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The wandering subspace problem for an analytic norm-increasing m m -isometry T T on a Hilbert space H \mathcal {H} asks whether every T T -invariant subspace of H \mathcal {H} can be generated by a wandering subspace. An affirmative solution to this problem for m = 1 m=1 is ascribed to Beurling-Lax-Halmos, while that for m = 2 m=2 is due to Richter. In this paper, we capitalize on the idea of weighted shift on a one-circuit directed graph to construct a family of analytic cyclic 3 3 -isometries which do not admit the wandering subspace property and which are norm-increasing on the orthogonal complement of a one-dimensional space. Further, on this one-dimensional space, their norms can be made arbitrarily close to 1 1 . We also show that if the wandering subspace property fails for an analytic norm-increasing m m -isometry, then it fails miserably in the sense that the smallest T T -invariant subspace generated by the wandering subspace is of infinite codimension.

We examine a variant of a Banach space $\mathfrak{X}_{0,1}$ defined by Argyros, Beanland, and the second named author that has the property that it admits precisely two spreading models in every infinite dimensional subspace. We prove that this space is asymptotically symmetric and thus it provides a negative answer to a problem of Junge, the first. named author, and Odell., 12 pages

In his celebrated paper "Generic projections", John Mather has shown that almost all linear projections from a submanifold of a vector space into a subspace are transverse with respect to a given modular submanifold. In this paper, an improvement of Mather's result is stated. Namely, we show that almost all linear perturbations of a smooth mapping from a submanifold of $\mathbb{R}^m$ into $\mathbb{R}^\ell$ yield a transverse mapping with respect to a given modular submanifold. Moreover, applications of this result are given., Comment: 11 pages, to appear in Proceedings of the American Mathematical Society

We prove the result stated in the title. It comes as a consequence of the fact that the space L p ∩ L ∞ L_p \cap L_{\infty } , 1 ≤ p > ∞ 1\leq p>\infty , p ≠ 2 p\neq 2 , does not contain a complemented subspace isomorphic to L p L_p . In particular, as a subproduct, we show that L p ∩ L ∞ L_p \cap L_{\infty } contains a complemented subspace isomorphic to ℓ 2 {\ell }_2 if and only if p = 2 p = 2 .

We prove that, provided $d > k$, every sufficiently large subset of $\mathbf{F}_q^d$ contains an isometric copy of every $k$-simplex that avoids spanning a nontrivial self-orthogonal subspace. We obtain comparable results for simplices exhibiting self-orthogonal behavior., 11 pages; improved and revised

We show that the predual of a JBW ∗ ^* -triple has the weak Banach-Saks property, that is, reflexive subspaces of a JBW ∗ ^* -triple predual are super-reflexive. We also prove that JBW ∗ ^* -triple preduals satisfy the Komlós property (which can be considered an abstract version of the weak law of large numbers). The results rely on two previous papers from which we infer the fact that, like in the classical case of L 1 L^1 , a subspace of a JBW ∗ ^* -triple predual contains ℓ 1 \ell _1 as soon as it contains uniform copies of ℓ 1 n \ell _1^n .

We study the weak bounded approximation property (weak BAP) of pairs and show that each of the spaces c0 and 1 has a subspace having the approximation property but failing the weak BAP.

We give a construction of the Gurarij space, analogous to Katetov's construction of the Urysohn space. The adaptation of Katetov's technique uses a generalisation of the Arens-Eells enveloping space to metric space with a distinguished normed subspace. This allows us to give a positive answer to a question of Uspenskij, whether the linear isometry group of the Gurarij space is a universal Polish group.

If S S , T T , R R , and K K are non-zero positive operators on a Banach lattice such that S ↔ T ↔ R ⩽ K S\leftrightarrow T\leftrightarrow R\leqslant K , where “ ↔ \leftrightarrow ” stands for the commutation relation, T T is non-scalar, and K K is compact, then S S has an invariant subspace.

In this short note we are interested in studying Banach spaces in which the range of a projection of norm one whose kernel is of finite dimension is the intersection of ranges of finitely many projections of norm one whose kernels are of dimension one. We show that for a certain class of Banach spaces X X , the natural duality between X X and X ∗ ∗ X^{\ast \ast } can be exploited when the range of the projection is of finite codimension. We show that if X ∗ X^\ast is isometric to L 1 ( μ ) L^1(\mu ) , then any central subspace of finite codimension is an intersection of central subspaces of codimension one. These results extend a recent result of Bandyopadhyay and Dutta which was proved for ranges of projections of norm one with finite dimensional kernel in continuous function spaces and unifies some earlier work of Baronti and Papini.

Two new applications of a technique for spaceability are given in this paper. For the first time this technique is used in the investigation of the algebraic genericity property of the weak form of Peano’s theorem on the existence of solutions of the ODE u ′ = f ( u ) u’=f(u) on c 0 c_0 . The space of all continuous vector fields f f on c 0 c_0 is proved to contain a closed c \mathfrak {c} -dimensional subspace formed by fields f f for which, except for the null field, the weak form of Peano’s theorem fails to be true. The second application generalizes known results on the existence of closed c \mathfrak {c} -dimensional subspaces inside certain subsets of ℓ p ( X ) \ell _p(X) -spaces, 0 > p > ∞ 0 > p > \infty , to the existence of closed subspaces of maximal dimension inside such subsets.

We consider convolution sampling and reconstruction of signals in certain reproducing kernel subspaces of L p , 1 ≤ p ≤ ∞ L^p, 1\le p\le \infty . We show that signals in those subspaces could be stably reconstructed from their convolution samples taken on a relatively separated set with small gap. Exponential convergence and error estimates are established for the iterative approximation-projection reconstruction algorithm.

In this paper we study an operator T T on a Banach space E E which is M M -hypercyclic for some subspace M M of E E . We give a sufficient condition for such an operator to be M M -hypercyclic and use it to answer negatively two questions asked by Madore and Martínez-Avendaño. We also give a sufficient condition for T T to be M M -hypercyclic for all finite co-dimensional subspaces M M in E E .

We study the almost Daugavet property, a generalization of the Daugavet property. It is analysed what kind of subspaces and sums of Banach spaces with the almost Daugavet property have this property as well. The main result of the paper is: if $Z$ is a closed subspace of a separable almost Daugavet space $X$ such that the quotient space $X/Z$ contains no copy of $\ell_1$, then $Z$ has the almost Daugavet property, too., 5 pages

In this paper, we show that Beurling’s theorem on analytic Hilbert spaces over the complex plane analogous to the Hardy space or the Bergman space does not hold, but for finite co-dimensional quasi-invariant subspaces, they are generated by their wandering subspace if and only if they are generated by z n z^n provided that the order of the reproducing kernels K λ ( z ) K_\lambda (z) is less than 2 but not equal to 1.

Let V, W be linear spaces over an algebraically closed field, and let S be an n―dimensional subspace of linear operators that maps V into W. We give a sharp upper bound for the dimension of the intersection of all images of nonzero operators from S, namely dim (∩ A∈S\{0} ImA ) ≤ dim V - n + 1. As an application, we also give a sharp upper bound for the dimension of the reflexivity closure Ref S of S, namely dim (RefS) ≤ n(n + 1)/2.

Let M be an invariant subspace of the Bergman space L 2 a (D) and S M be the compression of the coordinate multiplication operator M z to the co-invariant subspace L 2 a (D) ⊖ M. The present paper determines when the identity representation of C* (S M ) is a boundary representation for the Banach subalgebra B(S M ). The paper also considers boundary representations on the co-invariant subspaces of L 2 a (B n ).

In this paper we study spaces of nuclear operators Af(C(Q)) and spaces of compact operators K(C(Q)) on spaces of continuous functions C(Q), where Q is a countable compact metric space, in connection with the C. Bessaga and A. Pelczynski isomorphic classification of these spaces. We show that the spaces K(C(Q)) [resp. N(C(Q))] and K(C(Q')) [resp. N(C(Q'))] are isomorphic if, and only if, C(Q) and C(Q') are isomorphic. We show also that N(C(Q)) is not isomorphic to a subspace of K(C(Q)).

For real C(K) spaces, we show that being injected in a Hilbert space is a 3-space property. As a consequence, we obtain that, when K does not carry a strictly positive Radon measure, every quadratic continuous homogeneous real-valued polynomial on C(K) admits a linear zero subspace enjoying a property which implies non-separability.

For k > 2, we construct finite Z/2Z-CW complexes with one 2/22 cell in dimensions 0, 1 and k + 1. Using a theorem of Bruce Hughes, we show that these complexes are not homotopically stratified by orbit type in the sense of Quinn. Homotopically stratified sets were introduced by Quinn in [5] as a means of studying purely topological stratified phenomena. Quinn showed, under suitable conditions, that the orbit space of a finite group acting on a manifold, with the orbit type partition, is a homotopically stratified set ([5, Corollary 1.6]). In this paper we construct examples of 2/22-CW complexes having few 2/22-cells whose orbit spaces, with the orbit type partition, are not homotopically stratified. A closed subspace Y of a space X is forward tame in X if there exists a neigh borhood U of Y in X and a homotopy H: U x I -* X such that Ho is in clusion U -* X, HtIy is inclusion Y )-* X for every t E I, H1(U) = Y and H((U Y) x [0, 1)) C X Y. The homotopy link of Y in X is holink(X, Y) = {w c XI w (t) E Y if and only if t = 0}. A stratification of a space X consists of an indexed locally finite partition {Xi I i C I} of X by locally closed subspaces. We refer to X together with its stratification as a stratified space. Given a space X with an action of a group G, the orbit type corresponding to a subgroup H C G is the set of all points in X whose isotropy group is conjugate to H. The orbit type partition of X consists of the connected components of the orbit types of X. The orbits of these components give a partition of the orbit space G\X. A stratified space X is said to satisfy the frontier condition if for every i, j C 1, Xi n closure(Xj) $& 0 implies that Xi C closure(Xj). This induces a relation < on X, defined by i < j if and only if i 74 j and Xi C closure(Xj). The orbit type stratification of a finite G-CW complex need not satisfy the frontier condition. For example, let X = Si V S1 V S' be the wedge of three circles along a basepoint *. Express X * as the union of three disjoint 1-cells el, el, e3 whose closures are the corresponding S1 factors and give X the 2/22-CW complex structure: one 2/22-0 cell, the basepoint * (isotropy 2/22), and two 2/22-1-cells: el U el on which 2/22 acts by interchanging el and el (trivial isotropy) and el (isotropy 2/22). The orbit Received by the editors December 14, 2007. 2000 Mathematics Subject Classification. Primary 57N80, 57S17, 57N40.

We notice that Shelah's Strong Hypothesis is equivalent to the following reflection principle: Suppose (X, r) is a first-countable space whose density is a regular cardinal, κ. If every separable subspace of X is of cardinality at most κ, then the cardinality of X is κ.

We prove a dispersive estimate for the Schrodinger equation on the real line, mapping between weighted L P spaces with stronger time-decay (|t|-3 2 versus |t|-½ ) than is possible on unweighted spaces. To satisfy this bound, the long-term behavior of solutions must include transport away from the origin. Our primary requirements are that 3 V be integrable and -Δ+V not have a resonance at zero energy. If a resonance is present (for example, in the free case), similar estimates are valid after projecting away from a rank-one subspace corresponding to the resonance.

The reduction theorem for the Leray-Schauder degree provides an efficient tool to calculate the value of the degree in a suitable invariant subspace. We shall prove how the calculation of the value of the topological degree for a mapping of class ( S + ) (S_+) from a real separable reflexive Banach space X X into the dual space X ∗ X^* can be reduced into the calculation of degree of mapping from a closed subspace V ⊂ X V\subset X into V ∗ . V^*. Since the Leray-Schauder mappings are acting from X X to X X and we are dealing with mappings from X X to X ∗ , X^*, the standard ‘invariant subspace’ condition must be replaced by a less obvious one.

Let L(U, V) be the set of all linear transformations from U to V, where U and V are vector spaces over a field F. We show that every n-dimensional subspace of L(U, V) is algebraically [√2n]-reflexive, where [ t ] denotes the largest integer not exceeding t, provided n is less than the cardinality of F.

We characterize finite-dimensional normed linear spaces as strongly proximinal subspaces in all their superspaces. A connection between upper Hausdorff semi-continuity of metric projection and finite dimensionality of subspace is given.

Let G G be a finite group generated by unitary reflections in a Hermitian space V V , and let ζ \zeta be a root of unity. Let E E be a subspace of V V , maximal with respect to the property of being a ζ \zeta -eigenspace of an element of G G , and let C C be the parabolic subgroup of elements fixing E E pointwise. If χ \chi is any linear character of G G , we give a condition for the restriction of χ \chi to C C to be trivial in terms of the invariant theory of G G , and give a formula for the polynomial ∑ x ∈ G χ ( x ) T d ( x , ζ ) \sum _{x\in G}\chi (x)T^{d(x,\zeta )} , where d ( x , ζ ) d(x,\zeta ) is the dimension of the ζ \zeta -eigenspace of x x . Applications include criteria for regularity, and new connections between the invariant theory and the structure of G G .

A known, and easy to establish, fact in Best Approximation Theory is that, if the unit ball of a subspace G of a Banach space X is proximinal in X, then G itself is proximinal in X. We are concerned in this article with the reverse implication, as the knowledge of whether the unit ball is proximinal or not is useful in obtaining information about other problems. We show, by constructing a counterexample, that the answer is negative in general.

In this note, a new characterization of gaps between two subspaces of a Hilbert space is established.

The class of J-lattices was originally defined in the second author's thesis and subsequently by Longstaff, Nation, and Panaia. A subspace lattice £ on a Banach space X which is also a J-lattice is called a J-subspace lattice, abbreviated JSL. It is demonstrated that every single element of AlgL has rank at most one. It is also shown that AlgL has the strong finite rank decomposability property. Let L 1 and L 2 be subspace lattices that are also JSL's on the Banach spaces X 1 and X 2 , respectively. The two properties just referred to, when combined, show that every algebraic isomorphism between AlgL 1 and AlgL 2 preserves rank. Finally we prove that every algebraic isomorphism between AlgL 1 and AlgL 2 is quasi-spatial.

Let ∞ > p 1 > p 2 > ⋯ > 1 \infty > p_1 > p_2 > \cdots > 1 . We construct an easily determined 1 1 -symmetric basic sequence in ( ∑ n = 1 ∞ ⊕ ℓ p n ) 1 \Bigl ( \sum \limits _{n=1}^{\infty } \oplus \ell _{p_n} \Bigr )_1 , which spans a hereditarily ℓ 1 \ell _1 subspace without the Schur property. An immediate consequence is the existence of hereditarily ℓ 1 \ell _1 subspaces of L 1 L_1 without the Schur property.

We prove for some translation-invariant weighted spaces E E the following characterization: T T is a multiplier of E E if and only if T T leaves invariant every translation-invariant subspace of E E . This result is equivalent with the reflexivity of the multiplier algebra of E E .

We introduce the concept of the modular function for a shift-invariant subspace that can be represented by normalized tight frame generators for the shift-invariant subspace and prove that it is independent of the selections of the frame generators for the subspace. We shall apply it to study the connections between the dimension functions of wavelet frames for any expansive integer matrix A and the multiplicity functions for general multiresolution analysis (GMRA). Given a frame mutiresolution analysis (FMRA), we show that the standard construction formula for orthonormal multiresolution analysis wavelets does not yield wavelet frames unless the underlying FMRA is an MRA. A modified explicit construction formula for FMRA wavelet frames is given in terms of the frames scaling functions and the low-pass filters.

We present, two examples of WCG spaces that are not hereditarily WCG. The first is a space with an unconditional basis, and the second is a space X such that X** is WCG and X** does not contain 1 . The non-WCG subspace Y of X has the additional property that Y** is not WCG and X/Y is reflexive.

A Hermite-Biehler function E E gives rise to a de Branges Hilbert space H ( E ) \mathcal {H}(E) consisting of entire functions. We are going to show that for Hermite-Biehler functions of sufficiently small growth and a certain distribution of zeros every proper de Branges subspace of H ( E ) \mathcal {H}(E) coincides for some n ∈ N n\in \mathbb {N} with the ( n + 1 ) (n+1) -dimensional linear space of all polynomials of degree at most n n .

It is shown that if X is compact then every subspace of Cp(X) is a D-space in the sense of E. van Douwen, which positively answers Matveev's question. A connection between the D-property and Baturov's and Grothendieck's classical theorems about function spaces over compacta is established.

Let L \mathcal {L} be a subspace lattice on a normed space X X containing a nontrivial comparable element. If T T commutes with all the operators in Alg L \mbox {Alg}\mathcal {L} , then there exists a scalar λ \lambda such that ( T − λ I ) 2 = 0 (T-\lambda I)^2=0 . Furthermore, we classify the class of completely distributive subspace lattices into subclasses called Type I ( n ) I^{(n)} , Type I I ( n ) II^{(n)} and Type I I I III , respectively. It is shown that nontrivial nests or, more generally, completely distributive subspace lattices with a comparable element are Type I ( 1 ) I^{(1)} , and that nontrivial atomic Boolean subspace lattices are Type I I ( 0 ) II^{(0)} .

In this paper we prove the equivalence between the existence of perfectly normal, non-metrizable, non-archimedean spaces and the existence of “generalized Souslin lines", i.e., linearly ordered spaces in which every collection of disjoint open intervals is σ \sigma -discrete, but which do not have a σ \sigma -discrete dense set. The key ingredient is the observation that every first countable linearly ordered space has a dense non-archimedean subspace.

A subspace Y of a Banach space X is an almost constrained (AC) subspace of X if any family of closed balls centred at points of Y that intersects in X also intersects in Y. In this paper, we show that a subspace H of finite codimension in the space C(K) of continuous functions on a compact Hausdor space K is an AC-subspace if and only if H is the range of a norm one projection in C(K). We also give a simple proof that the implication "AC ) 1-complemented" holds for any subspace of the spaces c0() and c.

Let X be a subspace of the product of finitely many ordinals. If X is normal, then X is strongly zero-dimensional, collectionwise normal, and shrinking. The proof uses (κ 1 ,...,κ n )-stationary sets.

We prove that if a Banach space with a bimonotone shrinking basis does not contain l w 1 spreading models but every block sequence of the basis contains a further block sequence which is a c - l n 1 spreading model for every n E N, then every subspace has a further subspace which is arbitrarily distortable. We also prove that a mixed Tsirelson space T[(S n ,θ n ) n ], such that θ n ? 0, does not contain l w2 1 spreading models.

We prove that analytic operators satisfying certain series of operator inequalities possess the wandering subspace property. As a corollary, we obtain Beurling-type theorems for invariant subspaces in certain weighted l 2 l^2 and Bergman spaces.

In both the Bergman space A 2 and the Hardy space H 2 , the problem of determining which bounded univalent mappings of the unit disk have the wandering property is addressed. Generally, a function g in H∞ has the wandering property in X, where X denotes either A 2 or H 2 , provided that every g-invariant subspace M of X is generated by the orthocomplement of gM within M. It is known that essentially every function which has the wandering property in either space is the composition of a univalent mapping with a classical inner function, and that the identity mapping has this property in both spaces. Consequently, weak-star generators of H°° also have the wandering property in both settings. The present paper gives a partial converse to this, and shows that in both settings there is a large class of bounded univalent mappings which fail to have the wandering property.

It is shown that the span of {a i h i ○+ b i e i } n i=1 , where {h i } is the Haar system in Lp and {e i } the canonical basis of l p , is well isomorphic to a well complemented subspace of Lp, 2 < p < ∞. As a consequence we get that there is a rearrangement of the (initial segments of the) Haar system in L p , 2 < p < ∞, any block basis of which is well isomorphic to a well complemented subspace of L p .

It is shown that if every ≤ ω 1 -sized subspace of a (regular) space X of density ≤ ω 1 has a point-countable base, then so does X. Similar results hold for meta-Lindelofness. Dow's reflection theorem and a number of other results are deduced as corollaries and applications.

We show that every Banach space saturated with subsymmetric sequences contains a minimal subspace., Comment: 8 pages, minor corrections and redaction changes

A calculation formula is established for the codimension of the polynomial subspace in L 2 (R,dμ) with discrete indeterminate measure μ. We clarify how much the masspoint of the n-canonical solution of an indeterminate Hamburger moment problem differs from the masspoint of the corresponding N-extremal solution at a given point of the real axis.

In his famous paper The image of a derivation is contained in the radical, Marc Thomas establishes the (commutative) Singer-Wermer conjecture, showing that derivations from a commutative Banach algebra A to itself must map into the radical. The proof goes via first showing that the separating subspace of a derivation on A must lie in the radical of A. In this paper, we exhibit discontinuous derivations on a commutative unital Frechet algebra A such that the separating subspace is the whole of A. Thus, the situation on Frechet algebras is markedly different from that on Banach algebras.

We show that L∞ (μ, X* ) is a local dual of L 1 (μ,X ), and L 1 (μ, X*) is a local dual of L∞ (μ, X), where X is a Banach space. A local dual space of a Banach space Y is a subspace Z of Y* so that we have a local representation of Y* in Z satisfying the properties of the representation of X** in X provided by the principle of local reflexivity.

Let X be a real or complex Banach space and K > 1. Then X* contains a K-complemented, isometric copy of L 1 [0, 1] if and only if X* contains a K-complemented, isometric copy of C [0, 1]* if and only if X contains a subspace (1, K)-asymptotic to (l 1 ○+ Σ n l n ∞ ) 1 .

A consistent example of an uncountable Lindelof T 3 (and hence normal) space with no Lindelof subspace of size N 1 is constructed. It remains unsolved whether extra set-theoretic assumptions are necessary for the existence of such a space. However, our space has size N 2 and is a P-space, i.e., G δ 's are open, and for such spaces extra set-theoretic assumptions turn out to be necessary.