On Functions of Integrable Mean Oscillation

Given we denote by the modulus of mean oscillation given by where is an arc of , stands for the normalized length of , and . Similarly we denote by the modulus of harmonic oscillation given by where and stand for the Poisson kernel and the Poisson integral of respectively. It is shown that, for each , there exists such that

On the Unit Ball of Operator-valued H 2-functions

Let X be a complex Banach space and let H 2 (\( \mathbb{D} \), X) denote the space of X-valued analytic functions in the unit disc such that $$ \mathop {sup}\limits_{0 < r < 1} \int_0^{2\pi } {\left\| {F\left( {re^{it} } \right)} \right\|^2 \frac{{dt}} {{2\pi }} < \infty .} $$ It is shown that a function F belongs to the unit ball of H 2 ( \( \mathbb{D} \), X) if and only if there exist f∈H ∞ (\( \mathbb{D} \), X) and ϕ∈H ∞ (\( \mathbb{D} \)) such that $$ \left\| {f\left( z \right)} \right\|^2 + \left| {\varphi \left( z \right)} \right|^2 \leqslant 1 and F\left( z \right) = \frac{{f\left( z \right)}} {{1 - z\varphi \left( z \right)}} $$ for |z| < 1.

A bilinear version of Orlicz–Pettis theorem

Abstract Given three Banach spaces X, Y and Z and a bounded bilinear map B : X × Y → Z , a sequence x = ( x n ) n ⊆ X is called B -absolutely summable if ∑ n = 1 ∞ ‖ B ( x n , y ) ‖ Z is finite for any y ∈ Y . Connections of this space with l weak 1 ( X ) are presented. A sequence x = ( x n ) n ⊆ X is called B -unconditionally summable if ∑ n = 1 ∞ | 〈 B ( x n , y ) , z ∗ 〉 | is finite for any y ∈ Y and z ∗ ∈ Z ∗ and for any M ⊆ N there exists x M ∈ X for which ∑ n ∈ M 〈 B ( x n , y ) , z ∗ 〉 = 〈 B ( x M , y ) , z ∗ 〉 for all y ∈ Y and z ∗ ∈ Z ∗ . A bilinear version of Orlicz–Pettis theorem is given in this setting and some applications are presented.

Restricted weak type on maximal linear and multilinear integral maps

It is shown that multilinear operators of the form T ( f 1 , . . . , f k ) ( x ) T(f_1,...,f_k)(x) = ∫ R n K ( x , y 1 , . . . , y k ) f 1 ( y 1 ) . . . f k ( y k ) d y 1 . . . d y k =\!\int _{\mathbb {R}^n}\!K(x,y_1,...,y_k)f_1(y_1)... f_k(y_k)dy_1...dy_k of restricted weak type ( 1 , . . . , 1 , q ) (1,...,1,q) are always of weak type ( 1 , . . . , 1 , q ) (1,...,1,q) whenever the map x → K x x\to K_x is a locally integrable L 1 ( R n ) L^1(\mathbb {R}^n) -valued function.

Characterization of greedy bases in Banach spaces

Abstract We shall present a new characterization of greedy bases and 1-greedy bases in terms of certain functionals defined using distances to one dimensional subspaces generated by the basis. We also introduce a new property that unifies the notions of unconditionality and democracy and allows us to recover a better dependence on the constants.

Norm estimates for operators from Hp to ℓq

Abstract We give upper and lower estimates of the norm of a bounded linear operator from the Hardy space H p to l q in terms of the norm of the rows and the columns of its associated matrix in certain vector-valued sequence spaces.

Semigroups of composition operators and integral operators in spaces of analytic functions

We study the maximal spaces of strong continuity on BMOA and the Bloch space B for semigroups of composition operators. Characterizations are given for the cases when these maximal spaces are V MOA or the little Bloch B0. These characterizations are in terms of the weak compactness of the resolvent function or in terms of a specially chosen symbol g of an integral operator Tg. For the second characterization we prove and use an independent result, namely that the operators Tg are weakly compact on the above mentioned spaces if and only if they are compact.

Remarks on the semivariation of vector measures with respect to Banach spaces.

Suppose that and . It is shown that any Lp(µ)-valued measure has finite L2(v)-semivariation with respect to the tensor norm for 1 ≤ p &lt; ∞ and finite Lq(v)-semivariation with respect to the tensor norm whenever either q = 2 and 1 ≤ p ≤ 2 or q &gt; max{p, 2}. However there exist measures with infinite Lq-semivariation with respect to the tensor norm for any 1 ≤ q &lt; 2. It is also shown that the measure m (A) = χA has infinite Lq-semivariation with respect to the tensor norm if q &lt; p.

Bergman and Bloch spaces of vector-valued functions

We investigate Bergman and Bloch spaces of analytic vector-valued functions in the unit disc. We show how the Bergman projection from the Bochner-Lebesgue space Lp(, X) onto the Bergman space Bp(X) extends boundedly to the space of vector-valued measures of bounded p-variation Vp(X), using this fact to prove that the dual of Bp(X) is Bp(X*) for any complex Banach space X and 1 < p < ∞. As for p = 1 the dual is the Bloch space ℬ(X*). Furthermore we relate these spaces (via the Bergman kernel) with the classes of p-summing and positive p-summing operators, and we show in the same framework that Bp(X) is always complemented in p(X). (© 2003 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

Hölder inequality for functions that are integrable with respect to bilinear maps

Let $(\Omega, \Sigma, \mu)$ be a finite measure space, $1\le p&lt;\infty$, $X$ be a Banach space $X$ and $B:X\times Y \to Z$ be a bounded bilinear map. We say that an $X$-valued function $f$ is $p$-integrable with respect to $B$ whenever $\sup_{\|y\|=1} \int_\Omega \|B(f(w),y)\|^p\,d\mu&lt;\infty$. We get an analogue to Hölder's inequality in this setting.

LORENTZ SPACES OF VECTOR-VALUED MEASURES

The norm of the characteristic function of a set in the John‐Nirenberg space of exponent p

Lipschitz-type conditions on homogeneous Banach spaces of analytic functions

Abstract In this paper we deal with Banach spaces of analytic functions X defined on the unit disk satisfying that R t f ∈ X for any t > 0 and f ∈ X , where R t f ( z ) = f ( e i t z ) . We study the space of functions in X such that ‖ P r ( D f ) ‖ X = O ( ω ( 1 − r ) 1 − r ) , r → 1 − where D f ( z ) = ∑ n = 0 ∞ ( n + 1 ) a n z n and ω is a continuous and non-decreasing weight satisfying certain mild assumptions. The space under consideration is shown to coincide with the subspace of functions in X satisfying any of the following conditions: (a) ‖ R t f − f ‖ X = O ( ω ( t ) ) , (b) ‖ P r f − f ‖ X = O ( ω ( 1 − r ) ) , (c) ‖ Δ n f ‖ X = O ( ω ( 2 − n ) ) , or (d) ‖ f − s n f ‖ X = O ( ω …

Carleson's counterexample and a scale of Lorentz-BMO spaces on the bitorus

We introduce a full scale of Lorentz-BMO spaces BMO L p,q on the bidisk, and show that these spaces do not coincide for different values ofp andq. Our main tool is a detailed analysis of Carleson's construction in [C].

The Bergman projection on weighted spaces: L1and Herz spaces

We nd necessary and sucien t conditions on radial weights w on the unit disc so that the Bergman type projections of Forelli{Rudin are bounded on L 1 (w) and in the Herz spaces K q(w). 1. Introduction and preliminaries. The purpose of this paper is to study spaces of analytic functions on the unit disc D provided with a norm of a weighted Herz space. More precisely we consider the classical family of Bergman projections Ps, s > 1, and we give necessary and sucien t conditions on the weight making these projections continuous in the cor- responding weighted Herz space. We also consider the continuity of these

A space of projections on the Bergman space

We define a set of projections on the Bergman space A 2 , which is parameterized by an ane subset of a Banach space of holomorphic functions in the disk and which includes the classical Forelli-Rudin projections.

p-VARIATION OF VECTOR MEASURES WITH RESPECT TO BILINEAR MAPS

AbstractWe introduce the spaces Vℬp(X) (respectively 𝒱ℬp(X)) of the vector measures ℱ:Σ→X of bounded (p,ℬ)-variation (respectively of bounded (p,ℬ)-semivariation) with respect to a bounded bilinear map ℬ:X×Y →Z and show that the spaces Lℬp(X) consisting of functions which are p-integrable with respect to ℬ, defined in by Blasco and Calabuig [‘Vector-valued functions integrable with respect to bilinear maps’, Taiwanese Math. J. to appear], are isometrically embedded in Vℬp(X). We characterize 𝒱ℬp(X) in terms of bilinear maps from Lp′×Y into Z and Vℬp(X) as a subspace of operators from Lp′(Z*) into Y*. Also we define the notion of cone absolutely summing bilinear maps in order to describe t…

Some Aspects of Vector-Valued Singular Integrals

Let A, B be Banach spaces and \(1 < p < \infty. \; T\) is said to be a (p, A, B)- CalderoLon–Zygmund type operator if it is of weak type (p, p), and there exist a Banach space E, a bounded bilinear map \(u: E \times A \rightarrow B,\) and a locally integrable function k from \(\mathbb{R}^n \times \mathbb{R}^n \backslash \{(x, x): x \in \mathbb{R}^n\}\) into E such that $$T\;f(x) = \int u(k(x, y), f(y))dy$$ for every A-valued simple function f and \(x \notin \; supp \; f.\)

Modulus of continuity with respect to semigroups of analytic functions and applications

Abstract Given a complex Banach space E , a semigroup of analytic functions ( φ t ) and an analytic function F : D → E we introduce the modulus w φ ( F , t ) = sup | z | 1 ⁡ ‖ F ( φ t ( z ) ) − F ( z ) ‖ . We show that if 0 α ≤ 1 and F belongs to the vector-valued disc algebra A ( D , E ) , the Lipschitz condition M ∞ ( F ′ , r ) = O ( ( 1 − r ) 1 − α ) as r → 1 is equivalent to w φ ( F , t ) = O ( t α ) as t → 0 for any semigroup of analytic functions ( φ t ) , with φ t ( 0 ) = 0 and infinitesimal generator G , satisfying that φ t ′ and G belong to H ∞ ( D ) with sup 0 ≤ t ≤ 1 ⁡ ‖ φ ′ ‖ ∞ ∞ , and in particular is equivalent to the condition ‖ F − F r ‖ A ( D , E ) = O ( ( 1 − r ) α ) as r …

Atomic Decomposition of Weighted Besov Spaces

We find the atomic decomposition of functions in the weighted Besov spaces under certain factorization conditions on the weight. Introduction. After achieving the atomic decomposition of Hardy spaces (see [8,22, 33]), many of the function saces have been shown to admit similar decompositions. Let us mention the decomposition of B.M.O. (see [32, 25]), Bergman spaces (see [9, 23]), the predual of Bloch space (see [ 11]), Besov spaces (see [15, 4, 10]), Lipschitz spaces (see [18]), Triebel-Lizorkin spaces (see [16, 31]),... They are obtained by quite different methods, but there is a unified and beautiful approach to get the decomposition for most of the spaces. This is the use of a formula du…

Vector-valued analytic functions of bounded mean oscillation and geometry of Banach spaces

When dealing with vector-valued functions, sometimes is rather difficult to give non trivial examples, meaning examples which do not come from tensoring scalar-valued functions and vectors in the Banach space, belonging to certain classes. This is the situation for vector valued BMO. One of the objectives of this paper is to look for methods to produce such examples. Our main tool will be the vector-valued extension of the following result on multipliers, proved in [MP], which says that the space of multipliers between H and BMOA can be identified with the space of Bloch functions B, i.e. (H, BMOA) = B (see Section 3 for notation), which, in particular gives that g ∗ f ∈ BMOA whenever f ∈ H…

VECTOR-VALUED FUNCTIONS INTEGRABLE WITH RESPECT TO BILINEAR MAPS

Let $(\Omega, \Sigma, \mu)$ be a $\sigma-$finite measure space, $1\le p \lt \infty$, $X$ be a Banach space $X$ and ${\cal B} :X\times Y \to Z$ be a bounded bilinear map. We say that an $X$-valued function $f$ is $p-$integrable with respect to ${\cal B}$ whenever $\sup\{\int_\Omega\|{\cal B}(f(w),y)\|^pd\mu: \|y\|=1\}$ is finite. We identify the spaces of functions integrable with respect to the bilinear maps arising from H\"older's and Young's inequalities. We apply the theory to give conditions on $X$-valued kernels for the boundedness of integral operators $T_{{\cal B}}(f) (w)=\int_{\Omega'}{{\cal B}}(k(w,w'),$ $f(w'))d\mu'(w')$ from ${\mathrm L}^p(Y)$ into ${\mathrm L}^p(Z)$, extending t…

VECTOR MEASURES WITH VARIATION IN A BANACH FUNCTION SPACE

Let E be a Banach function space and X be an arbitrary Banach space. Denote by E(X) the Kothe-Bochner function space defined as the set of measurable functions f : Ω → X such that the nonnegative functions ‖f‖X : Ω → [0,∞) are in the lattice E. The notion of E-variation of a measure —which allows to recover the pvariation (for E = Lp), Φ-variation (for E = LΦ) and the general notion introduced by Gresky and Uhl— is introduced. The space of measures of bounded E-variation VE(X) is then studied. It is shown, among other things and with some restriction of absolute continuity of the norms, that (E(X))∗ = VE′ (X ∗), that VE(X) can be identified with space of cone absolutely summing operators fr…

Norm continuity and related notions for semigroups on Banach spaces

We find some conditions on a c0-semigroup on a Banach space and its resolvent connected with the norm continuity of the semigroup. We use them to get characterizations of norm continuous, eventually norm continuous and eventually compact semigroups on Hilbert spaces in terms of the growth of the resolvent of their generator.

Equivalences involving (p,q)-multi-norms

Remarks on p-summing multipliers.

Let X and Y be Banach spaces and 1 ≤ p < ∞, a sequence of operators (Tn) from X into Y is called a p-summing multiplier if (Tn(xn)) belongs to lp(Y) whenever (xn) satisfies that ((x*, xn)) belongs to lp for all x* ∈ X*. We present several examples of p-summing multipliers and extend known results for p-summing operators to this setting. We get, using almost summing and Rademacher bounded operators, some sufficient conditions for a sequence to be a p-summing multiplier between spaces with some geometric properties.

Norm estimates for operators from Hp to ℓq

AbstractWe give upper and lower estimates of the norm of a bounded linear operator from the Hardy space Hp to ℓq in terms of the norm of the rows and the columns of its associated matrix in certain vector-valued sequence spaces.

Summing multi-norms defined by Orlicz spaces and symmetric sequence space

We develop the notion of the \((X_1,X_2)\)-summing power-norm based on a~Banach space \(E\), where \(X_1\) and \(X_2\) are symmetric sequence spaces. We study the particular case when \(X_1\) and \(X_2\) are Orlicz spaces \(\ell_\Phi\) and \(\ell_\Psi\) respectively and analyze under which conditions the \((\Phi, \Psi)\)-summing power-norm becomes a~multinorm. In the case when \(E\) is also a~symmetric sequence space \(L\), we compute the precise value of \(\|(\delta_1,\cdots,\delta_n)\|_n^{(X_1,X_2)}\) where \((\delta_k)\) stands for the canonical basis of \(L\), extending known results for the \((p,q)\)-summing power-norm based on the space \(\ell_r\) which corresponds to \(X_1=\ell_p\), …

Spaces of Operator-valued Functions Measurable with Respect to the Strong Operator Topology

Let X and Y be Banach spaces and (Ω, Σ, μ) a finite measure space. In this note we introduce the space L p /μ; ℒ(X, Y)] consisting of all (equivalence classes of) functions Φ:Ω↦ℒ(X, Y) such that ω↦Φ(ω)x is strongly μ-measurable for all x∈X and ω↦Φ(ω)f(ω) belongs to L 1(μ; Y) for all f∈L p′ (μ; X), 1/p+1/p′=1. We show that functions in L p /μ; ℒ(X, Y)] define operator-valued measures with bounded p-variation and use these spaces to obtain an isometric characterization of the space of all ℒ(X, Y)-valued multipliers acting boundedly from L p (μ; X) into L q (μ; Y), 1≤q<p<∞.

Rate of growth of frequently hypercyclic functions

AbstractWe study the rate of growth of entire functions that are frequently hypercyclic for the differentiation operator or the translation operator. Moreover, we prove the existence of frequently hypercyclic harmonic functions for the translation operator and we study the rate of growth of harmonic functions that are frequently hypercyclic for partial differentiation operators.

On Taylor coefficients of entire functions integrable against exponential weights

On coefficients of vector-valued Bloch functions

Holomorphic Hölder‐type spaces and composition operators

Type and Cotype in Vector-Valued Nakano Sequence Spaces

AbstractGiven a sequence of Banach spaces {Xn}n and a sequence of real numbers {pn}n in [1,∞), the vector-valued Nakano sequence spaces ℓ({pn},{Xn}) consist of elements {xn}n in ∏nXn for which there is a constant λ>0 such that ∑n(‖xn‖/λ)pn<∞. In this paper we find the conditions on the Banach spaces Xn and on the sequence {pn}n for the spaces ℓ({pn},{Xn}) to have cotype q or type p.

Commutators of linear and bilinear Hilbert transforms

Let α ∈ R \alpha \in \mathbb {R} , and let H α ( f , g ) ( x ) = 1 π p . v . ∫ f ( x − t ) g ( x − α t ) d t t H_\alpha (f,g)(x)=\frac {1}{\pi } p.v. \int f(x-t)g(x-\alpha t)\frac {dt}{t} and H f ( x ) = 1 π p . v . ∫ f ( x − t ) d t t Hf(x)= \frac {1}{\pi } p.v.\int f(x-t)\frac {dt}{t} denote the bilinear and linear Hilbert transforms, respectively. It is proved that, for 1 &gt; p &gt; ∞ 1&gt;p&gt;\infty and α 1 ≠ α 2 \alpha _1\ne \alpha _2 , H α 1 − H α 2 H_{\alpha _1}-H_{\alpha _2} maps L p × B M O L^p\times BMO into L p L^{p} and it maps B M O × L p BMO \times L^p into L p L^{p} if and only if sign ⁡ ( α 1 ) = sign ⁡ ( α 2 ) \operatorname {sign}(\alpha _1)=\operatorname {sign}(\alpha _2…

Multipliers on Vector Valued Bergman Spaces

AbstractLet X be a complex Banach space and let Bp(X) denote the vector-valued Bergman space on the unit disc for 1 ≤ p &lt; ∞. A sequence (Tn)n of bounded operators between two Banach spaces X and Y defines a multiplier between Bp(X) and Bq(Y) (resp. Bp(X) and lq(Y)) if for any function we have that belongs to Bq(Y) (resp. (Tn(xn))n ∈ lq(Y)). Several results on these multipliers are obtained, some of them depending upon the Fourier or Rademacher type of the spaces X and Y. New properties defined by the vector-valued version of certain inequalities for Taylor coefficients of functions in Bp(X) are introduced.

New spaces of matrices with operator entries

In this paper, we will consider matrices with entries in the space of operators $\mathcal{B}(H)$, where $H$ is a separable Hilbert space and consider the class of matrices that can be approached in the operator norm by matrices with a finite number of diagonals. We will use the Schur product with Toeplitz matrices generated by summability kernels to describe such a class and show that in the case of Toeplitz matrices it can be identified with the space of continuous functions with values in $\mathcal B(H)$. We shall also introduce matriceal versions with operator entries of classical spaces of holomorphic functions such as $H^\infty(\mathbb{D})$ and $A(\mathbb{D})$ when dealing with upper t…

Embedding of analytic function spaces with given mean growth of the derivative

MSC (2000) 30D55 If φ is a positive function defined in (0,1) and 0 <p< ∞, we consider the space L(p, φ) which consists of all functions f analytic in the unit disc D for which the integral means of the derivative Mp(r, f � )= " 1 2π Rπ −π þ f � (re iθ ) þ p dθ "1/p , 0 <r< 1, satisfy M p(r, f � )=O (φ(r)) ,a sr → 1. In this paper, for any given p ∈ (0,1), we characterize the functions φ, among a certain class of weight functions, to be able to embedd L(p, φ) into classical function spaces. These results complement other previously obtained by the authors for p ≥ 1. c

Remarks on (Q, P, Y)-Summing Operators

Abstract unavailable at this time... Mathematics Subject Classification (1991): 47B10. Key words: Summing operators; injective tensor product. Quaestiones Mathematicae 26(2003), 97-103

Notes on bilinear multipliers on Orlicz spaces

Let $\Phi_1 , \Phi_2 $ and $ \Phi_3$ be Young functions and let $L^{\Phi_1}(\mathbb{R})$, $L^{\Phi_2}(\mathbb{R})$ and $L^{\Phi_3}(\mathbb{R})$ be the corresponding Orlicz spaces. We say that a function $m(\xi,\eta)$ defined on $\mathbb{R}\times \mathbb{R}$ is a bilinear multiplier of type $(\Phi_1,\Phi_2,\Phi_3)$ if \[ B_m(f,g)(x)=\int_\mathbb{R} \int_\mathbb{R} \hat{f}(\xi) \hat{g}(\eta)m(\xi,\eta)e^{2\pi i (\xi+\eta) x}d\xi d\eta \] defines a bounded bilinear operator from $L^{\Phi_1}(\mathbb{R}) \times L^{\Phi_2}(\mathbb{R})$ to $L^{\Phi_3}(\mathbb{R})$. We denote by $BM_{(\Phi_1,\Phi_2,\Phi_3)}(\mathbb{R})$ the space of all bilinear multipliers of type $(\Phi_1,\Phi_2,\Phi_3)$ and inve…

On operator valued sequences of multipliers and R-boundedness

AbstractIn recent papers (cf. [J.L. Arregui, O. Blasco, (p,q)-Summing sequences, J. Math. Anal. Appl. 274 (2002) 812–827; J.L. Arregui, O. Blasco, (p,q)-Summing sequences of operators, Quaest. Math. 26 (2003) 441–452; S. Aywa, J.H. Fourie, On summing multipliers and applications, J. Math. Anal. Appl. 253 (2001) 166–186; J.H. Fourie, I. Röntgen, Banach space sequences and projective tensor products, J. Math. Anal. Appl. 277 (2) (2003) 629–644]) the concept of (p,q)-summing multiplier was considered in both general and special context. It has been shown that some geometric properties of Banach spaces and some classical theorems can be described using spaces of (p,q)-summing multipliers. The p…

Vector-valued Hardy inequalities and B-convexity

Inequalities of the form $$\sum\nolimits_{k = 0}^\infty {|\hat f(m_k )|/(k + 1) \leqslant C||f||_1 } $$ for allf∈H 1, where {m k } are special subsequences of natural numbers, are investigated in the vector-valued setting. It is proved that Hardy's inequality and the generalized Hardy inequality are equivalent for vector valued Hardy spaces defined in terms ff atoms and that they actually characterizeB-convexity. It is also shown that for 1<q<∞ and 0<α<∞ the spaceX=H(1,q,γa) consisting of analytic functions on the unit disc such that $$\int_0^1 {(1 - r)^{q\alpha - 1} M_1^q (f,r) dr< \infty } $$ satisfies the previous inequality for vector valued functions inH 1 (X), defined as the space ofX…

Convolution of three functions by means of bilinear maps and applications

When dealing with spaces of vector-valued analytic functions there is a natural way to understand multipliers between them. If X and Y are Banach spaces and L(X,Y ) stands for the space of linear and continuous operators we may consider the convolution of L(X,Y )-valued analytic functions, say F (z) = ∑ n=0∞ Tnz , and X-valued polynomials, say f(z) = ∑m n=0 xnz , to get the Y -valued function F ∗ f(z) = ∑ Tn(xn)z. The second author considered such a definition and studied multipliers between H(X) and BMOA(Y ) in [5]. When the functions take values in a Banach algebra A then the natural extension of multiplier is simply that if f(z) = ∑ anz n and g(z) = ∑ bnz , then f ∗ g(z) = ∑ an.bnz n whe…

COMPLEX CONVEXITY AND VECTOR-VALUED LITTLEWOOD–PALEY INEQUALITIES

Let 2 p 0s uch thatfHp(X) (� f(0)� p + λ (1 −| z| 2 ) p−1 � f � (z)� p dA(z)) 1/p ,f or all f ∈ H p (X). Applications to embeddings between vector-valued BMOA spaces defined via Poisson integral or Carleson measures are provided.

(p,q)-summing sequences

Abstract A sequence (x j ) in a Banach space X is (p,q) -summing if for any weakly q -summable sequence (x j ∗ ) in the dual space we get a p -summable sequence of scalars (x j ∗ (x j )) . We consider the spaces formed by these sequences, relating them to the theory of (p,q) -summing operators. We give a characterization of the case p=1 in terms of integral operators, and show how these spaces are relevant for a general question on Banach spaces and their duals, in connection with Grothendieck theorem.

The Bohr Radius of a Banach Space

Following the scalar-valued case considered by Djakow and Ramanujan (A remark on Bohr’s theorem and its generalizations 14:175–178, 2000) we introduce, for each complex Banach space X and each \(1\le p0\). We study the p-Bohr radius of the Lebesgue spaces \(L^q(\mu )\) for different values of p and q. In particular we show that \(r_p(L^q(\mu ))=0\) whenever \(p<2\) and \(dim(L^q(\mu ))\ge 2\) and \(r_p(L^q(\mu ))=1\) whenever \(p\ge 2\) and \(p'\le q\le p\). We also provide some lower estimates for \(r_2(L^q(\mu ))\) for the values \(1\le q<2\).

Boundedness of composition operators in holomorphic Hölder type spaces

(p,q)-Summing Sequences of Operators

Abstract unavailable at this time... Mathematics Subject Classification (2000): 47B10. Key words: Summing operators, vector-valued multipliers. Quaestiones Mathematicae 26(2003), 441–452

Bloch functions on the unit ball of an infinite dimensional Hilbert space

The Bloch space has been studied on the open unit disk of C and some ho- mogeneous domains of C n . We dene Bloch functions on the open unit ball of a Hilbert space E and prove that the corresponding space B(BE) is invariant under composition with the automorphisms of the ball, leading to a norm that- modulo the constant functions - is automorphism invariant as well. All bounded analytic functions on BE are also Bloch functions. ones, resulting the fact that if for a given n; the restrictions of the function to the n-dimensional subspaces have their Bloch norms uniformly bounded, then the function is a Bloch one and conversely. We also introduce an equivalent norm forB(BE) obtained by repla…