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On Power Bounded Operators

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Abstract

The main result asserts that, for any contraction T on an arbitrary Banach space X, ∥ Tn − Tn + 1 ∥ → 0 as n → ∞, if and only if the spectrum of T has no points on the unit circle except perhaps z = 1. This theorem is extended for ϑ(T)Tn, where ϑ is a function of spectral synthesis on the unit circle. As an application, we generalize the so-called “zero-two” law of Ornstein and Sucheston and Zaharopol to positive contraction on a very large class of Banach lattices.

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... The goal of this article is to considerably improve various complex Tauberian theorems for Laplace transforms and power series. In particular, we shall refine and extend a number of results from [1,2,19,20,22,24,26,27,36]. Most of the theorems from those articles can be considered as generalizations of the classical version of the Fatou-Riesz theorem for Laplace transforms by Ingham [19] that we state below, or as extensions of the Katznelson-Tzafriri theorem [22] for power series which we will generalize in Section 6 (Theorem 6.4). ...
... In particular, we shall refine and extend a number of results from [1,2,19,20,22,24,26,27,36]. Most of the theorems from those articles can be considered as generalizations of the classical version of the Fatou-Riesz theorem for Laplace transforms by Ingham [19] that we state below, or as extensions of the Katznelson-Tzafriri theorem [22] for power series which we will generalize in Section 6 (Theorem 6.4). Our improvements consist, on the one hand, in relaxing the boundary behavior of Laplace transforms (power series) to local pseudofunction behavior, with possibly exceptional null sets of boundary singularities, and, on the other hand, by simultaneously considering one-sided Tauberian conditions on the functions (sequences). ...
... Our improvements consist, on the one hand, in relaxing the boundary behavior of Laplace transforms (power series) to local pseudofunction behavior, with possibly exceptional null sets of boundary singularities, and, on the other hand, by simultaneously considering one-sided Tauberian conditions on the functions (sequences). It should be pointed out that the use of pseudofunctions in Tauberian theory was initiated by the seminal work of Katznelson and Tzafriri [22]. More recently, Korevaar has written a series of papers [24,26,27] that emphasize the role of local pseudofunction boundary behavior as optimal boundary condition in complex Tauberian theorems for Laplace transforms, see also his book [25]. ...
Preprint
We provide several Tauberian theorems for Laplace transforms with local pseudofunction boundary behavior. Our results generalize and improve various known versions of the Ingham-Fatou-Riesz theorem and the Wiener-Ikehara theorem. Using local pseudofunction boundary behavior enables us to relax boundary requirements to a minimum. Furthermore, we allow possible null sets of boundary singularities and remove unnecessary uniformity conditions occurring in earlier works; to this end, we obtain a useful characterization of local pseudofunctions. Most of our results are proved under one-sided Tauberian hypotheses; in this context, we also establish new boundedness theorems for Laplace transforms with pseudomeasure boundary behavior. As an application, we refine various results related to the Katznelson-Tzafriri theorem for power series.
... and we are able to use the coefficients a 11 = 1, a 12 = 0, a 11 = −1, a 12 = 1 to conclude by Theorem 6 that lim m→∞ T m − P op = 0, 2 where the finite-rank projection P : C([0, 1]) → ker(T − I) is defined for f ∈ C([0, 1]) by P f = (a 11 δ 0 (f ) + a 12 δ 1 (f )) · 1 + (a 21 δ 1 (f ) + a 22 δ 0 (f )) · x = δ 0 (f ) ...
... Under the assumption of Theorem 1, the projection operator P defined by (11) has finite-rank and is bounded by ...
... The first case has already been characterized by Katznelson and Tzafriri [11], who have been shown that for every linear operator T on a Banach space X with T op ≤ 1 the limit ...
Preprint
It is well known that iterates of quasi-compact operators converge towards a spectral projection, whereas the explicit construction of the limiting operator is in general hard to obtain. Here, we show a simple method to explicitly construct this projection operator, provided that the fixed points of the operator and its adjoint are known which is often the case for operators used in approximation theory. We use an approach related to Riesz-Schauder and Fredholm theory to analyze the iterates of operators on general Banach spaces, while our main result remains applicable without specific knowledge on the underlying framework. Applications for Markov operators on the space of continuous functions C(X) are provided, where X is a compact Hausdorff space.
... On the other hand, in the infinite dimensional case, by harmonic analysis and operator theory, many results on the asymp-totic behavior of solutions of Eq. (1.1) have been obtained, see e.g. [1,3,5,6,7,8,11,14,18,19,20]. Among many interesting results in this direction is a famous theorem due to Katznelson-Tzafriri (see [11]) saying that if T is a bounded operator in a Banach space X such that There are a lot of extensions and improvements of this result as well as simple proofs of it, see e.g. ...
... [1,3,5,6,7,8,11,14,18,19,20]. Among many interesting results in this direction is a famous theorem due to Katznelson-Tzafriri (see [11]) saying that if T is a bounded operator in a Banach space X such that There are a lot of extensions and improvements of this result as well as simple proofs of it, see e.g. [1,3,6,8,11,14,18,19,20] and the references therein. ...
... Among many interesting results in this direction is a famous theorem due to Katznelson-Tzafriri (see [11]) saying that if T is a bounded operator in a Banach space X such that There are a lot of extensions and improvements of this result as well as simple proofs of it, see e.g. [1,3,6,8,11,14,18,19,20] and the references therein. ...
Article
In this paper using a transform defined by the translation operator we introduce the concept of spectrum of sequences that are bounded by a given polynomial. We apply this spectral theory to study the asymptotic behavior of solutions of fractional linear difference equations. One of the obtained results is an extension of a famous Katznelson-Tzafriri Theorem, saying that the α-resolvent operator that is associated with the fractional equation, satisfies an asymptotic estimate of Katznelson-Tzafriri type, provided that it is bounded by the polynomial, and the spectrum of the fractional equation on the unit circle is either empty or consists of only one element 1. Three concrete examples are also included to illustrate the obtained results.
... Katznelson and Tzafriri proved the following result in [49,Theorem 5]. ...
... (i) In order for the theorem to be non-trivial, σ u (T ) must be contained in the zero set of f in T for some non-zero f ∈ W + (D). This implies that σ u (T ) must be of measure zero with respect to Lebesgue measure on T. (ii) If E is a countable closed subset of T, then E is a set of spectral synthesis for W (T). (iii) There exist closed subsets E of T which have measure zero and are not of spectral synthesis for W (T). The proof of Theorem 2.1 in [49] used methods from harmonic analysis. Vũ [86] gave a very short proof based on a functional analytic construction and an application of spectral theory for an invertible isometry. ...
... In this section, we consider the case when f (z) = 1 − z. This was the case which inspired the authors of [49] to obtain Theorem 2.1, and it is the case which is most often used. ...
Article
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This paper is a survey article on developments arising from a theorem proved by Katznelson and Tzafriri in 1986 showing that limnTn(IT)=0{\rm lim}_{n \rightarrow \infty} ||T^{n}(I-T)|| = 0 if T is a power-bounded operator on a Banach space and σ(T)T{1}\sigma (T)\cap \mathbb{T} \subseteq\{1\}. Many variations and consequences of the original theorem have been proved subsequently, and we provide an account of this branch of operator theory.
... In 1986, Katznelson and Tzafriri [11] proved a theorem concerning asymptotics of the discrete semigroup (T n ) n≥0 for a power-bounded operator T on a complex Banach space X. They showed that lim n→∞ T n (I − T ) = 0 if σ(T ) ∩ T ⊆ {1}. ...
... Let W (T) be the space of all functions on T of the form g(z) = ∞ k=−∞ b k z k , where g W (T) := ∞ k=−∞ |b k | < ∞. It was shown in [11] that lim n→∞ T n f (T ) = 0 if f ∈ W + (D) and is of spectral synthesis in W (T) with respect to σ(T ) ∩ T. This assumption means that there exist functions (g k ) k≥1 in W (T) such that each g k vanishes on a neighbourhood U k of σ(T ) ∩ T in T and lim k→∞ g k − f W (T) = 0. ...
... The following theorem is an analogue of the original theorem for bounded C 0 -semigroups, and it was proved in [9] and [17]. Their proofs were quite different from the proofs in [11] and from each other. For a discussion of other proofs, see [4,Section 3.1]. ...
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We prove a continuous-parameter version of the recent theorem of Katznelson-Tzafiri type for power-bounded operators which have a bounded calculus for analytic Besov functions.
... This paper is a survey article on developments arising from a theorem proved by Katznelson and Tzafriri in 1986 showing that limn→∞ T n (I − T ) = 0 if T is a power-bounded operator on a Banach space and σ(T ) ∩ T ⊆ {1}. Many variations and consequences of the original theorem have been proved subsequently, and we provide an account of this branch of operator theory. ...
... The main theorem. Katznelson and Tzafriri proved the following result in [48,Theorem 5]. ...
... The proof of Theorem 2.1 in [48] used methods from harmonic analysis. Vũ [85] gave a very short proof based on a functional analytic construction and an application of spectral theory for an invertible isometry. ...
Preprint
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This paper is a survey article on developments arising from a theorem proved by Katznelson and Tzafriri in 1986 showing that limnTn(IT)=0\lim_{n\to\infty} \|T^n(I-T)\| =0 if T is a power-bounded operator on a Banach space and \sigma(T) \cap \T \subseteq \{1\}. Many variations and consequences of the original theorem have been proved subsequently, and we provide an account of this branch of operator theory.
... On the other hand, in the infinite dimensional case, by Harmonic Analysis and Operator Theory, many results on the asymptotic behavior of solutions of Eq. (1.1) have been obtained, see e.g. [1,3,4,5,6,7,9,11,14,15,16]. Among many interesting results in this direction is a famous theorem due to Katznelson-Tzafriri (see [9]) saying that if T is a bounded operator in a Banach space X such that There are a lot of extensions and improvements of this result as well as simple proofs of it, see e.g. ...
... [1,3,4,5,6,7,9,11,14,15,16]. Among many interesting results in this direction is a famous theorem due to Katznelson-Tzafriri (see [9]) saying that if T is a bounded operator in a Banach space X such that There are a lot of extensions and improvements of this result as well as simple proofs of it, see e.g. [1,3,5,7,9,11,14,15,16] and the references therein. ...
... Among many interesting results in this direction is a famous theorem due to Katznelson-Tzafriri (see [9]) saying that if T is a bounded operator in a Banach space X such that There are a lot of extensions and improvements of this result as well as simple proofs of it, see e.g. [1,3,5,7,9,11,14,15,16] and the references therein. ...
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In this paper using a transform defined by the translation operator we introduce the concept of spectrum of sequences that are bounded by nνn^\nu, where ν\nu is a natural number. We apply this spectral theory to study the asymptotic behavior of solutions of fractional difference equations of the form Δαx(n)=Tx(n)+y(n)\Delta^\alpha x(n)=Tx(n)+y(n), nNn\in \N, where 0<α10<\alpha\le 1. One of the obtained results is an extension of a famous Katznelson-Tzafriri Theorem, saying that if the α\alpha-resolvent operator SαS_\alpha satisfies supnNSα(n)/nν<\sup_{n\in\N} \| S_\alpha (n)\| /n^\nu <\infty and the set of z_0\in \C such that (zk~α(z)T)1(z-\tilde k^\alpha (z)T)^{-1} exists, and together with k~α(z)\tilde k^\alpha (z), is holomorphic in a neighborhood of z0z_0 consists of at most 1, where k~α(z) \tilde k^\alpha (z) is the Z-transform of kα(n):=Γ(α+n)/(Γ(α)Γ(n+1))k^\alpha (n):= \Gamma (\alpha +n)/(\Gamma (\alpha )\Gamma (n+1)), then \begin{align*} \lim_{n\to \infty} \frac{1}{n^\nu} \sum_{k=0}^{\nu+1} \frac{(\nu+1)!}{k!(\nu+1-k)!} (-1)^{\nu+1+k} S_\alpha (n+k) =0. \end{align*}
... Let us here mention an interesting result of Katznelson and Tzafriri, which can be regarded as a dual of Theorem 8.1. (see [5,13,14]). Analogous results concerning G = R can be obtained. ...
... Katznelson and Tzafriri, 1986) Let {a n } n≥0 be a bounded sequence of complex numbers. Suppose that the analytic function f (z) 1 boundary behavior on the unit circle except for z = 1. ...
Article
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We study a summability method called almost convergence for bounded measurable functions defined on a locally compact abelian group. We define almost convergence using topologically invariant means and exhibit two different kinds of necessary and sufficient conditions, one is analytic and the other is functional analytic, for a given function to be almost convergent. As an application, we show complex Tauberian theorems for almost convergence on the integers and the real numbers. These results are closely related to some of the classical Tauberian theorems like the Ingham–Karamata and Katznelson–Tzafriri theorems.
... In most cases, f ∈ A(D) and then σ u (T ) is of measure zero, unless f is the zero function. The fundamental version, proved by Katznelson and Tzafriri in [20,Theorem 5], is as follows. ...
... The case when σ u (T ) ⊆ {1} and f (z) = 1 − z is a particularly important corollary of Theorem 1.1, and it was stated in [20,Theorem 1]. ...
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Let T be a power-bounded operator on a Banach space X, A\mathcal{A} be a Banach algebra of bounded holomorphic functions on the unit disc D\mathbb{D}, and assume that there is a bounded functional calculus for the operator T, so there is a bounded algebra homomorphism mapping functions fAf \in \mathcal{A} to bounded operators f(T) on X. Theorems of Katznelson-Tzafriri type establish that limnTnf(T)=0\lim_{n\to\infty} \|T^n f(T)\| = 0 for functions fAf \in \mathcal{A} whose boundary functions vanish on the unitary spectrum σ(T)T\sigma(T)\cap \mathbb{T} of T, or sometimes satisfy a stronger assumption of spectral synthesis. We consider the case when A\mathcal{A} is a standard Banach algebra B(D)\mathcal{B}(\mathbb{D}) of analytic Besov functions on D\mathbb{D}. We set out the relevant properties of the algebra B(D)\mathcal{B}(\mathbb{D}), and we define the B(D)\mathcal{B}(\mathbb{D})-calculus in an explicit way. We show in particular that the existence of a bounded B(D)\mathcal{B}(\mathbb{D})-calculus for T is equivalent to an integral condition for the resolvent of T. The B(D)\mathcal{B}(\mathbb{D})-calculus has been briefly touched on in some previous papers, but our approach is new and it follows the analogous theory for bounded C0C_0-semigroups developed in two recent papers. This approach enables us to prove a Katznelson-Tzafriri theorem for the B(D)\mathcal{B}(\mathbb{D})-calculus which extends several previous results.
... In [6] Katznelson and Tzafriri studied the asymptotic behavior of the sequence {T n } ∞ n=1 , where T is a power bounded operator in a Banach space X (that is sup n∈N T n < ∞). A famous result of the paper [6, Theorem 1] says that lim n→∞ (T n+1 − T n ) = 0 if and only if σ(T ) ∩ {z ∈ C : |z| = 1} ⊂ {1}. ...
... Moreover, as in [19, p. 414] the eigenvalues of A on iR are determined from the set of solutions of the equations This is the "if" part of [6, Theorem 1]. The reader can find various proofs of this result in [2,6,9,16]. ...
Article
We consider a Katznelson-Tzafriri type theorem for linear difference equations of the form x(n+1)=Tx(n)+y(n) ()x(n+1)=Tx(n)+y(n)\ (*), where T is a bounded operator in a Banach space \X and {y(n)}n=1\{y(n)\}_{n=1}^\infty is a bounded sequence that is asymptotically constant, that is, limn[y(n+1)y(n)]=0\lim_{n\to\infty} [y(n+1)-y(n)]=0. A sequence {u(n)}n=1\{u(n)\}_{n=1}^\infty is said to be an asymptotic solution to ()(*) if u(n+1)=Tu(n)+y(n)+ϵ(n)u(n+1)=Tu(n)+y(n)+\epsilon (n), where limnϵ(n)=0\lim_{n\to\infty}\epsilon (n)=0. We show that if 1 is either not in \sigma (T)\cap \{z\in \C:\ |z|=1\}, or is its isolated element, then if ()(*) has a bounded asymptotic solution, it has an asymptotic solution that is asymptotically constant. Furthermore, if \sigma (T)\cap \{z\in \C:\ |z|=1\}\subset \{ 1\}, then every asymptotic solution of ()(*) is asymptotic constant. An application to the evolution periodic equations is given.
... We point out that the formulated result is true for p = 2 and other symmetric spaces (see [13]). Extensions to general Banach lattices were made in [23,24] and [15]. ...
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In recent years, great attention has been given to abstract state spaces. Also, there are many papers dedicated to zero-two law. In the present paper, we will investigate generalizations of the dominated and uniform zero-two law results for Markov operators defined on abstract state spaces. As a consequence of the obtained results, we can produce several applications of the main theorems.
... It follows from Theorem 2.1 that the boundary ∂D of D satisfies σ(C γ ) ∩ ∂D = {1} = σ(C γ,0 ) ∩ ∂D. Since both C γ and C γ,0 are power bounded, it follows from Theorem 1 and the Remark on p.317 of [22] that lim n→∞ C n+1 γ − C n γ = 0 = lim n→∞ C n+1 γ,0 − C n γ,0 . (ii) Let γ > 1. Theorem 2.1 shows that λ = 1 is an isolated singularity of the resolvent map of both C γ and C γ,0 . ...
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The Cesaro operator C\mathsf{C}, when acting in the classical growth Banach spaces AγA^{-\gamma} and A0γA_0^{-\gamma}, for γ>0\gamma > 0 , of analytic functions on D\mathbb{D}, is investigated. Based on a detailed knowledge of their spectra (due to A. Aleman and A.-M. Persson) we are able to determine the norms of these operators precisely. It is then possible to characterize the mean ergodic and related properties of C\mathsf{C} acting in these spaces. In addition, we determine the largest Banach space of analytic functions on D\mathbb{D} which C\mathsf{C} maps into AγA^{-\gamma} (resp. into A0γA_0^{-\gamma}); this optimal domain space always contains AγA^{-\gamma} (resp. A0γA_0^{-\gamma}) as a proper subspace.
... From now on, we focus on stability results for the Cesàro mean differences of size n and n + 1 for bounded operators. In 1986, Y. Katznelson and L. Tzafriri proved that if T ∈ B(X) is power bounded, then lim n→∞ T n − T n+1 = 0 if and only if σ(T ) ∩ ∂D ⊂ {1}, see [13,Theorem 1]. If T is (C, 1)-Cesàro bounded and σ(T ) ∩ ∂D = {1}, but T is not power bounded, then T n − T n+1 need not converge to zero. ...
Preprint
In this paper, we extend the concept of absolutely Ces\`aro boundedness to the fractional case. We construct a weighted shift operator belonging to this class of operators, and we prove that if T is an absolutely Ces\`{a}ro bounded operator of order α\alpha with 0<α1,0<\alpha\le 1, then Tn=o(nα)\| T^n\|=o(n^{\alpha}), generalizing the result obtained for α=1\alpha =1. Moreover, if α>1\alpha > 1, then Tn=O(n)\|T^{n}\|= O(n). We apply such results to get stability properties for the Ces\`aro means of bounded operators.
... Our analysis in the next sections makes extensive use of complex Tauberian theorems for Laplace transforms, which we collect here together with some background material on related concepts for the reader's convenience. These Tauberian theorems are in terms of local pseudofunction boundary behavior [7,21,19,20,25], which turns out to be an optimal assumption on the Laplace transform, in the sense that it often leads to "if and only if" results. See also [12,28] for an L 1 loc -approach to "if and only if" type complex Tauberian theorems. ...
Preprint
In classical prime number theory several asymptotic relations are considered to be "equivalent" to the prime number theorem. In the setting of Beurling generalized numbers, this may no longer be the case. Under additional hypotheses on the generalized integer counting function, one can however still deduce various equivalences between the Beurling analogues of the classical PNT relations. We establish some of the equivalences under weaker conditions than were known so far.
... It follows that L n+1 − L n tends to zero (in norm, say) as n → ∞; and that obviously implies condition (iii) of Theorem 6.1. We remark that the asymptotic behavior of T n+1 − T n for contractions T on Banach spaces has been much-studied; see [KT86] and references therein. ...
Preprint
Starting with a unit-preserving normal completely positive map L: M --> M acting on a von Neumann algebra - or more generally a dual operator system - we show that there is a unique reversible system \alpha: N --> N (i.e., a complete order automorphism \alpha of a dual operator system N) that captures all of the asymptotic behavior of L, called the {\em asymptotic lift} of L. This provides a noncommutative generalization of the Frobenius theorems that describe the asymptotic behavior of the sequence of powers of a stochastic n x n matrix. In cases where M is a von Neumann algebra, the asymptotic lift is shown to be a W*-dynamical system (N,\mathbb Z), whick we identify as the tail flow of the minimal dilation of L. We are also able to identify the Poisson boundary of L as the fixed point algebra of (N,\mathbb Z). In general, we show the action of the asymptotic lift is trivial iff L is {\em slowly oscillating} in the sense that limnρLn+1ρLn=0,ρM. \lim_{n\to\infty}\|\rho\circ L^{n+1}-\rho\circ L^n\|=0,\qquad \rho\in M_* . Hence \alpha is often a nontrivial automorphism of N.
... Moreover, L M (∇ ω , μ ω ) has a weak unit. Then, due to Theorem 1.3 (see also [24,Theorem 6] ...
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The present paper deals with L0L0L_0-valued measures and the associated Orlicz–Kantorovich lattice. The considered Orlicz–Kantorovich lattice, endowed with the Luxemburg norm, is represented as a measurable bundle of classical Orlicz spaces associated with scalar measures. This kind of representation allows us to investigate positive contractions and apply the corresponding zero-two laws on the classical Orlicz spaces, to prove vector versions of “zero-two” laws on the considered Orlicz–Kantorovich space.
... We recall that the operator sequence {T n+1 − T n : n ≥ 0} was studied by Esterle [9], who proved that if T is power bounded, i.e., ∥T n ∥ ≤ C for all n ≥ 1, for some C > 0, and σ(T ) = {1}, then this sequence is always stable in operator norm. Some years later Katznelson and Tzafriri [23] provided a complete characterization for power bounded operators: T n+1 − T n → 0 in operator norm if and only if σ(T ) ⊆ D ∪ {1} (for further results see, for instance, Léka [25], Batty [3] and Esterle, Strouse and Zoukia [10], Mustafayev [30,31] and Zarrabi [41]). In the following, for f (z) = n≥0 a n z n , we set f (z) = n≥0 a n n! z n , and all norms of functions are taken in A 2,α q (D). ...
... If T is doubly power bounded operator, then σ (T ) ⊆ T. The classical Gelfand's theorem [12] states that if T is a doubly power bounded operator with σ (T ) = {1} , then T = I . A more general result has been proved by Katznelson and Tzafriri [6]: If T is a power bounded operator with σ u (T ) = {1}, then lim n→∞ T n+1 − T n = 0. ...
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In this note, we present some results concerning intertwining properties of two isometries on Banach spaces. In this connection, we obtain also some Katznelson–Tzafriri type results for power bounded operators.
... There are many classic results in this direction that are models for subsequent researches. Among many such works we refer the reader to very well known results by Massera [20] and Katznelson-Tsafriri [13] due to their very different techniques of study. ...
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In this paper we study the asymptotic behavior of solutions of fractional differential equations of the form DCαu(t)=Au(t)+f(t),u(0)=x,0<α1,() D^{\alpha}_Cu(t)=Au(t)+f(t), u(0)=x, 0<\alpha\le1, ( *) where DCαu(t)D^{\alpha}_Cu(t) is the derivative of the function u in the Caputo's sense, A is a linear operator in a Banach space \X that may be unbounded and f satisfies the property that limt(f(t+1)f(t))=0\lim_{t\to \infty} (f(t+1)-f(t))=0 which we will call asymptotic 1-periodicity. By using the spectral theory of functions on the half line we derive analogs of Katznelson-Tzafriri and Massera Theorems. Namely, we give sufficient conditions in terms of spectral properties of the operator A for all asymptotic mild solutions of Eq. (*) to be asymptotic 1-periodic, or there exists an asymptotic mild solution that is asymptotic 1-periodic.
... The open unit disc and the unit circle in the complex plane will be denoted by D and T, respectively. If T ∈ B (X) is power bounded then clearly, σ (T ) ⊆ D, where σ (T ) is the spectrum of T. The classical Katznelson-Tzafriri theorem [13] states ...
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Let G be a locally compact group with the left Haar measure mGm_{G}. A probability measure μ{\mu} on G is said to be strictly aperiodic if the support of μ{\mu} is not contained in a proper closed left coset of G. In this paper, we prove weight ergodic theorems for strictly aperiodic measures.
... Now we mention an interesting result of Katznelson and Tzafriri related to this theorem, which can be regarded as a dual of the above theorem. (see [7], [8]). Analogous results concerning G = R can be obtained. ...
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We study a summability method called almost convergence for bounded measurable functions defined on a locally compact abelian group. We define almost convergence using topologically invariant means and exhibit two different kinds of necessary and sufficient conditions, one is analytic and the other is functional analytic, for a given function to be almost convergent. As an application, we show complex Tauberian theorems for almost convergence on the integers and the real numbers. In particular, the latter one can be viewed as an analogue of the Wiener-Ikehara theorem.
... We point out that the formulated result is true for p = 2 and other symmetric spaces (see [10]). Extensions to general Banach lattices were made in [19,20] and [13]. ...
Article
In this paper, we generalize the zero-two law for positive contractions acting on Lp-spaces. Moreover, an application of the main result is provided.
... T. The classical Gelfand's theorem [12] states that if T is a doubly power bounded operator with (T ) = f1g ; then T = I: A more general result has been proved by Katznelson and Tzafriri [6]: If T is a power bounded operator with u (T ) = f1g, then lim n!1 T n+1 T n = 0: In [10], it was proved that if T is a Hilbert space contraction, then ...
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In this note, we present some results concerning intertwining properties of two isometries on Banach space. In this connections, we obtain also some Katznelson-Tzafriri type results for power bounded operators.
... Asymptotic periodicity in the sense we just defined above has a close relation with a property discussed in the classic result by Katznelson-Tzafriri (see e.g. [12]) saying that for a contraction T : X → X with σ (T ) ∩ ⊂ {1} one has lim n→∞ (T n+1 − T n ) = 0. In fact, this is equivalent to say that the sequence {x n } n∈N = {T n } n∈N is "asymptotic 1-periodic", that is, lim n→∞ (x n+1 − x n ) = 0. ...
Article
We present an analog of Massera Theorem for asymptotic periodic solutions of linear equations x′(t)=A(t)x(t)+f(t),t≥0(⁎), where the family of linear operators A(t) generates a 1-periodic process (U(t,s))t≥s≥0 in a Banach space X and f is asymptotic 1-periodic in the sense that limt→∞⁡(f(t+1)−f(t))=0. The main result says that if 1 is isolated in σ(U(1,0)) on the unit circle Γ, then (⁎) has an asymptotic 1-periodic mild solution if and only if it has an asymptotic mild solution that is bounded and uniformly continuous with precompact range. If 1∉σ(U(1,0))∩Γ, such an asymptotic 1-periodic mild solution always exists and unique within a function g(t) with limt→∞⁡g(t)=0. Our study relies on a spectral theory of functions on the half line and the evolution semigroups associated with linear equations. The obtained results seem to be new, even in the finite dimensional case.
... An operator T ∈ L(X) is power bounded if sup n≥1 T n < ∞. Y. Katznelson and L. Tzafriri proved in 1986 that for a power bounded T , we have σ(T ) ⊂ D ∪ {1} if and only if lim n→∞ (I − T )T n = 0, see [8]. Related to this, J. Zemánek asked in 1992 whether (1.1) implies lim n→∞ (I − T )T n = 0, too. ...
... Also Elke Wolf studied when weighted composition operators acting between weighted Banach spaces are power bounded. [6] is a good source to study about power bounded operators. ...
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In this paper, we investigate about power boundedness of weighted composition operators on Bloch space and we give some necessarily and sufficient conditions under which a weighted composition operator is power bounded on Bloch space.
... The theorem of Katznelson-Tzafriri [13] mentioned in the abstract is: Theorem 1.1 Let X be a Banach space and T : X → X a power bounded linear operator (that is, Sup n≥1 ||T n || < ∞). Then lim n→∞ ||T n+1 − T n || → 0 iff σ(T) ∩ Π ⊆ {1}. ...
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... We obtain the following interesting spectral condition for relatively compact orbits: If T ∈ L (E) is a power-bounded operator on a reflexive Banach space E with peripheral spectrum σ(T ) ∩ T ⊆ {1}, then T has relatively compact orbits. Indeed, this follows from (a) and the Katznelson-Tzafriri theorem, the latter stating that under this spectral condition one has T n (I − T ) → 0 as n → ∞, see [13]. ...
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We investigate for a bounded semigroup of linear operators S on a Banach space E and a vector xEx \in E, when relative compactness of S(IT)xS(I-T)x for every TST \in S implies relative compactness of the orbit Sx. In particular, we derive characterizations of separable Banach spaces not containing c0\mathrm{c}_0 and of reflexivity of Banach spaces with an unconditional Schauder basis in terms of such compactness results.
... In this section, we deal with sublinear functionals F ∞ induced by a functional F through infinite iteration. We need the following result from operator theory (see [13]): for any contraction U on a Banach space X, we put ...
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... The classical Katznelson-Tzafriri theorem [18] may be stated as follows. Here T = {λ ∈ C : |λ| = 1} denotes the unit circle and a bounded linear operator T is said to be power-bounded if sup n≥0 T n < ∞. ...
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The Katznelson-Tzafriri theorem is a central result in the asymptotic theory of discrete operator semigroups. It states that for a power-bounded operator T on a Banach space we have ||Tn(I−T)‖→0 if and only if σ(T)∩T⊆{1}. The main result of the present paper gives a sharp estimate for the rate at which this decay occurs for operators on Hilbert space, assuming the growth of the resolvent norms ‖R(eiθ,T)‖ as |θ|→0 satisfies a mild regularity condition. This significantly extends an earlier result by the second author, which covered the important case of polynomial resolvent growth. We further show that, under a natural additional assumption, our condition on the resolvent growth is not only sufficient but also necessary for the conclusion of our main result to hold. By considering a suitable class of Toeplitz operators we show that our theory has natural applications even beyond the setting of normal operators, for which we in addition obtain a more general result.
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We prove the quantum Zeno effect in open quantum systems whose evolution, governed by quantum dynamical semigroups, is repeatedly and frequently interrupted by the action of a quantum operation. For the case of a quantum dynamical semigroup with a bounded generator, our analysis leads to a refinement of existing results and extends them to a larger class of quantum operations. We also prove the existence of a novel strong quantum Zeno limit for quantum operations for which a certain spectral gap assumption, which all previous results relied on, is lifted. The quantum operations are instead required to satisfy a weaker property of strong power-convergence. In addition, we establish, for the first time, the existence of a quantum Zeno limit for the case of unbounded generators. We also provide a variety of physically interesting examples of quantum operations to which our results apply.
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Following Bermúdez et al. [5], we study the rate of growth of the norms of the powers of a linear operator, under various resolvent conditions or Cesàro boundedness assumptions. In Hilbert spaces, we prove that if T satisfies the Kreiss condition, ‖Tn‖=O(n/log⁡n); if T is absolutely Cesàro bounded, ‖Tn‖=O(n1/2−ε) for some ε>0 (which depends on T); if T is strongly Kreiss bounded, then ‖Tn‖=O((log⁡n)κ) for some κ>0. We show that a Kreiss bounded operator on a reflexive space is Abel ergodic, and its Cesàro means of order α converge strongly when α>1.
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The Ornstein-Sucheston "zero-two" law for Markov operators is extended, and its proof simplified.
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