Type norms with respect to characters of compact abelian groups and algebraic relations

Let A _n =(a ₁,...,a _n) be a system of characters of a compact abelian group A _n with normalized Haar measure μ and let T be a bounded linear operator from a Banach space X into a Banach space Y. The type norm τ(T| A _n ) of T with respect to A _n is the least constant c such that

Hereditarily indecomposable Banach algebras of diagonal operators

We provide a characterization of the Banach spaces X with a Schauder basis (e _n ) _n∈ℕ which have the property that the dual space X* is naturally isomorphic to the space L _diag(X) of diagonal operators with respect to (e _n ) _n∈ℕ. We also construct a Hereditarily Indecomposable Banach space $ \mathfrak{X} $ \mathfrak{X} _D with a Schauder basis (e _n ) _n∈ℕ such that $ \mathfrak{X} $ \mathfrak{X} *_D is isometric to L _diag($ \mathfrak{X} $ \mathfrak{X} _D) with these Banach algebras being Hereditarily Indecomposable. Finally, we show that every T ∈ L _diag($ \mathfrak{X} $ \mathfrak{X} _D) is of the form T = λI + K, where K is a compact operator.     

Limit points of eigenvalues of truncated unbounded tridiagonal operators

Let T be a self-adjoint tridiagonal operator in a Hilbert space H with the orthonormal basis {e _n } _n=1 ^∞ , σ(T) be the spectrum of T and Λ(T) be the set of all the limit points of eigenvalues of the truncated operator T _N . We give sufficient conditions such that the spectrum of T is discrete and σ(T) = Λ(T) and we connect this problem with an old problem in analysis.      

Entropy of C(K)-Valued Operators

We investigate how the entropy numbers (e_n(T)) of an arbitrary Hölder-continuous operator T: E→C(K) are influenced by the entropy numbers (_n(K)) of the underlying compact metric space K and the geometry of E. We derive diverse universal inequalities relating finitely many _n(K)'s with finitely many e_n(T)'s which yield statements about the asymptotically optimal behaviour of the sequence (e_n(T)) in terms of the sequence (_n(K)). As an application we present new methods for estimating the entropy numbers of a precompact and convex subset in a Banach space E, provided that the entropy numbers of its extremal points are known.     

Triangular finite elements of HCT type and classC ρ

Let τ be some triangulation of a planar polygonal domain Ω. Given a smooth functionu, we construct piecewise polynomial functionsv∈C ^ρ(Ω) of degreen=3 ρ for ρ odd, andn=3ρ+1 for ρ even on a subtriangulation τ₃ of τ. The latter is obtained by subdividing eachT∈ρ into three triangles, andv/T is a composite triangular finite element, generalizing the classicalC ¹ cubic Hsieh-Clough-Tocher (HCT) triangular scheme. The functionv interpolates the derivatives ofu up to order ρ at the vertices of τ. Polynomial degrees obtained in this way are minimal in the family of interpolation schemes based on finite elements of this type.     

Universal non-completely-continuous operators

A bounded linear operator between Banach spaces is calledcompletely continuous if it carries weakly convergent sequences into norm convergent sequences. Isolated is a universal operator for the class of non-completely-continuous operators fromL ₁ into an arbitrary Banach space, namely, the operator fromL ₁ into ⊆_∞ defined byT ₀(f) = (∫r _n f d μ) _n>-0, wherer _n is thenth Rademacher function. It is also shown that there does not exist a universal operator for the class of non-completely-continuous operators between two arbitrary Banach spaces. The proof uses the factorization theorem for weakly compact operators and a Tsirelson-like space. Supported in part by NSF grant DMS-9306460. Participant, NSF Workshop in Linear Analysis & Probability, Texas A&M University (supported in part by NSF grant DMS-9311902). Supported in part by NSF grant DMS-9003550.     

Weak Operator Topology, Operator Ranges and Operator Equations via Kolmogorov Widths

Let K be an absolutely convex infinite-dimensional compact in a Banach space χ. The set of all bounded linear operators T on χ satisfying TK ⊃ K is denoted by G(K). Our starting point is the study of the closure WG(K) of G(K) in the weak operator topology. We prove that WG(K) contains the algebra of all operators leaving [`(lin(K))]\overline{{\rm lin}(K)} invariant. More precise results are obtained in terms of the Kolmogorov n-widths of the compact K. The obtained results are used in the study of operator ranges and operator equations.     

Euler operator and homogeneous hida distributions

Let (S)⊄L ²(S′(∔),μ)⊄(S)^* be the Gel'fand triple over the white noise space (S′(∔),μ). Let (e _n ,n>-0) be the ONB ofL ²(∔) consisting of the eigenfunctions of the s.a. operator . In this paper the Euler operator Δ _E is defined as the sum , where ∂ _i stands for the differential operatorD _{e i}. It is shown that Δ _E is the infinitesimal generator of the semigroup (T _t ), where (T _t ϕ)(x)=ϕ(e ^t x) for ϕ∈(S). Similarly to the finite dimensional case, the λ-order homogeneous test functionals are characterized by the Euler equation: Δ _Eϕ =λ_ϕ. Via this characterization the λ-order homogeneous Hida distributions are defined and their properties are worked out. Supported by the National Natural Science Foundation of China.     

Compact operators on a Banach space into the space of almost periodic functions

LetS be a topological semigroup andAP(S) the space of continous complex almost periodic functions onS. We obtain characterizations of compact and weakly compact operators from a Banach spaceX into AP(S). For this we use the almost periodic compactification ofS obtained through uniform spaces. For a bounded linear operatorT fromX into AP(S), letT ₅, be the translate ofT bys inS defined byT ₅(x)=(Tx) ₅ . We define topologies on the space of bounded linear operators fromX into AP(S) and obtain the necessary and sufficient conditions for an operatorT to be compact or weakly compact in terms of the uniform continuity of the maps→T ₅. IfS is a Hausdorff topological semigroup, we also obtain characterizations of compact and weakly compact multipliers on AP(S) in terms of the uniform continuity of the map S→μ_s, where μ_s denotes the unique vector measure corresponding to the operatorT ₅.     

Spectral Criteria of Abstract Sequences; Application to Difference Equations

We suppose that M is a closed subspace of l ^∞(J, X), the space of all bounded sequences {x(n)} _n?J ? X, where J ? {Z⁺,Z} and X is a complex Banach space. We define the M-spectrum σ_M (u) of a sequence u ? l ^∞(J,X). Certain conditions will be supposed on both M and σ_M (u) to insure the existence of u ? M. We prove that if u is ergodic, such that σ_M (u,) is at most countable and, for every λ ? σ_M (u), the sequence e^?iλnu(n) is ergodic, then u ? M. We apply this result to the operator difference equationu(n + 1) = Au(n) + ψ(n), n ? J,and to the infinite order difference equation Σ ^r _k=1 a_k (u(n + k) ? u(n)) + Σ _{s ? Z}?(n ? s)u(s) = h(n), n?J, where ψ?l ^∞(Z,X) such that ψ| _J ? M, A is the generator of a C ₀-semigroup of linear bounded operators {T(t)} _t>0 on X, h ? M, ? ? l ¹(Z) and a_k ?C. Certain conditions will be imposed to guarantee the existence of solutions in the class M.     

Some divisibilities amongst the terms of linear recurrences

Let (u_n)n≥0 be a non-degenerate linear recurrence sequence of integers. We show that the set of positive integersn such that either ω)(n) orΩ(n) dividesu _n is of asymptotic density zero, where ω(n) and Ω(n) are the numbers of prime and prime power divisors ofn, respectively. The same also holds for the set of positive integersn such that τ(n)u _n , where τ(n) is the number of the positive integer divisors of n, provided thatu _n satisfies some mild technical conditions.     

Averaging operators on normed köthe spaces

Summary Under study is the existence of averaging operators determined by measurable maps φ from a measure space (S, Σ, μ) into an arbitrary Hausdorff topological space T. The map φ induces a continuous map φ^e from the space C_b(T) into the normed (Banach) function space L_ϱ = L_ϱ(S, Σ, μ) defined by φ^e(f)=foφ for all f ε C_b(T). An integral representation for such operators is first studied. The existence is then determined by the existence of an averaging operator U₁ for the restriction of φ to a certain measurable subset B₁ of S. Utilizing a representation of L_ϱ(S, Σ, μ) as a Banach function space over a compact extremally disconnected Hausdorff space Ŝ, we are able to give a definition for the concept of plural points and irreducible map. A significant upper bound is given for the operator U₁. Finally conditions are considered under which no bounded projection from L_ϱ onto the range of φ^e may exist. From a topological point of view the development is pursued in a general setting. Averaging operators have recently been used for the study of injective Banach spaces of the type C_b(T) and in non-linear prediction and approximation theory relative to Tshebyshev subspaces of L_ϱ. Entrata in Redazione l’ll settembre 1975.     

On the Property (gw)

In this note we introduce and study the property (gw), which extends property (w) introduced by Rakoc̆evic in [23]. We investigate the property (gw) in connection with Weyl type theorems. We show that if T is a bounded linear operator T acting on a Banach space X, then property (gw) holds for T if and only if property (w) holds for T and Π _a (T) = E(T), where Π _a (T) is the set of left poles of T and E(T) is the set of isolated eigenvalues of T. We also study the property (gw) for operators satisfying the single valued extension property (SVEP). Classes of operators are considered as illustrating examples. The second author was supported by Protars D11/16 and PGR- UMP.     

A remark on the range of elementary operators

Let L(H) denote the algebra of all bounded linear operators on a separable infinite dimensional complex Hilbert space H into itself. Given A ∈ L(H), we define the elementary operator Δ _A : L(H) → L(H) by Δ _A (X) = AXA − X. In this paper we study the class of operators A ∈ L(H) which have the following property: ATA = T implies AT*A = T* for all trace class operators T ∈ C ₁(H). Such operators are termed generalized quasi-adjoints. The main result is the equivalence between this character and the fact that the ultraweak closure of the range of Δ _A is closed under taking adjoints. We give a characterization and some basic results concerning generalized quasi-adjoints operators.     

Computation of Topological Indices of Some Graphs

Let G=(V,E) be a simple connected graph with vertex set V and edge set E. The Wiener index of G is defined by W(G)=∑_{x,y}⊆V d(x,y), where d(x,y) is the length of the shortest path from x to y. The Szeged index of G is defined by Sz(G)=∑ _e=uv∈E n _u (e|G)n _v (e|G), where n _u (e|G) (resp. n _v (e|G)) is the number of vertices of G closer to u (resp. v) than v (resp. u). The Padmakar–Ivan index of G is defined by PI(G)=∑ _e=uv∈E [n _eu (e|G)+n _ev (e|G)], where n _eu (e|G) (resp. n _ev (e|G)) is the number of edges of G closer to u (resp. v) than v (resp. u). In this paper we find the above indices for various graphs using the group of automorphisms of G. This is an efficient method of finding these indices especially when the automorphism group of G has a few orbits on V or E. We also find the Wiener indices of a few graphs which frequently arise in mathematical chemistry using inductive methods.     

Stabilization of bilinear control systems in Hilbert space with nonquadratic feedback

We consider bilinear control systems of the form y′(t) = Ay(t) + u(t)By(t) where A generates a strongly continuous semigroup of contraction (e ^{t A} ) _t⩾0 on an infinite-dimensional Hilbert space Y whose scalar product is denoted by 〈.,.〉. The function u denotes the scalar control. We suppose that B is a linear bounded operator from the state Y into itself. Tacking into account the control saturation, we study the problem of stabilization by feedback of the form u(t)=−f(〈By(t), y(t)〉). Application to the heat equation is considered.      

Possible limit laws for entrance times of an ergodic aperiodic dynamical system

LetG denote the set of decreasingG: ℝ→ℝ withGэ1 on ]−∞,0], and ƒ ₀ ^∞ G(t)dt⩽1. LetX be a compact metric space, andT: X→X a continuous map. Let μ denone aT-invariant ergodic probability measure onX, and assume (X, T, μ) to be aperiodic. LetU⊂X be such that μ(U)>0. Let τ _U (x)=inf{k⩾1:T ^k xεU}, and defineG _U (t)=1/u(U)u({xεU:u(U)τU(x)>t),tεℝ We prove that for μ-a.e.x∈X, there exists a sequence (U _n ) _n≥1 of neighbourhoods ofx such that {x}=∩ _n U _n , and for anyG ∈G, there exists a subsequence (n _k ) _k≥1 withG _{U n k} ↑U weakly. We also construct a uniquely ergodic Toeplitz flowO(x ^∞,S, μ), the orbit closure of a Toeplitz sequencex ^∞, such that the above conclusion still holds, with moreover the requirement that eachU _n be a cylinder set. In memory of Anzelm Iwanik     

Strong Arens irregularity of Beurling algebras with a locally convex topology

Let G be a locally compact group with a weight function ω. Recently, we have shown that the Banach space L₀^∞ (G,1/ω) can be identified with the strong dual of L¹(G, ω)equipped with some locally convex topologies τ. Here we use this duality to introduce an Arens multiplication on (L¹(G, ω), τ)**, and prove that the topological center of (L¹(G, ω), τ)** is (L¹(G, ω); this enables us to conclude that (L¹(G, ω), τ) is Arens regular if and only if G is discrete. We also give a characterization for Arens regularity of L₀^∞ (G, 1/ω)¹. Received: 8 March 2005     

A local variational relation and applications

In [BGH] the authors show that for a given topological dynamical system (X,T) and an open coveru there is an invariant measure μ such that infh _μ(T,ℙ)≥h _top(T,U) where infimum is taken over all partitions finer thanu. We prove in this paper that if μ is an invariant measure andh _μ(T,ℙ) > 0 for each ℙ finer thanu, then infh _μ(T,ℙ > 0 andh _top(T,U) > 0. The results are applied to study the topological analogue of the Kolmogorov system in ergodic theory, namely uniform positive entropy (u.p.e.) of ordern (n≥2) or u.p.e. of all orders. We show that for eachn≥2 the set of all topological entropyn-tuples is the union of the set of entropyn-tuples for an invariant measure over all invariant measures. Characterizations of positive entropy, u.p.e. of ordern and u.p.e. of all orders are obtained. We could answer several open questions concerning the nature of u.p.e. and c.p.e.. Particularly, we show that u.p.e. of ordern does not imply u.p.e. of ordern+1 for eachn≥2. Applying the methods and results obtained in the paper, we show that u.p.e. (of order 2) system is weakly disjoint from all transitive systems, and the product of u.p.e. of ordern (resp. of all orders) systems is again u.p.e. of ordern (resp. of all orders). Project supported by one hundred talents plan and 973 plan.