Analytic Radon-Nikodym Property Research Articles

On the complexity of some classes of Banach spaces and non-universality

These notes are dedicated to the study of the complexity of several classes of separable Banach spaces. We compute the complexity of the Banach-Saks property, the alternating Banach-Saks property, the complete continuous property, and the LUST property. We also show that the weak Banach-Saks property, the Schur property, the Dunford-Pettis property, the analytic Radon-Nikodym property, the set of Banach spaces whose set of unconditionally converging operators is complemented in its bounded oper- ators, the set of Banach spaces whose set of weakly compact operators is complemented in its bounded operators, and the set of Banach spaces whose set of Banach-Saks opera- tors is complemented in its bounded operators, are all non Borel in SB. At last, we give several applications of those results to non-universality results.

Applications of semi-embeddings to the study of the projective tensor product of Banach spaces

In this paper, we use the theory of semi-embeddings to show that if $E$ is a Banach lattice and $X$ is a Banach space then $E\,\hat{\otimes}\,X$, the projective tensor product of $E$ and $X$, has, respectively, the near Radon–Nikodym property, the analytic Radon–Nikodym property, the analytic complete continuity property, and the property of non-containment of a copy of $c_{0}$ whenever both $E$ and $X$ have the same property.

Some Properties of the Injective Tensor Product of Banach Spaces

Let X and Y be Banach spaces such that X has an unconditional basis. Then X $$ \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{ \otimes } $$ Y , the injective tensor product of X and Y , has the Radon–Nikodym property (respectively, the analytic Radon–Nikodym property, the near Radon–Nikodym property, non-containment of a copy of c 0, weakly sequential completeness) if and only if both X and Y have the same property and each continuous linear operator from the predual of X to Y is compact.

Stability of Banach Space Properties In The Projective Tensor Product

Let X be a real or complex Banach space and 1 < p < ∞. We show that L p [0, 1] ⊗ X has a property (P) whenever X has property (P), provided that (P) has certain characteristics. Examples of property (P) are the Radon-Nikodym property, the analytic Radon-Nikodym property, the near Radon-Nikodym property, the complete continuity property, the analytic complete continuity property, and the property of not containing a copy of c 0.

Types of Radon-Nikodym properties for the projective tensor product of Banach spaces

Let $X$ and $Y$ be Banach spaces such that $X$ has a boundedly complete basis. Then $X\hat{\otimes}Y$, the projective tensor product of $X$ and $Y$, has the Radon-Nikodym property (resp.~the analytic Radon-Nikodym property, the near Radon-Nikodym property, contains no copy of $c_0$) if and only if $Y$ has the same property.

Some properties of the injective tensor product of Lp[0,1] and a Banach space

Let X be a (real or complex) Banach space and 1< p, p′<∞ such that 1/ p+1/ p′=1. Then L p[0,1] ⊗ ̌ X , the injective tensor product of L p [0,1] and X, has the Radon–Nikodym property (resp. the analytic Radon–Nikodym property, the near Radon–Nikodym property, contains no copy of c 0, is weakly sequentially complete) if and only if X has the same property and each continuous linear operator from L p′ [0,1] to X is compact.

Observations About the Projective Tensor Product of Banach Spaces, Iii – LP [0, 1] ⊗ X, 1 < P < ∞

Let X be a complex Banach space and 1 < p < ∞. Then the projective tensor product L p [0, 1]⊗ X has the analytic Radon-Nikodym property (respectively, the near Radon-Nikodym property) if and only if X does. Also, L p[0, 1]⊗ X does not contain a copy of c 0 if and only if X does not contain a copy of c 0.

THE EXISTENCE OF RADIAL LIMITS OF ANALYTIC FUNCTIONS IN BANACH SPACES

Let X be a complex Banach space without the analytic Radon-Nikodym property. The author shows that G={f∈H∞(D, X) : there exists ∊>0, such that for almost all θ∈[0, 2π], $\lim\sup\limits_{\!\!\!\!\!\!r, s\uparrow1}||f(re^{i\theta})-f(se^{i\theta})||\geq\epsilon\}$ is a dense open subset of H∞(D, X). It is also shown that for every open subset B of T, there exists F∈H∞(D, X), such that F has boundary values everywhere on Bc and F has radial limits nowhere on B. When A is a measurable subset of T with positive measure, there exists f∈H∞(D, X), such that f has nontangential limits almost everywhere on Ac and f has radial limits almost nowhere on A.

VECTOR-VALUED HOLOMORPHIC FUNCTIONS ON THE COMPLEX BALL AND THE ANALYTIC RADON-NIKODYM PROPERTY

The complex Banach spaces X with values in which every bounded holomorphic function in the unit ball B of Cd(d > 1) has boundary limits almost surely are exactly the spaces with the analytic Radon-Nikodym property. The proof is based on inner Hardy martingales introduced here. The inner Hardy martingales are constructed in terms of inner functions in B and are reasonable discrete approximations for the image processes of the holomorphic Brownian motion under X-valued holomorphic functions in B.

A new characterisation of the analytic Radon-Nikodym property

We show that a separable complex Banach space X X has the analytic Radon-Nikodym property if and only if there exists 1 ≤ p &gt; ∞ 1\leq p &gt;\infty , such that the space consisting of all L p L^{p} -bounded X X -valued analytic martingales is separable.

B-Convexity, the Analytic Radon–Nikodym Property, and Individual Stability ofC0-Semigroups

LetT={T(t)}t≥0be aC0-semigroup on a Banach spaceX, with generatorAand growth bound ω. Assume thatx0∈Xis such that the local resolvent λ↦R(λ,A)x0admits a bounded holomorphic extension to the right half-plane {Reλ>0}. We prove the following results:•(i) Ifhas Fourier type∈(1,2], then lim‖()(λ−)‖=0 for all β>1/and λ>ω.•(ii) Ifhas the analytic RNP, then lim‖()(λ−)‖=0 for all β>1 and λ>ω.•(iii) Ifis arbitrary, then weak-lim()(λ−)=0 for all β>1 and λ>ω.As an application we prove a Tauberian theorem for the Laplace transform of functions with values in aB-convex Banach space.

THE ANALYTIC KREIN-MILMAN PROPERTY IN BANACH SPACES

In the papaer, it is defined the analytic Krein-Milman property for plurisubharmonic convex subsets in complex Banach spaces and studied the relation between it and the analytic Radon-Nikodym property.

On the Analytic Radon-Nikodym Property for Bounded Subsets in Banach Spaces

We define the analytic Radon–Nikodym property for bounded subsets in complex Banach spaces and show the equivalence between the analytic Radon–Nikodym property, convergence of a class of analytic (or alternatively, Hardy) martingales related to the subset and the existence of radial limits for a class of analytic functions related to the subset.

Some Applications of Convolution of Operators on Banach Spaces

Using convolution between functions and cone absolutely summing operators, we obtain characterizations of Banach spaces with the Radon-Nikodym property, the analytic Radon-Nikodym property and Banach spaces not containing a copy of ${c_0}$.

Approximation of Jensen Measures by Image Measures Under Holomorphic Functions and Applications

We show that Jensen measures defined on ${\mathbb {C}^n}$ or more generally on a complex Banach space $X$ can be approximated by the image of Lebesgue measure on the torus under $X$-valued polynomials defined on $\mathbb {C}$. We give similar characterizations for Jensen measures in terms of analytic martingales and Hardy martingales. The results are applied to approximate plurisubharmonic martingales by Hardy martingales, which enables us to give a characterization of the analytic Radon-Nikodym property of Banach spaces in terms of convergence of plurisubharmonic martingales, thus solving a problem of G. A. Edgar.

Approximation of Jensen measures by image measures under holomorphic functions and applications

We show that Jensen measures defined on ${\mathbb {C}^n}$ or more generally on a complex Banach space $X$ can be approximated by the image of Lebesgue measure on the torus under $X$-valued polynomials defined on $\mathbb {C}$. We give similar characterizations for Jensen measures in terms of analytic martingales and Hardy martingales. The results are applied to approximate plurisubharmonic martingales by Hardy martingales, which enables us to give a characterization of the analytic Radon-Nikodym property of Banach spaces in terms of convergence of plurisubharmonic martingales, thus solving a problem of G. A. Edgar.

Some applications of convolution of operators on Banach spaces

Using convolution between functions and cone absolutely summing operators, we obtain characterizations of Banach spaces with the Radon-Nikodym property, the analytic Radon-Nikodym property and Banach spaces not containing a copy of c 0 {c_0} .

Duals of vector-valued Hp-spaces for 0

The Hardy spaces Hp(X) of holomorphic functions on the unit discDin the complex plane with values in a complex Banach space X are studied. For an arbitrary X and 0<p<1, a representation of the continuous linear functionals on Hp(X) is obtained. It is proved that if 0<p<1 and X does not have the Analytic Radon Nikodym Property, then the space of all X-valued polynomials is not dense but is weakly dense in Hp(X). Moreover, if X contains a copy of c0 then Hp(X) does not have the Orlicz-Pettis property.

Analytic functions with values in lattices and symmetric spaces of measurable operators

AbstractLet 0 &lt; p,pi ≤ ∞, 0 &lt; q,qi &lt; ∞ (i = 1, 2) such thatLet E be a quasi-Banach lattice which fails to contain c0 and whose α-convexity constant is equal to 1 for some 0 &lt; α &lt; ∞. Then for every f∈H(E(q)) there exist g∈Hp, 0(E(q0)), h∈Hp1(E(q1)) such thatConsequently, E is q-concave for some finite q if and only if E is uniformly H1-convexifiable in the sense of [24]. Analogous results are also obtained for symmetric spaces of measurable operators. Another result proved in the paper says that if E is a symmetric quasi-Banach function space on (0, ∞) having the analytic Radon–Nikodym property then LE(M, τ) also possesses this property for any semifinite von Neumann algebra (M, τ).

Factorization of Analytic Functions with Values in Non-Commutative L1-spaces and Applications

Let X be a Banach space such that X* is a von Neumann algebra. We prove that X has the analytic Radon-Nikodym property (in short: ARNP). More precisely we show that for any function ƒ in H1(X) we have This implies the ARNP for X as well as for all the Banach spaces which are finitely representable in X. The proof uses a C*-algebraic formulation of the classical factorization theorems for matrix valued H1-functions. As a corollary we prove (for instance) that if A ⊂ B is a C*-subalgebra of a C*-algebra B, then every operator from A into H∞ extends to an operator from B into H∞ with the same norm. We include some remarks on the ARNP in connection with the complex interpolation method.