Contributions to the theory of the classical Banach spaces

The paper contains several results on the linear topological structure of the spaces C(K), K compact metric, and L_p(0, 1), 1 ? p < ∞. The topics which are studied include: complemented subspaces, special Schauder bases, and equivalent norms in these spaces.     

A note on the sequence spacel (p v )

In this paper, the linear isometry of the sequence space l(p_v) into itself is specified as the automorphism of l(p_v) onto itself, when (p_v) satisfies the conditions, (i) 0 < p_v? 1, (ii) 1 +d ? p_v ? p < ∞,q < q_v < 1+d/d,d > o When (p_v) satisfies condition (ii),l (p_v) andl (q_v) are proved to be perfect spaces in the sense of Kothe and Toeplitz. A similar result connecting linear isometry and automorphism has been noted in the case of a non-normable complete linear metric space whose conjugate space is also determined.     

Duality for Lipschitz p-summing operators

Building upon the ideas of R. Arens and J. Eells (1956) [1] we introduce the concept of spaces of Banach-space-valued molecules, whose duals can be naturally identified with spaces of operators between a metric space and a Banach space. On these spaces we define analogues of the tensor norms of Chevet (1969) [3] and Saphar (1970) [14], whose duals are spaces of Lipschitz p-summing operators. In particular, we identify the dual of the space of Lipschitz p-summing operators from a finite metric space to a Banach space — answering a question of J. Farmer and W.B. Johnson (2009) [6] — and use it to give a new characterization of the non-linear concept of Lipschitz p-summing operator between metric spaces in terms of linear operators between certain Banach spaces. More generally, we define analogues of the norms of J.T. Lapresté (1976) [11], whose duals are analogues of A. Pietsch?s (p,r,s)-summing operators (A. Pietsch, 1980 [12]). As a special case, we get a Lipschitz version of (q,p)-dominated operators.     

A characterization of pointwise-lipschitz-continuous selections for the metric projection

In a series of papers the last two authors have obtained a complete characterization of those finite-dimensional subspaces G of C[a,b] for which there exists a continuous selection for the metric projection onto G. By showing that all continuous selections constructed there are even pointwise-Lipschitz-continuous and quasi-linear, we get a complete characterization concerning selections for the metric projection with this stronger property.     

A characterization of projective subspaces of codimension two as quasi-symmetric designs with good blocks

Consider an incidence structure whose points are the points of a PG_n(n+2,q) and whose block are the subspaces of codimension two, where n?2. Since every two subspaces of codimension two intersect in a subspace of codimension three or codimension four, it is easily seen that this incidence structure is a quasi-symmetric design. The aim of this paper is to prove a characterization of such designs (that are constructed using projective geometries) among the class of all the quasi-symmetric designs with correct parameters and with every block a good block. The paper also improves an earlier result for the special case of n=2 and obtains a Dembowski-Wagner-type result for the class of all such quasi-symmetric designs.     

Invariant subspaces and extremum problems in spaces of vector-valued functions

In this article we obtain, for 1 ? p ? ∞, a characterization of the invariant subspaces of spaces of vector-valued L^p functions defined on the unit circle—i.e., of those subspaces invariant under multiplication by e^ix. This result is then applied to extend, to the corresponding Hardy classes of vector-valued functions, the known characterizations of the extreme points of the unit ball in the scalar Hardy classes H¹ and H^∞. Finally, it is shown that the characterization of the closure of the set of extreme points of the unit ball in H¹ changes significantly when we pass from the scalar to the vector case.     

Block scaling with optimal Euclidean condition

Let $\text{M}$ denote the set of all complex n×n matrices whose columns span certain given linear subspaces. The minimal Euclidean condition number of matrices in $\text{M}$ is given in terms of the canonical angles between the linear subspaces, and optimal matrices in $\text{M}$ are described. The result is also stated in terms of norms of certain projections.     

Shape fibrations and strong shape theory

The notion of shape fibration was introduced by Marde?i? and Rushing. In this paper we use ‘fibrant space’ techniques in strong shape theory to prove that every shape fibration p:E → B of compact metric spaces is contained in a map of fibrant spaces p′:E′→B′ which enjoys a certain lifting property and whose homotopy properties reflect the strong shape properties of the map p. Standard methods for studying Hurewicz fibrations are readily applied to the map p' and in this way we obtain a number of strong shape generalizations of results of Marde?i? and Rushing. We also prove the following theorem which answers a question of Rushing: A shape fibration of compact metric spaces which is a strong shape equivalence is an hereditary shape equivalence. Since the converse was known, this gives a characterization of hereditary shape equivalences.     

Fixräume regulärer operatoren und ein Korovkin-Satz

S. J. Bernau has introduced the notion of an exchange subspace of an L_p-space and has shown that the range of a contractive linear projection on an L_p-space (1 ? p < ∞, p ≠ 2) is an exchange subspace. In the present paper we define this notion for real Banach lattices with order continuous norm and prove among other things that fixed spaces of special regular operators on these spaces are exchange subspaces. As application we give a Korovkin theorem for sequences of contractions on real Banach lattices with an uniformly monotone norm.     

Lindelöf type of generalization of separability in Banach spaces

We will introduce the countable separation property (CSP) of Banach spaces X, which is defined as follows: X has CSP if each family E of closed linear subspaces of X whose intersection is the zero space contains a countable subfamily E₀ with the same intersection. All separable Banach spaces have CSP and plenty of examples of non-separable CSP spaces are provided. Connections of CSP with Marku?evi?-bases, Corson property and related geometric issues are discussed.     

On the non-uniqueness of minimal projection in spaces

The main objective of this note is to exhibit a simple example of subspaces UL_p(μ) (p≠2) that admit two different projections with minimal norm. While for p=1,∞, such subspaces are well-known [W. Odyniec, G. Lewicki, Minimal Projections in Banach Spaces, in: Lecture Notes in Mathematics, vol. 1449, Springer-Verlag, Berlin, 1990. Problems of existence and uniqueness and their application], for 1<p<∞ their existence was open.     

Nonuniform behavior and robustness

We establish the robustness of linear cocycles in Banach spaces admitting a nonuniform exponential dichotomy. We first obtain robustness results for positive and negative time, by establishing exponential behavior along certain subspaces, and showing that the associated sequences of projections have bounded exponential growth. We then establish a robustness result in Z by constructing explicitly appropriate projections on the stable and unstable subspaces. We emphasize that in general these projections may be different from those obtained separately from the robustness for positive and negative time. We also consider the case of strong nonuniform exponential dichotomies.     

Weighted L p-Approximation of Derivatives by the Method of Gammaoperators

We consider Gammaoperators G _n on suitable Sobolev type subspaces of L _p(0, ∞) and characterize the global rate of approximation of derivatives f ^(r) through corresponding derivatives (G _n f)^(r) in an appropriate weighted L _p — metric by the rate of Ditzian and Totik’s second order weighted modulus of smoothness.     

度量射影和余度量射影的线性选择

本文讨论了度量射影和余度量射影的关系,并在L_p(T)和C(T)空间中讨论了它们的线性选择的性质.     

On a subspace metric based on matrix rank

The metric between subspaces M,N⊆C_n,1, defined by δ(M,N)=rk(P_M-P_N), where rk(·) denotes rank of a matrix argument and P_M and P_N are the orthogonal projectors onto the subspaces M and N, respectively, is investigated. Such a metric takes integer values only and is not induced by any vector norm. By exploiting partitioned representations of the projectors, several features of the metric δ(M,N) are identified. It turns out that the metric enjoys several properties possessed also by other measures used to characterize subspaces, such as distance (also called gap), Frobenius distance, direct distance, angle, or minimal angle.     

Ergodic Banach spaces

We show that any Banach space contains a continuum of non-isomorphic subspaces or a minimal subspace. We define an ergodic Banach space X as a space such that E₀ Borel reduces to isomorphism on the set of subspaces of X, and show that every Banach space is either ergodic or contains a subspace with an unconditional basis which is complementably universal for the family of its block-subspaces. We also use our methods to get uniformity results. We show that an unconditional basis of a Banach space, of which every block-subspace is complemented, must be asymptotically c₀ or ?_p, and we deduce some new characterisations of the classical spaces c₀ and ?_p.     

On the totality of sequences in topological vector spaces

Price and Zink [Ann. of Math.82 (1965), 139–145] gave necessary and sufficient conditions for the existence of a multiplier m so that {mφ_n}₁^∞ is total; that is, the linear span is dense in L²[0, 1], thus answering a question raised by Boas and Pollard [Bull. Amer. Math. Soc.54 (1948), 512–522]. Using techniques similar to those of Price and Zink, it is shown that this result can be extended to more general spaces. Indeed, if X is either a separable Fréchet space or a complete separable p-normed space (0 < p ? 1), then the existence of a continuous linear operator A so that {Aφ_n}₁^∞ spans a dense subspace is implied by the existence of a nested, equicontinuous family of commuting projections which in addition has the properties that the union of their ranges is dense and that, for each projection, the projection of the original sequence is total in the projected space. Conversely, in a Banach space, it is shown that if such an operator exists and is 1-1 and scalar, then such a family of projections also exists. Further, it is shown that the above considerations extend the first half of the Price-Zink result to L^p[0, 1] (0 < p < ∞) and the other half to L^p[0, 1] (1 ? p < ∞).     

Some properties of frames of subspaces obtained by operator theory methods

We study the relationship among operators, orthonormal basis of subspaces and frames of subspaces (also called fusion frames) for a separable Hilbert space H. We get sufficient conditions on an orthonormal basis of subspaces E={Ei}_i∈I of a Hilbert space K and a surjective T∈L(K,H) in order that {T(Ei)}_i∈I is a frame of subspaces with respect to a computable sequence of weights. We also obtain generalizations of results in [J.A. Antezana, G. Corach, M. Ruiz, D. Stojanoff, Oblique projections and frames, Proc. Amer. Math. Soc. 134 (2006) 1031-1037], which relate frames of subspaces (including the computation of their weights) and oblique projections. The notion of refinement of a fusion frame is defined and used to obtain results about the excess of such frames. We study the set of admissible weights for a generating sequence of subspaces. Several examples are given.     

Continuous selections and convergence of best Lp –approximations in subspaces of spline functions

The chief purpose of this paper is to study the problem of existence of continuous selections for the metric projection and of convergence of best L_p–approximations in subspaces of polynomial spline functions defined on a real compact interval I. Nürnberger-Sommer [8] have shown that there exists a continuous selection s if and only if the numberof knots k is less than or equal to the order m of the splines. Using their construction of s the author [12] has proved that the sequence of best L_p–approximations of f converges to s(f) as ρ→∞ for every continuous function f. The main results of this paper say that also in the case when k>m there exists always a continuous selection s (it is even pointwise-Lipschitz-continuous and quasi-linear) provided that the approximation problem is restricted to certain subsets I_epsilon; of I. In addition it is shown that anologously as for k≤m the sequence of best L_papproximations of f converges to s(f) for every continuous function f on Iε