keyword(s):  
Banach Space ◽  
Linear Operator ◽  
Invariant Subspace ◽  
Bounded Linear Operator ◽  
Invariant Subspaces ◽  
Complex Banach Space ◽  
Open Cover ◽  
Bounded Linear ◽  
Image Position ◽  
Finite Open Cover

Let X be a complex Banach space and let T be a bounded linear operator on X. Then T is decomposable if for every finite open cover of σ(T) there are invariant subspaces Yi(i= 1, 2, …, n) such that(An invariant subspace Y is spectral maximal [for T] if it contains every invariant subspace Z for which σ(T|Z) ⊂ σ(T|Y).).

scholarly journals A result on hermitian operators

1989 ◽  
Vol 31 (1) ◽  
pp. 71-72
Author(s):  
J. E. Jamison ◽  
Pei-Kee Lin
Keyword(s):  
Banach Space ◽  
Hilbert Space ◽  
Linear Operator ◽  
Bounded Linear Operator ◽  
Numerical Range ◽  
Complex Banach Space ◽  
Bounded Linear ◽  
Image Position

Let X be a complex Banach space. For any bounded linear operator T on X, the (spatial) numerical range of T is denned as the setIf V(T) ⊆ R, then T is called hermitian. Vidav and Palmer (see Theorem 6 of [3, p. 78] proved that if the set {H + iK:H and K are hermitian} contains all operators, then X is a Hilbert space. It is natural to ask the following question.

scholarly journals The n-inverses of a matrix

Filomat ◽  
2017 ◽  
Vol 31 (12) ◽  
pp. 3801-3813
Author(s):  
Caixing Gu ◽  
Heidi Keas ◽  
Robert Lee
Keyword(s):  
Banach Space ◽  
Linear Operator ◽  
Bounded Linear Operator ◽  
Tensor Products ◽  
Complex Banach Space ◽  
Spectral Characterization ◽  
Bounded Linear ◽  
Nilpotent Operators ◽  
Nilpotent Matrices

The concept of a left n-inverse of a bounded linear operator on a complex Banach space was introduced recently. Previously, there have been results on products and tensor products of left n-inverses, and the representation of left n-inverses as the sum of left inverses and nilpotent operators was being discussed. In this paper, we give a spectral characterization of the left n-inverses of a finite (square) matrix. We also show that a left n-inverse of a matrix T is the sum of the inverse of T and two nilpotent matrices.

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