Perturbations of Generalized Dual Banach Frames

Almost overcomplete and almost overtotal sequences in Banach spaces

Let E be a Banach sequence space with the property that if (αi) ∈ E and |βi|≦|αi| for all i then (βi) ∈ E and ‖(βi)‖E≦‖(αi)‖E. For example E could be co, lp or some Orlicz sequence space. If (Xn) is a sequence of real or complex Banach spaces, then E can be used to construct a vector sequence space which we will call the E sum of the Xn's and symbolize by ⊕EXn. Specifically, ⊕EXn = {(xn)|(xn)∈Xn and (‖xn‖)∈E}. The E sum is a Banach space with norm defined by: ‖(xn)‖ = ‖(‖xn‖)‖E. This type of space has long been the source of examples and counter-examples in the geometric theory of Banach spaces. For instance, Day [7] used E=lp and Xk=lqk, with appropriate choice of qk, to give an example of a reflexive Banach space not isomorphic to any uniformly conves Banach space. Recently VanDulst and Devalk [33] have considered Orlicz sums of Banach spaces in their studies of Kadec-Klee property.

Given an analytic multiple gap $$\Gamma = \{\Gamma _i{: }\,i<n\}$$ , we study the family $$\mathfrak {B}(\Gamma )$$ of the sets $$A\subset n$$ for which there is a restriction $$\{\Gamma _i|_a{: }\,i\in A\}$$ which is still a multiple gap, while $$a\in \Gamma _i^\perp $$ for $$i\not \in A$$ . This family always contains at least two sets of cardinality 2, and every set of cardinality k is contained in a set from $$\mathfrak {B}(\Gamma )$$ of cardinality J(k), a number that grows as $$\frac{3}{8\sqrt{2\pi k}}\cdot 9^k$$ . All these results can be stated in terms of the topology of the Cech–Stone remainder $$\omega ^*$$ and in terms of sequences in Banach spaces. For example, for any finite family of analytic open sets of $$\omega ^*$$ with non-disjoint closures there is always a point that lies in exactly two closures. And given a sequence $$\{x_n\}_{n<\omega }$$ of vectors in a Banach space that contains subsequences equivalent to $$\ell _1,\ell _2,\ldots ,\ell _n$$ in a way that cannot be separated, it always contains a subsequence $$\{x_{n_k}\}_{k<\omega }$$ where the $$\ell _1$$ and $$\ell _2$$ subsequences cannot be separated, while there are at most 6 (and this is sharp) of the remaining p’s for which $$\{x_{n_k}\}_{k<\omega }$$ contains subsequences equivalent to $$\ell _p$$ .

We introduce a geometric property on Banach spaces, the E-property, that is implied by the \( \lambda \)-property and that implies the Bade property, although these properties are different. By mean of the E-property we characterize the topological dimension of compact metric spaces. If \( (X_n)_{n\in {\Bbb N}} \) is a sequence of Banach spaces and \( p \in [1,+ \infty) \) we relate the E-property of \( (\oplus X_n) _p \) with the E-property of every Xn. Finally, if K is a compact Hausdorff space and X is a Banach space, we study the E-property on the dual space of C (K, X) .

On shrinking basic sequences in Banach spaces

Generalized framing and Riesz-dual sequences in Banach spaces

We address some phenomena about the interaction between lower semicontinuous submeasures on N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb {N}}$$\\end{document} and Fσ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$F_{\\sigma }$$\\end{document} ideals. We analyze the pathology degree of a submeasure and present a method to construct pathological Fσ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$F_{\\sigma }$$\\end{document} ideals. We give a partial answers to the question of whether every nonpathological tall Fσ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$F_{\\sigma }$$\\end{document} ideal is Katětov above the random ideal or at least has a Borel selector. Finally, we show a representation of nonpathological Fσ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$F_{\\sigma }$$\\end{document} ideals using sequences in Banach spaces.

We prove generalized convergence theorems and Tauberian theorems for vector-valued functions and sequences of growth order γ − 1 with γ > 0 and for positive functions and sequences in Banach lattices. Then the general results are applied to obtain some interesting particular Tauberian results for various examples of operator semigroups. Among them are mean ergodic theorems for Cesaro-mean-bounded semigroups (discrete and continuous) of operators and for semigroups of positive operators.

The strong and uniform ball-covering properties

A family of local atoms in a Banach space has been introduced and it has been generalized to an atomic system for operators in Banach spaces, which has been further led to introduce new frames for operators by Dastourian and Janfada, by making use of semi-inner products. Unlike the traditional way of considering sequences in the dual space, sequences in the original space are considered to study them. Appropriate changes have been made in the definitions of atomic systems and frames for operators to fit them for sequences in the dual space without using semi-inner products so that the new notion for Banach spaces can be thought of as a generalization of Banach frames. With some crucial assumptions, we show that frames for operators in Banach spaces share nice properties of frames for operators in Hilbert spaces.

A classical result due to M. Eidelheit and B. Yood states that the standard algebra norm on the algebra of bounded linear operators on a Banach space is minimal, in the sense that the norm must be less than a multiple of any other submultiplicative norm on the same algebra. This definition does not assume that the arbitrary algebra norm is complete. In cases when the standard algebra norm is, in addition, maximal, it is therefore unique up to equivalence. More recently, M. Meyer showed that the Calkin algebras of a very restricted class of Banach spaces also have unique algebra norms. We generalise the Eidelheit-Yood method of proof, to show that the conventional quotient norm on a larger class of Calkin algebras is minimal. Since maximality of the norm is a presumed property for the class, the norm is also unique. We thus extend the result of Meyer. In particular, we establish that the Calkin algebras of canonical Banach spaces such as James’ space and Tsirelson’s space have unique algebra norms, without assuming completeness. We also prove uniqueness of norm for quotients of the algebras of operators on classical non-separable spaces, the closed ideals of which were previously studied by M. Daws. One aspect of the Eidelheit-Yood method is a dependence on the uniform boundedness principle. As a component of our generalisation, we prove an analogue of that principle which applies to Calkin algebra elements rather than bounded linear operators. In order to translate the uniform boundedness principle into this new setting, we take the perspective that non-compact operators map certain wellseparated sequences to other well-separated sequences. We analyse the limiting separation of such sequences, using these values to measure the non-compactness of operators and define the requisite notion of a bounded set of non-compact operators. In the cases when the underlying Banach space has a Schauder basis, we are able to restrict attention to seminormalised block basic sequences. As a consequence, our main uniqueness of norm result for Calkin algebras relies on the existence of bounded mappings between, and projections onto, the spans of block basic sequences in the relevant Banach spaces.

This paper contains a survey of the most important basic properties of certain measures of noncompactness on bounded sets of complete metric spaces and Banach spaces, and of the Hausdorff measure of noncompactness of operators between Banach spaces. We also demonstrate how the theory of measures of noncompactness can be applied in fixed point theory, the theory of differential and integral equations, and the characterizations of classes of compact operators between certain Banach spaces.

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.

Banach frames in coorbit spaces consisting of elements which are invariant under symmetry groups

It is a well-known problem in Gabor analysis how to construct explicitly given dual frames associated with a given frame. In this paper we will consider a class of window functions for which approximately dual windows can be calculated explicitly. The method makes it possible to get arbitrarily close to perfect reconstruction by allowing the modulation parameter to vary. Explicit estimates for the deviation from perfect reconstruction are provided for some of the standard functions in Gabor analysis, e.g., the Gaussian and the two-sided exponential function.