Uniformly bounded composition operators in the Banach space of absolutely continuous functions

Weakly compact composition operators on vector-valued BMOA

We present a non-weak supercyclicity criterion for vectors in infinite dimensional Banach spaces. Also, we give sufficient conditions under which a class of weighted composition operators on a Banach space of analytic functions is not weakly supercyclic. In particular, we show that the semigroup of linear isometries on the spaces $S^p$ ($p>1$), is not weakly supercyclic. Moreover, we observe that every composition operator on some Banach space of analytic functions such as the disc algebra or the analytic Lipschitz space is not weakly supercyclic.

LetVbe an arbitrary system of weights on an open connected subsetGofℂN(N≥1)and letB(E)be the Banach algebra of all bounded linear operators on a Banach spaceE. LetHVb(G,E)andHV0(G,E)be the weighted locally convex spaces of vector-valued analytic functions. In this survey, we present a development of the theory of multiplication operators and composition operators from classical spaces of analytic functionsH(G)to the weighted spaces of analytic functionsHVb(G,E)andHV0(G,E).

An algebra of bounded linear operators on a Banach space is said to be strongly compact if its unit ball is precompact in the strong operator topology, and a bounded linear operator on a Banach space is said to be strongly compact if the algebra with identity generated by the operator is strongly compact. Our interest in this notion stems from the work of Lomonosov on the existence of invariant subspaces. A spectral condition is provided for an operator to be strongly compact and for its commutant to be a strongly compact algebra. This condition is applied to test strong compactness for some classes of operators, namely, bilateral weighted shifts, Cesaro operators, and composition operators. Some tools from function theory like the Mellin transform, the principle of analytic continuation, and the Weierstrass approximation theorem arise naturally in the pursuit of this program.

Function spaces and its various applications are subjects of investigation in many research centers and we are glad that this special issue has received the remarkable attention by researchers in these areas. We have received papers from wide range of topics of function spaces such as interpolation methods for stochastic processes spaces, vector optimization problems in Banach spaces, Marcinkiewicz integral operator in Morrey spaces, Hardy’s operator in the variable exponent Lebesgue spaces, and differential equations of weakly parabolic type in Banach spaces. A number of papers are devoted to investigating different kinds of problems in Hardy spaces or Hardy subspaces, for example, Calderon-Zygumnd singular integral operators in Hardy spaces, weighted Hardy spaces related to the Schrodinger operator, and composition operator in hyperbolic Bloch-type spaces. Geometric problems such as modulus of convexity in some Banach sequence spaces, p-uniform convexity, and quniform smoothness in C spaces are also covered by this issue.

We prove that the symmetric convexified Tsirelson space is of weak cotype 2 but not of cotype 2. Introduction Weak type 2 and weak cotype 2 spaces were originally introduced and investigated by V.D. Milman and G. Pisier in [10] and weak Hilbert spaces by Pisier in [13]. A further detailed investigation can be found in Pisier’s book [14]. The first example of a weak Hilbert space which is not isomorphic to a Hilbert space is the 2-convexified Tsirelson space (called the convexified Tsirelson space in this paper). This follows from the results of W.B. Johnson in [4]. For a detailed study of the original Tsirelson space we refer to [3]. Let X be a Banach space with a symmetric basis. It was proved in [14] that if X is a weak Hilbert space, then it is isomorphic to a Hilbert space and this has lead to the belief that if X is just of weak cotype 2, then it is of cotype 2. However, this turns out not necessarily to be the case. The main result of this paper states that the symmetric convexified Tsirelson space is of weak cotype 2 but not of cotype 2. We now wish to discuss the arrangement of this paper in greater detail. In Section 1 we give some basic facts on properties related to weak type 2 and weak cotype 2 while Section 2 is devoted to a review of some results on the convexified Tsirelson space which we need for our main result. Most of these results are stated without proofs since they can be proved in a similar manner as the corresponding results for the original Tsirelson space. In Section 3 we make the construction of the symmetric convexified Tsirelson space, investigate its basic properties and prove our main result stated above. 02000 Mathematics Subject Classification: 46B03, 46B07. ∗Supported by NSF grants DMS 9706108 and 0102686. †Supported by the Danish Natural Science Research Council, grant 9801867.

The purpose of this paper is to propose an algorithm for solving the split common fixed point problem for strict quasi-ϕ-pseudocontractive mappings in Banach spaces. It is proved that the sequence generated by the proposed iterative algorithm converges strongly to a solution of the split common fixed point problem. Then, the main result is used to study the split common null point problem and the split quasi-inclusion problem. Finally, a numerical example is provided to illustrate our main result. The results presented in this paper extend and improve some recent corresponding results.

We characterize the compact and weakly compact weighted composition operators on the Banach space CP(T, X) of those continuous functions from a completely regular Hausdorff space T into a Banach space X which have relatively compact range.

This paper deals with the study of mixed variational-like inclusions involving Yosida approximation operator (MVLIYAO) and η-proximal mapping in Banach spaces. It is investigated that (MVLIYAO) is equivalent to fixed point problems in Banach spaces. Using this equivalence, a new iterative algorithm is proposed to find the solution of (MVLIYAO). A numerical example is provided to support our main result by using the MATLAB program.

A Banach space X has the Dunford–Pettis property (DPP) if and only if whenever (xn) and (pn) are weakly null sequences in X and X*, respectively, we have pn(xn)→ 0. Freedman introduced a stricly weaker version of the DPP called the alternative Dunford–Pettis property (DP1). A Banach space X has the DP1 if whenever xn ! x weakly in X, with kxnk = kxk, and (xn) is weakly null in X*, we have that xn(xn)→ 0. The authors study the DP1 on projective tensor products of C*-algebras and JB*-triples. Their main result, Theorem 3.5, states that if X and Y are Banach spaces such that X contains an isometric copy of c0 and Y contains an isometric copy of C[0, 1], then Xˆ_Y , the projective tensor product of X and Y , does not have the DP1. As a corollary, they get that if X and Y are JB*-triples such that X is not reflexive and Y contains `1, then Xˆ_Y does not have the DP1. Furthermore, if A and B are infinite-dimensional C*-algebras, then Aˆ_B has the DPP if and only if Aˆ_B has the DP1 if and only if both A and B have the DPP and do not contain `1.

The main result is that if $\{ T(t):t \geqslant 0\}$ is a strongly continuous semigroup of scalar type operators on a weakly complete Banach space $X$ and if the resolutions of the identity for $T(t)$ are uniformly bounded in norm, then the infinitesimal generator is scalar type. Moreover, there exists a countably additive spectral measure $K( \cdot )$ such that $T(t) = \smallint \exp (\lambda t)dK(\lambda )$, for $t \geqslant 0$. This is a direct generalization of the well-known theorem of Sz.-Nagy about semigroups of normal operators on a Hilbert space. Similar spectral representations are given for representations of locally compact abelian groups and for semigroups of unbounded operators. Connections with the theory of hermitian and normal operators on Banach spaces are established. It is further shown that $R$ is the infinitesimal generator of a semigroup of hermitian operators on a Banach space if and only if iR is the generator of a group of isometries.

If E and F are real Banach spaces let Cp,q(E, F) O ≤ q ≤ p ≤ ∞, denote those maps from E to F which have p continuous Frechet derivatives of which the first q derivatives are bounded. A Banach space E is defined to be Cp,q smooth if Cp,q(E,R) contains a nonzero function with bounded support. This generalizes the standard Cp smoothness classification. If an Lp space, p ≥ 1, is Cq smooth then it is also Cq,q smooth so that in particular Lp for p an even integer is C∞,∞ smooth and Lp for p an odd integer is Cp-1,p-1 smooth. In general, however, a Cp smooth B-space need not be Cp,p smooth. Co is shown to be a non-C2,2 smooth B-space although it is known to be C∞ smooth. It is proved that if E is Cp,1 smooth then Co(E) is Cp,1 smooth and if E has an equivalent Cp norm then co(E) has an equivalent Cp norm. Various consequences of Cp,q smoothness are studied. If f ϵ Cp,q(E,F), if F is Cp,q smooth and if E is non-Cp,q smooth, then the image under f of the boundary of any bounded open subset U of E is dense in the image of U. If E is separable then E is Cp,q smooth if and only if E admits Cp,q partitions of unity; E is Cp,psmooth, p ˂∞, if and only if every closed subset of E is the zero set of some CP function. f ϵ Cq(E,F), 0 ≤ q ≤ p ≤ ∞, is said to be Cp,q approximable on a subset U of E if for any ϵ ˃ 0 there exists a g ϵ Cp(E,F) satisfying sup/xϵU, O≤k≤q ‖ Dk f(x) - Dk g(x) ‖ ≤ ϵ. It is shown that if E is separable and Cp,q smooth and if f ϵ Cq(E,F) is Cp,q approximable on some neighborhood of every point of E, then F is Cp,q approximable on all of E. In general it is unknown whether an arbitrary function in C1(l2, R) is C2,1 approximable and an example of a function in C1(l2, R) which may not be C2,1 approximable is given. A weak form of C∞,q, q≥1, to functions in Cq(l2, R) is proved: Let {Uα} be a locally finite cover of l2 and let {Tα} be a corresponding collection of Hilbert-Schmidt operators on l2. Then for any f ϵ Cq(l2,F) such that for all α sup ‖ Dk(f(x)-g(x))[Tαh]‖ ≤ 1. xϵUα,‖h‖≤1, 0≤k≤q

In this paper, we investigate the boundedness of composition operators defined on a quasi-Banach space continuously included in the space of smooth functions on a manifold. We prove that the boundedness of a composition operator strongly limits the behavior of the original map, and it provides an effective method to investigate the properties of composition operators using the theory of dynamical system. Consequently, we prove that only affine maps can induce bounded composition operators on any quasi-Banach space continuously included in the space of entire functions of one variable if the function space contains a nonconstant function. We also prove that any polynomial automorphisms except affine transforms cannot induce bounded composition operators on a quasi-Banach space composed of entire functions in the two-dimensional complex affine space under several mild conditions.

We present a novel generalization, in Banach spaces, of the celebrated Helly’s principle selection. Specifically, our main result is a quantitative version of such principle selection. Our main tool is the so called Degree of Nondensifiability, which measures (in the specified sense) the distance of a given convex subset of a Banach space to the class of its Peano Continua. As application of our results, we analyze the solvability of certain Volterra integral equations.

We consider the existence of mild solutions for fractional semilinear differential inclusions involving a nonconvex set-valued map in Banach spaces. First, we study the continuous property of the solution map for an auxiliary fractional differential equation. Then the main result is obtained by using this solution map, selection theorems from multivalued analysis and Schauder’s fixed point theorem. Finally an example to illustrate the applications of the main result is also given.