Rates of decay in the classical Katznelson-Tzafriri theorem

The Katznelson-Tzafriri Theorem states that, given a power-bounded operator T, ∥Tⁿ(I ? T)∥ → 0 as n → ∞ if and only if the spectrum σ(T) of T intersects the unit circle T in at most the point 1. This paper investigates the rate at which decay takes place when σ(T) ∩ T = {1}. The results obtained lead, in particular, to both upper and lower bounds on this rate of decay in terms of the growth of the resolvent operator R(e^iθ, T) as θ → 0. In the special case of polynomial resolvent growth, these bounds are then shown to be optimal for general Banach spaces but not in the Hilbert space case.     

Cantor-Lebesgue theorem for double trignometric series

Let ∥·∥ be a norm in R² and let γ be the unit sphere induced by this norm. We call a segment joining points x,y ε R² rational if (x₁ ? y₁)/(x₂ ? y₂) or (x₂ ? y₂)/(x₁ ? y₁) is a rational number. Let γ be a convex curve containing no rational segments. Satisfaction of the condition $$T_\nu (x) = \sum\nolimits_{\parallel n\parallel = \nu } {c_n e^{2\pi i(n_1 x_1 + n_2 x_2 )} } \to 0(\nu \to \infty )$$ in measure on the set e? [- 1/2,1/2)×[- 1/2, 1/2) =T² of positive planar measure implies ∥T _v ∥L₄ (T²) → 0(v → ∞). if, however, γ contains a rational segment, then there exist a sequence of polynomials {T _v } and a set E ? T², ¦E¦ > 0, such that T _v (x) → 0(v → ∞) on E; however, ¦c_n¦ ? 0 for ∥n∥ → ∞.     

On a problem of Nirenberg concerning expanding maps

Let X be a Banach space and T:X → X a continuous map, which is expanding (i.e., ∥Tu ? Tv∥ ? ∥u ? v∥ for all u, v?X) and such that T(X) has a nonempty interior. Does this guarantee that T is onto? We give a counterexample in the case of X=L¹(N).     

The alternative Daugavet property of C *‐algebras and JB *‐triples

A Banach space X is said to have the alternative Daugavet property if for every (bounded and linear) rank‐one operator T: X → X there exists a modulus one scalar ω such that ∥Id+ωT ∥ = 1 + ∥T ∥. We give geometric characterizations of this property in the setting of C *‐algebras, JB *‐triples, and of their isometric preduals. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Inner derivations on ultraprime normed real algebras

We show that the set of all inner derivations of an ultraprime real Banach algebra is closed within all bounded derivations. More concretely, we show that for such an algebra A there exists a positive number γ (depending only on the “constant of ultraprimeness” of A) satisfying γ ∥ a+Z(A) ∥≦∥ D _a ∥ for all a in A, where Z(A) denotes the centre of A and D _a denotes the inner derivation on A induced by a. This result is an extension of the corresponding complex version obtained by the authors in [Proc. Amer. Math. Soc., to appear]. The proof relies on the following theorem: ultraproducts of a family of central ultraprime real Banach algebras with a unit and with constant of ultraprimeness greater than or equal to a fixed positive constant K are central ultraprime Banach algebras with a unit. This fact is obained via a general result for real Banach algebras that reads as follows: If A is a central real Banach algebra with a unit 1, then for every a in A satisfying ∥ 1+a ² ∥<1 we have [1+√1?||1+1a ²||]²≦2(|?l+M _a ||+||D _a ||) where M _a denotes the two-sided multiplication operator by a on A.     

SPECIAL AUTOMORPHISMS WITH CONTINUOUS SPECTRUM

Abstract

In this paper conditions are given for the primitive automorphism of a cyclic KS approximation to have continuous spectrum. If T: X → X admits a cyclic KS approximation with speed o(1/n) it is then shown that for a dense set of measurable sets A € 𝔉, T^A: X^A → X^A is weakly mixing, i.e. has continuous spectrum. In particular it is shown that if T_α is an irrational rotation on the unit circle there exists an uncountable dense set of measurable sets for which (T^α)^A has continuous spectrum.     

A limit theorem for order preserving nonexpansive operators inL 1

We prove the following theorem:Let T be an order preserving nonexpansive operator on L ₁ (μ) (or L ₁ ⁺ ) of a σ-finite measure, which also decreases theL _∞-norm, and let S=tI+(1?t)T for 0<t<1. Then for everyf ∈ L_p (1<p<∞),the sequence S ⁿf converges weakly in L_p. (The assumptions do not imply thatT is nonexpansive inL _p for anyp>1, even ifμ is finite.) For the proof we show that ∥S ⁿ⁺¹ f?S ⁿf∥ _p → 0 for everyf ∈L _p, 1<p<∞, and apply toS the following theorem:Let T be order preserving and nonexpansive in L ₁ ⁺ , and assume that T decreases theL _∞-norm. Then forg ∈L _p (1<p<∞) Tⁿg is weakly almost convergent. If forf ∈ L_p we have T ⁿ⁺¹ f?T ⁿ f → 0weakly, then T ⁿf converges weakly in L_p (1<p<∞).     

On the Sidon-Telyakovskii integrability class for cosine series

Various reformulations of the classical Sidon integrability condition for cosine series are considered. Two apparent generalizations are shown to be equivalent to Sidon's condition. As an offshoot of these results it is shown that if the cosine Fourier coefficients satisfy $\text{n¦Δa}{}_{\mathrm{n}}\text{¦ = o(1) (n → ∞)}$, then ∥S_n(f) ?f ∥ = o(1) (n → ∞) is equivalent to a_nlgn = o(1) (n → ∞).     

Some generalizations of the first Fredholm theorem to multivalued A-proper mappings with applications to nonlinear elliptic equations

Let X and Y be real normed spaces with an admissible scheme Γ = {E_n, V_n; F_n, W_n} and T: X → 2^YA-proper with respect to Γ such that dist(y, A(x)) < kc(∥ x ∥) for all y in T(x) with ∥ x ∥ ? R for some R > 0 and k > 0, where c: R⁺ → R⁺ is a given function and A: X → 2^Y a suitable possibly not A-proper mapping. Under the assumption that either T or A is odd or that (u, Kx) ? 0 for all u in T(x) with $\text{∥ x ∥ ? r > 0}\text{and some}\text{K: X → Y}{}^1$, we obtain (in a constructive way) various generalizations of the first Fredholm theorem. The unique approximation-solvability results for the equation T(x) = f with T such that T(x) ? T(y) ?A(x ? y) for x, y in X or T is Fréchet differentiable are also established. The abstract results for A-proper mappings are then applied to the (constructive) solvability of some boundary value problems for quasilinear elliptic equations. Some of our results include the results of Lasota, Lasota-Opial, Hess, Ne?as, Petryshyn, and Babu?ka.     

Invariant subspaces for polynomially bounded operators

Let T be a polynomially bounded operator on a Banach space X whose spectrum contains the unit circle. Then T∗ has a nontrivial invariant subspace. In particular, if X is reflexive, then T itself has a nontrivial invariant subspace. This generalizes the well-known result of Brown, Chevreau, and Pearcy for Hilbert space contractions.     

A spectral radius theorem for matrix seminorms

A matrix seminorm ∥·∥ is called supspectral if it satisfies the condition that the spectral radius of a square matrix A is lim sup ∥Aⁿ∥^1/n as n→∞. This property is shown to be equivalent to each of two conditions on ∥·∥, one characterizing behavior on idempotent A, and the other characterizing behavior on non-nilpotent A. Examples of supspectral seminorms are provided.     

The approximation of flows

Let U, V be two strongly continuous one-parameter groups of bounded operators on a Banach space $\text{X}$ with corresponding infinitesimal generators S, T. We prove the following: ∥U_t, ? V_t ∥ = O(t), t → 0, if and only if U = V; ∥U_t ? V_t∥ = O(t^α), t → 0; with 0 ? α ? 1, if and only if $\text{S = Ω(T + P)Ω}{}^{\mathrm{?1}}$, where Ω, P, are bounded operators on $\text{X}$ such that $\text{∥U}{}_{\mathrm{t}}\text{Ω ? ΩU}{}_{\mathrm{t}}\text{∥ = O(t}{}^{\mathrm{\alpha }}\text{), ∥U}{}_{\mathrm{t}}\text{P ? PU}{}_{\mathrm{t}}\text{∥ = ?O(t}{}^{\mathrm{\alpha }}\text{), t → 0; ∥U}{}_{\mathrm{t}}\text{? V}{}_{\mathrm{t}}\text{∥ = O(t)}$ if and only if $\text{S}{}^1\text{? T}{}^1$ has a bounded extension to $\text{X}$¹. Further results of this nature are inferred for semigroups, reflexive spaces, Hilbert spaces, and von Neumann algebras.     

Norm estimates related to self-commulators

For a given self-adjoint matrix H with ∥H∥ = 1 and tr(H) = 0, we consider the number γ(H) which is defined to be the minimum of ∥T∥² for those T satisfying [T¹,T] = H. We show that 1 ⩽ γ(H) ⩽ 2 and that γ(H) is close to 2 if H is suitably chosen.     

Computing projections via Householder transformations and Gram–Schmidt orthogonalizations

Let x * denote the solution of a linear least‐squares problem of the form where A is a full rank m × n matrix, m > n. Let r *= b ‐ A x * denote the corresponding residual vector. In most problems one is satisfied with accurate computation of x *. Yet in some applications, such as affine scaling methods, one is also interested in accurate computation of the unit residual vector r */∥ r *∥₂. The difficulties arise when ∥ r *∥₂ is much smaller than ∥ b ∥₂. Let x? and r? denote the computed values of x * and r *, respectively. Let εdenote the machine precision in our computations, and assume that r? is computed from the equality r? = b ‐A x? . Then, no matter how accurate x? is, the unit residual vector û = r? /∥ r? ∥₂ contains an error vector whose size is likely to exceed ε∥ b ∥₂/∥ r* ∥₂. That is, the smaller ∥ r* ∥₂ the larger the error. Thus although the computed unit residual should satisfy A^T û = 0 , in practice the size of ∥A^T û ∥₂ is about ε∥A∥₂∥ b ∥₂/∥ r* ∥₂. The methods discussed in this paper compute a residual vector, r? , for which ∥A^T r? ∥₂ is not much larger than ε∥A∥₂∥ r? ∥₂. Numerical experiments illustrate the difficulties in computing small residuals and the usefulness of the proposed safeguards. Copyright © 2004 John Wiley & Sons, Ltd.     

Isometric factorization of weakly compact operators and the approximation property

Using an isometric version of the Davis, Figiel, Johnson, and Pe?czyński factorization of weakly compact operators, we prove that a Banach spaceX has the approximation property if and only if, for every Banach spaceY, the finite rank operators of norm ≤1 are dense in the unit ball ofW(Y,X), the space of weakly compact operators fromY toX, in the strong operator topology. We also show that, for every finite dimensional subspaceF ofW(Y,X), there are a reflexive spaceZ, a norm one operatorJ:Y→Z, and an isometry Φ :F →W(Y,X) which preserves finite rank and compact operators so thatT=Φ(T) oJ for allT∈F. This enables us to prove thatX has the approximation property if and only if the finite rank operators form an ideal inW(Y,X) for all Banach spacesY.     

Paranormal Elements in Normed Algebra

For a normed algebra A and natural numbers k we introduce and investigate the ∥ · ∥ closed classes P _k (A). We show that P₁(A) is a subset of P _k (A) for all k. If T in P₁(A), then Tⁿ lies in P₁(A) for all natural n. If A is unital, U, V ∈ A are such that ∥U∥ = ∥V∥ = 1, VU = I and T lies in P _k (A), then UTV lies in P _k (A) for all natural k. Let A be unital, then 1) if an element T in P₁(A) is right invertible, then any right inverse element T^?1 lies in P₁(A); 2) for ßßIßß = 1 the class P₁(A) consists of normaloid elements; 3) if the spectrum of an element T, T ∈ P₁(A) lies on the unit circle, then ∥TX∥ = ∥X∥ for all X ∈ A. If A = B(H), then the class P₁(A) coincides with the set of all paranormal operators on a Hilbert space H.     

Applications of the theory of semi-embeddings to Banach space theory

Let X and Y be Banach spaces and T:X → Y an injective bounded linear operator. T is called a semi-embedding if T maps the closed unit ball of X to a closed subset of Y. (This concept was introduced by Lotz, Peck, and Porta, Proc. Edinburgh Math. Soc.22 (1979), 233–240.) It is proved that if X semi-embeds in Y, and X is separable, then X has the Radon-Nikodym property provided Y does. It is shown that if L¹ semi-embeds in Y, then Y fails the Schur property and contains a subspace isomorphic to l¹. As a consequence of the proof, it is shown that if X is a subspace of L¹, either L¹ embeds in X or l¹ embeds in $\text{L}{}^1\text{X}$. The simpler result that L¹ does not semi-embed in c₀ is treated separately. This result is used to deduce the classic result of Menchoff that there exists a singular probability measure on the circle with Fourier coefficients vanishing at infinity. Some generalizations of the notion of semi-embedding are given, and several complements and open questions are discussed.     

Shift-type invariant subspaces of contractions

Using the Sz.-Nagy-Foias functional model it was shown in [L. Kérchy, Injection of unilateral shifts into contractions with non-vanishing unitary asymptotes, Acta Sci. Math. (Szeged) 61 (1995) 443-476] that under certain conditions on a contraction T the natural embedding of a Hardy space of vector-valued functions into the corresponding L² space can be factored into the product of two transformations, intertwining T with a unilateral shift and with an absolutely continuous unitary operator, respectively. The norm estimates in the Factorization Theorem of this paper are sharpened to their best possible form by essential improvements in the proof. As a consequence we obtain that if the residual set of a contraction covers the whole unit circle then those invariant subspaces, where the restriction is similar to the unilateral shift with a similarity constant arbitrarily close to 1, span the whole space. Furthermore, the hyperinvariant subspace problem for asymptotically non-vanishing contractions is reduced to these special circumstances.     

Idempotence preserving maps on spaces of triangular matrices

SupposeF is an arbitrary field. Let |F| be the number of the elements ofF. LetT _n (F) be the space of allnxn upper-triangular matrices overF. A map Ψ: T _N (F) → T _N (F) is said to preserve idempotence ifA - λ B is idempotent if and only if Ψ(A) - λΨ(B) is idempotent for anyA, B ∈ T _n (F) and λ ∈ F. It is shown that: when the characteristic ofF is not 2, |F|>3 and n ≥ 3, Ψ:T _n (F) → T _n (F) is a map preserving idempotence if and only if there exists an invertible matrixP τ T _n (F) such that either ?(A) = PAP ^?1 for everyA ∈ T _n (F) or Ψ(A) = PJA ^t JP ^?1 for everyA ∈ T _n (F), whereJ = ∑ _n=1 ⁿ E _i,n+1?i and E_ij is the matrix with 1 in the (i,j)th entry and 0 elsewhere.     

Borel properties of linear operators

Given an injective bounded linear operator T:X→Y between Banach spaces, we study the Borel measurability of the inverse map T−1:TX→X. A remarkable result of Saint-Raymond (Ann. Inst. Fourier (Grenoble) 26 (1976) 211-256) states that if X is separable, then the Borel class of T⁻¹ is α if, and only if, X∗ is the αth iterated sequential weak∗-closure of T∗Y∗ for some countable ordinal α. We show that Saint-Raymond's result holds with minor changes for arbitrary Banach spaces if we assume that T has certain property named co-σ-discreteness after Hansell (Proc. London Math. Soc. 28 (1974) 683-699). As an application, we show that the Borel class of the inverse of a co-σ-discrete operator T can be estimated by the image of the unit ball or the restrictions of T to separable subspaces of X. Our results apply naturally when X is a WCD Banach space since in this case any injective bounded linear operator defined on X is automatically co-σ-discrete.