Uniqueness of symmetric basis in quasi-Banach spaces

We show that if X is a nonlocally convex natural quasi-Banach space with symmetric basis whose Banach envelope is isomorphic to ?₁, then all symmetric bases of X are equivalent. The scope of this result is quite ample since the Banach envelopes of natural quasi-Banach spaces with basis always exhibit an ?₁-like behavior, in the sense that they contain copies of 's uniformly complemented.     

Uniqueness of unconditional basis in Lorentz sequence spaces

We show that the Lorentz sequence spaces with and have unique unconditional basis. This completely settles the question of uniqueness of unconditional basis in Lorentz sequence spaces, and solves a problem raised by Popa in 1981 and Nawrocki and Ortyński in 1985.

     

Uniqueness of unconditional bases in Banach spaces

We prove a general results on complemented unconditional basic sequences in Banach lattices and apply it to give some new examples of spaces with unique unconditional basis. We show that Tsirelson space and certain Nakano spaces have unique unconditional bases. We also construct an example of a space with a unique unconditional basis with a complemented subspace failing to have a unique unconditional basis. Both authors were supported by NSF Grant DMS-9201357.     

Uniqueness of unconditional bases in quasi-banach spaces with applications to Hardy spaces

We prove some general results on the uniqueness of unconditional bases in quasi-Banach spaces. We show in particular that certain Lorentz spaces have unique unconditional bases answering a question of Nawrocki and Ortynski. We then give applications of these results to Hardy spaces by showing the spacesH _p (T ⁿ ) are mutually non-isomorphic for differing values ofn when 0<p<1. The research of the first two authors was partially supported by NSF-grant DMS 8901636.     

Uniqueness questions for two-dimensional surfaces in Euclidean spaces

A geometric approach to the solution of uniqueness problems for two-dimensional surfaces with nonzero Gaussian curvature in E⁴ is presented.Translated from Itogi Nauki i Tekhniki. Problemy Geometrii, Vol. 8, pp. 243–256, 1977.The author thanks Professor É. G. Poznyak profoundly for help.     

Uniqueness of unconditional bases in quasi-banach spaces with applications to hardy spaces, II

We prove that a wide class of quasi-Banach spaces has a unique up to a permutation unconditional basis. This applies in particular to Hardy spacesH _p forp<1. We also investigate the structure of complemented subspaces ofH _p (D). The proofs use in essential way matching theory. This research was supported in part by KBN grant N. 2P301004.06 and by the Overseas Visiting Scholarship of St. John’s College, Cambridge.     

Characterization of quasi-Banach spaces which coarsely embed into a Hilbert space

We show that a quasi-Banach space coarsely embeds into a Hilbert space if and only if it is isomorphic to a linear subspace of for some probability space .

     

Absolutely summing polynomials on Banach spaces with unconditional basis

If X is a Banach space with a normalized unconditional Schauder basis (xn), we define whenever and obtain estimates for μX,(xn) when every continuous m-homogeneous polynomial from X into Y is absolutely (q,1) summing. Our results provide new information on coincidence situations between the space of absolutely summing m-homogeneous polynomials and the whole space of continuous m-homogeneous polynomials. In particular, when m=1, we obtain new contributions to the linear theory of absolutely summing operators.     

Quasi-minimality and tightness by range in spaces with unconditional basis

We present a reflexive Banach space with an unconditional basis which is quasi-minimal and tight by range, i.e., of type (4) in the Ferenczi-Rosendal list within the framework of Gowers’ classification program of Banach spaces. The space is an unconditional variant of the Gowers Hereditarily Indecomposable space with an asymptotically unconditional basis.     

A class of bornological barrelled spaces which are not ultrabornological

N. Bourbaki, [1, p. 35], notices that it is not known if every bornological barrelled space is ultrabornological. In this paper we prove that if E is the topological product of an infinite family of bornological barrelled spaces, of non-zero dimension, there exists an infinite number of bornological barrelled subspaces ofE, which are not ultrabornological. We also give some examples of barrelled normable spaces which are not ultrabornological.Supported in part by the Patronato para el Fomento de la Investigación en la Universidad.     

A characterization of 2-trivial Banach spaces with unconditional basis

Let X be a Banach space with an unconditional basis such that each operator from X into ² is 2-absolutely summing. Then X is isomorphic either to c_o or to ¹ or to c_o¹.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 157, pp. 76–87, 1987.The author is grateful to I. A. Komarchev for a series of useful marks and for the permission to publish the proof of Lemma 1.     

An unconditional basis in periodic spaces with dominating mixed smoothness properties

В статье построен без условный базис в функ циональных пространствах с доми нирующей смешанной гладкость ю типа Никольского — А манова:S _p,q ^r (Т ₂),p = (р ₁,р ₂,q=(q ₁,q ₂),r=(r ₁,r ₂), 1 <p _i , < ∞, 1 <q _i < ∞, -∞<r _i <∞,i=1, 2;T ₂ — двухмерный тор. С помощью этого ба зиса исследование собств енных чисел некоторых инте гральных операторов редуцируется к соотв етствующей задаче для матричных операт оров.     

Weighted Bloch spaces which are Banach spaces

Let $\mathcal{B }_\omega $ be a weighted Bloch space on the open unit disc which is a Banach space. In this paper, we study $\mathcal{B }_\omega $ by using four operators, that is, a point derivation, a point evaluation, a composition operator and an integral operator.     

Spaceability in Banach and quasi-Banach sequence spaces

Let X be a Banach space. We prove that, for a large class of Banach or quasi-Banach spaces E of X-valued sequences, the sets E-?_q∈Γ?_q(X), where Γ is any subset of (0,∞], and E-c₀(X) contain closed infinite-dimensional subspaces of E (if non-empty, of course). This result is applied in several particular cases and it is also shown that the same technique can be used to improve a result on the existence of spaces formed by norm-attaining linear operators.