Abelian ergodic theorems for vector-valued functions

This note contains extensions of the Abelian ergodic theorems in [3] and [6] to functions which take their values in a Banach space. The results are based on an adaptation of Rota's maximal ergodic theorem for Abel limits [8]. Convergence theorems for continuous parameter semigroups are deduced by the approximation technique developed in [3], [6]. A direct application of the resolvent equation also enables us to deduce a convergence theorem for pseudo-resolvents.

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POINTWISE CONVERGENCE AND SEMIGROUPS ACTING ON VECTOR-VALUED FUNCTIONS

Bulletin of the Australian Mathematical Society ◽

10.1017/s0004972710002030 ◽

2011 ◽

Vol 84 (1) ◽

pp. 44-48 ◽

Cited By ~ 3

Author(s):

MICHAEL G. COWLING ◽

MICHAEL LEINERT

Keyword(s):

Banach Space ◽

Pointwise Convergence ◽

Function Spaces ◽

Semigroup Of Operators ◽

Almost Everywhere ◽

Vector Valued Function ◽

Vector Valued Functions ◽

Complex Valued ◽

Vector Valued

AbstractA submarkovian C0 semigroup (Tt)t∈ℝ+ acting on the scale of complex-valued functions Lp(X,ℂ) extends to a semigroup of operators on the scale of vector-valued function spaces Lp(X,E), when E is a Banach space. It is known that, if f∈Lp(X,ℂ), where 1<p<∞, then Ttf→f pointwise almost everywhere. We show that the same holds when f∈Lp(X,E) .

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The spectrum and eigenspaces of a meromorphic operator-valued function

Proceedings of the Royal Society of Edinburgh Section A Mathematics ◽

10.1017/s0308210500026871 ◽

1997 ◽

Vol 127 (5) ◽

pp. 1027-1051 ◽

Cited By ~ 4

Author(s):

Robert Magnus

Keyword(s):

Banach Space ◽

Riemann Surface ◽

Spectral Theory ◽

Approximation Theorem ◽

Meromorphic Mapping ◽

Bounded Operators ◽

Single Operator ◽

Definition Of ◽

Vector Valued Functions ◽

Vector Valued

SynopsisIt is shown how to associate eigenvectors with a meromorphic mapping defined on a Riemann surface with values in the algebra of bounded operators on a Banach space. This generalises the case of classical spectral theory of a single operator. The consequences of the definition of the eigenvectors are examined in detail. A theorem is obtained which asserts the completeness of the eigenvectors whenever the Riemann surface is compact. Two technical tools are discussed in detail: Cauchy-kernels and Runge's Approximation Theorem for vector-valued functions.

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Nuclear operators on spaces of continuous vector-valued functions

Glasgow Mathematical Journal ◽

10.1017/s0017089500008259 ◽

1991 ◽

Vol 33 (2) ◽

pp. 223-230 ◽

Cited By ~ 6

Author(s):

Paulette Saab ◽

Brenda Smith

Keyword(s):

Banach Space ◽

Vector Measure ◽

Type Theorem ◽

Representing Measure ◽

Bounded Linear ◽

Continuous Vector ◽

Nuclear Operators ◽

Nikodym Property ◽

Vector Valued Functions ◽

Vector Valued

Let Ω: be a compact Hausdorff space, let E be a Banach space, and let C(Ω, E) stand for the Banach space of continuous E-valued functions on Ω under supnorm. It is well known [3, p. 182] that if F is a Banach space then any bounded linear operator T:C(Ω, E)→ F has a finitely additive vector measure G defined on the σ-field of Borel subsets of Ω with values in the space ℒ(E, F**) of bounded linear operators from E to the second dual F** of F. The measure G is said to represent T. The purpose of this note is to study the interplay between certain properties of the operator T and properties of the representing measure G. Precisely, one of our goals is to study when one can characterize nuclear operators in terms of their representing measures. This is of course motivated by a well-known theorem of L. Schwartz [5] (see also [3, p. 173]) concerning nuclear operators on spaces C(Ω) of continuous scalar-valued functions. The study of nuclear operators on spaces C(Ω, E) of continuous vector-valued functions was initiated in [1], where the author extended Schwartz's result in case E* has the Radon-Nikodym property. In this paper, we will show that the condition on E* to have the Radon-Nikodym property is necessary to have a Schwartz's type theorem. This leads to a new characterization of dual spaces E* with the Radon-Nikodym property. In [2], it was shown that if T:C(Ω, E)→ F is nuclear than its representing measure G takes its values in the space (E, F) of nuclear operators from E to F. One of the results of this paper is that if T:C(Ω, E)→ F is nuclear then its representing measure G is countably additive and of bounded variation as a vector measure taking its values in (E, F) equipped with the nuclear norm. Finally, we show by easy examples that the above mentioned conditions on the representing measure G do not characterize nuclear operators on C(Ω, E) spaces, and we also look at cases where nuclear operators are indeed characterized by the above two conditions. For all undefined notions and terminologies, we refer the reader to [3].

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SCALAR BOUNDEDNESS OF VECTOR-VALUED FUNCTIONS

Glasgow Mathematical Journal ◽

10.1017/s0017089511000620 ◽

2011 ◽

Vol 54 (2) ◽

pp. 325-333 ◽

Cited By ~ 1

Author(s):

MATÍAS RAJA ◽

JOSÉ RODRÍGUEZ

Keyword(s):

Banach Space ◽

Dual Space ◽

Sufficient Conditions ◽

Vector Integration ◽

Vector Valued Functions ◽

Vector Valued

AbstractWe provide sufficient conditions for a Banach space-valued function to be scalarly bounded, which do not require to test on the whole dual space. Some applications in vector integration are also given.

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Vector-valued holomorphic and harmonic functions

Concrete Operators ◽

10.1515/conop-2016-0007 ◽

2016 ◽

Vol 3 (1) ◽

pp. 68-76

Author(s):

Wolfgang Arendt

Keyword(s):

Banach Space ◽

Convergence Theorem ◽

Harmonic Functions ◽

Dual Space ◽

Holomorphic Functions ◽

Bounded Functions ◽

Vector Valued ◽

Separating Subspace

AbstractHolomorphic and harmonic functions with values in a Banach space are investigated. Following an approach given in a joint article with Nikolski [4] it is shown that for bounded functions with values in a Banach space it suffices that the composition with functionals in a separating subspace of the dual space be holomorphic to deduce holomorphy. Another result is Vitali’s convergence theorem for holomorphic functions. The main novelty in the article is to prove analogous results for harmonic functions with values in a Banach space.

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Rotundity In Köthe Spaces of Vector-Valued Functions

Canadian Journal of Mathematics ◽

10.4153/cjm-1989-030-2 ◽

1989 ◽

Vol 41 (4) ◽

pp. 659-675 ◽

Cited By ~ 10

Author(s):

A. Kamińska ◽

B. Turett

Keyword(s):

Banach Space ◽

Analogous Result ◽

Function Spaces ◽

Sequence Spaces ◽

Uniform Rotundity ◽

Köthe Spaces ◽

Bochner Space ◽

Köthe Space ◽

Vector Valued Functions ◽

Vector Valued

In this paper, Köthe spaces of vector-valued functions are considered. These spaces, which are generalizations of both the Lebesgue-Bochner and Orlicz-Bochner spaces, have been studied by several people (e.g., see [1], [8]). Perhaps the earliest paper concerning the rotundity of such Köthe space is due to I. Halperin [8]. In his paper, Halperin proved that the function spaces E(X) is uniformly rotund exactly when both the Köthe space E and the Banach space X are uniformly rotund; this generalized the analogous result, due to M. M. Day [4], concerning Lebesgue-Bochner spaces. In [20], M. Smith and B. Turett showed that many properties akin to uniform rotundity lift from X to the Lebesgue-Bochner space LP(X) when 1 < p < ∞. A survey of rotundity notions in Lebesgue-Bochner function and sequence spaces can be found in [19].

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On upcrossing inequalities for subadditive superstationary processes

Ergodic Theory and Dynamical Systems ◽

10.1017/s0143385700003035 ◽

1985 ◽

Vol 5 (3) ◽

pp. 409-416

Author(s):

Michael Krawczak

Keyword(s):

Ergodic Theorem ◽

Convergence Theorem ◽

Stationary Processes ◽

Convergence Theorems ◽

Discrete Parameter ◽

Common Generalization ◽

Continuous Parameter ◽

Birkhoff’S Ergodic Theorem ◽

Unpublished Thesis ◽

Parameter Case

AbstractBishop [2]has given a proof of Birkhoff's ergodic theorem by establishing upcrossing inequalities similar to those of Doob. Such inequalities can be considered as quantitative improvements of convergence theorems: while convergence a.e. means that the number of upcrossings of any interval is a.e. finite, they assert integrability and prove bounds for the integrals. The main point of this paper is to prove upcrossing inequalities for the class of subadditive superstationary processes introduced by Abid [1] as a common generalization of Kingman's [5] subadditive stationary processes and Krengel's [6] superstationary processes. We make use of ideas of Smeltzer [7] who handled the subadditive stationary discrete parameter case in his unpublished thesis. In the continuous parameter case our upcrossing inequality requires more restrictive conditions than the corresponding convergence theorem, due to Hachem [3]. We actually show by example that the number of upcrossings need not be integrable under the assumptions of Hachem even for additive stationary processes.

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On Measurability for Vector-Valued Functions

Canadian Journal of Mathematics ◽

10.4153/cjm-1963-062-4 ◽

1963 ◽

Vol 15 ◽

pp. 613-621 ◽

Cited By ~ 1

Author(s):

D. O. Snow

Keyword(s):

Banach Space ◽

Infinite Series ◽

Unconditional Convergence ◽

Integration Theory ◽

Bochner Integral ◽

Norm Topology ◽

Vector Valued Functions ◽

Vector Valued

The problem of developing an abstract integration theory has been approached from many angles (6). The most general of several definitions based on the norm topology is that of Birkhoff (1), which includes the well-known and widely used Bochner integral (3).The original Birkhoff formulation was based on the notion of unconditional convergence of an infinite series of elements in a Banach space and the closed convex extensions of certain approximating sums.

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Transitivity in spaces of vector-valued functions

Proceedings of the Edinburgh Mathematical Society ◽

10.1017/s0013091508000540 ◽

2010 ◽

Vol 53 (3) ◽

pp. 601-608 ◽

Cited By ~ 1

Author(s):

Félix Cabello Sánchez

Keyword(s):

Banach Space ◽

Maximal Ideal ◽

Real Banach Space ◽

Cantor Set ◽

Maximal Ideal Space ◽

Ideal Space ◽

Vector Valued Functions ◽

Vector Valued

AbstractWe exhibit a real Banach space M such that C(K,M) is almost transitive if K is the Cantor set, the growth of the integers in its Stone–Čech compactification or the maximal ideal space of L∞. For finite K, the space C(K,M) = M|K| is even transitive.

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A Choquet theorem for general subspaces of vector-valued functions

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004100063490 ◽

1985 ◽

Vol 98 (2) ◽

pp. 323-326 ◽

Cited By ~ 1

Author(s):

Paulette Saab ◽

Michel Talagrand

Keyword(s):

Banach Space ◽

Hausdorff Space ◽

Linear Subspace ◽

Vector Measure ◽

Complex Banach Space ◽

Supremum Norm ◽

Bounded Linear ◽

Vector Valued Functions ◽

Bounded Linear Functional ◽

Vector Valued

Let X be a compact Hausdorff space, let E be a (real or complex) Banach space, and let C(X, E) stand for the Banach space of all continuous E-valued functions defined on X under the supremum norm. If A is an arbitrary linear subspace of C(X, E), then it is shown that each bounded linear functional l on A can be represented by a boundary E*-valued vector measure μ on X that has the same norm as l. This result constitutes an extension to vector-valued functions of the so-called analytic version of Choquet's integral representation theorem.

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