Schauder decompositions and functional calculus

We establish a functional calculus, with nice properties, for one and several continuous operators on some non-normable locally convex spaces, more specifically, for operators on the space of entire functions and on other power series spaces. In particular we obtain spectral mapping theorems. The calculus rests on Schauder decompositions for the spaces under consideration, which are of independent interest.     

Constructing unconditional finite dimensional decompositions

Primariness of a Banach space is almost always obtained through the use of the Pelczynski decomposition method. In this paper we show that it is possible to directly construct UFDD’s in many cases from which the primariness can be deduced. We give applications tol _p andX _p. Research supported in part by NSF grant DMS-8602395.     

Isomorphic Schauder decompositions in certain Banach spaces

We extend a theorem of Kato on similarity for sequences of projections in Hilbert spaces to the case of isomorphic Schauder decompositions in certain Banach spaces. To this end we use ?_ψ-Hilbertian and ∞-Hilbertian Schauder decompositions instead of orthogonal Schauder decompositions, generalize the concept of an orthogonal Schauder decomposition to the case of Banach spaces and introduce the class of Banach spaces with Schauder-Orlicz decompositions. Furthermore, we generalize the notions of type, cotype, infratype and M-cotype of a Banach space and study the properties of unconditional Schauder decompositions in Banach spaces possessing certain geometric structure.     

Characterizations of Schauder decompositions in Banach spaces

Acta Mathematica Hungarica -     

Unconditional Schauder decompositions and stopping times in the Lebesgue-Bochner spaces

We extend Troitsky's study of martingales in Banach lattices to include stopping times. Results from the theory of unconditional Schauder decompositions and multipliers are used to derive an optional stopping theorem for unbounded stopping times. We also apply these techniques to convergent nets of stopped processes, as well as to unconditional Schauder decompositions in vector-valued Lp-spaces (1<p<∞).     

Unconditional bases and unconditional finite-dimensional decompositions in Banach spaces

LetX be a Banach space with an unconditional finite-dimensional Schauder decomposition (E _n). We consider the general problem of characterizing conditions under which one can construct an unconditional basis forX by forming an unconditional basis for eachE _n. For example, we show that if sup _n dimE _n<∞ andX has Gordon-Lewis local unconditional structure thenX has an unconditional basis of this type. We also give an example of a non-Hilbertian spaceX with the property that wheneverY is a closed subspace ofX with a UFDD (E _n) such that sup _n dimE _n<∞ thenY has an unconditional basis, showing that a recent result of Komorowski and Tomczak-Jaegermann cannot be improved. Both authors were supported by NSF Grant DMS-9201357.     

Schauder decompositions and the Fremlin projective tensor product of Banach lattices

In this paper, we first introduce a lattice decomposition and finite-dimensional lattice decomposition (FDLD) for Banach lattices. Then we show that for a Banach lattice with FDLD, the following are equivalent: (i) it has the Radon-Nikodym property; (ii) it is a KB-space; (iii) it is a Levi space; and (iv) it is a σ-Levi space. We then give a sequential representation of the Fremlin projective tensor product of an atomic Banach lattice with a Banach lattice. Using this sequential representation, we show that if one of the Banach lattices X and Y is atomic, then the Fremlin projective tensor product has the Radon-Nikodym property (or, respectively, is a KB-space) if and only if both X and Y have the Radon-Nikodym property (or, respectively, are KB-spaces).     

Characterization of Besov-Triebel-Lizorkin spaces via approximation by Whittaker's cardinal series and related unconditional Schauder bases

Let

Non-sequential weak supercyclicity and hypercyclicity

A bounded linear operator T acting on a Banach space B is called weakly hypercyclic if there exists x∈B such that the orbit is weakly dense in B and T is called weakly supercyclic if there is x∈B for which the projective orbit is weakly dense in B. If weak density is replaced by weak sequential density, then T is said to be weakly sequentially hypercyclic or supercyclic, respectively. It is shown that on a separable Hilbert space there are weakly supercyclic operators which are not weakly sequentially supercyclic. This is achieved by constructing a Borel probability measure μ on the unit circle for which the Fourier coefficients vanish at infinity and the multiplication operator Mf(z)=zf(z) acting on L₂(μ) is weakly supercyclic. It is not weakly sequentially supercyclic, since the projective orbit under M of each element in L₂(μ) is weakly sequentially closed. This answers a question posed by Bayart and Matheron. It is proved that the bilateral shift on ?p(Z), 1?p<∞, is weakly supercyclic if and only if 2<p<∞ and that any weakly supercyclic weighted bilateral shift on ?p(Z) for 1?p?2 is norm supercyclic. It is also shown that any weakly hypercyclic weighted bilateral shift on ?p(Z) for 1?p<2 is norm hypercyclic, which answers a question of Chan and Sanders.     

Weak compactness, transfinite Schauder bases and differentiability

We provide a characterization of the class of nonseparable Banach spaces that contain a nonseparable weakly compact set (respectively a relatively weakly compact transfinite basic sequence) in terms of differentiability properties of those spaces.     

Topologically transitive skew-products of backward shift operators and hypercyclicity

In this article we look at skew-products of multiples of the backward shift and examine conditions under which the skew-product is topologically transitive or hypercyclic in the second coordinate. We also give an application of the theory to iterated function systems of multiples of backward shift operators.

     

On the supercyclicity and hypercyclicity of the operator algebra

Let B（X） be the operator algebra for a separable infinite dimensional Hilbert space H, endowed with the strong operator topology or ＊-strong topology. We give sufficient conditions for a continuous linear mapping L ： B（X） →B（X） to be supercyclic or ,-supercyclic. In particular our condition implies the existence of an infinite dimensional subspace of supercyclic vectors for a mapping T on H. Hypercyclicity of the operator algebra with strong operator topology was studied＇ by Chan and here we obtain an analogous result in the case of ＊-strong operator topology.     

Directed graphs,decompositions, and spatial linkages

The decomposition of a linkage into minimal components is a central tool of analysis and synthesis of linkages. In this paper we prove that every pinned d$d$-isostatic (minimally rigid) graph (grounded linkage) has a unique decomposition into minimal strongly connected components (in the sense of directed graphs), or equivalently into minimal pinned isostatic graphs, which we call d$d$-Assur graphs. We also study key properties of motions induced by removing an edge in a d$d$-Assur graph — defining a sharper subclass of strongly d$d$-Assur graphs by the property that all inner vertices go into motion, for each removed edge. The strongly 3-Assur graphs are the central building blocks for kinematic linkages in 3-space and the 3-Assur graphs are components in the analysis of built linkages. The d$d$-Assur graphs share a number of key combinatorial and geometric properties with the 2-Assur graphs, including an associated lower block-triangular decomposition of the pinned rigidity matrix which provides modular information for extending the motion induced by inserting one driver in a bottom Assur linkage to the joints of the entire linkage. We also highlight some problems in combinatorial rigidity in higher dimensions (d≥3$d\ge 3$) which cause the distinction between d$d$-Assur and strongly d$d$-Assur which did not occur in the plane.     

Nonlinear initial-value problems and Schauder bases

In this work we obtain an approximation of the solution of a nonlinear initial-value problem, by means of certain Schauder basis associated in a natural way with that differential problem.     

Circulants,displacements and decompositions of matrices

In this paper are suggested new formulas for representation of matrices and their inverses in the form of sums of products of factor circulants, which are based on the analysis of the factor cyclic displacement of matrices. The results in applications to Toeplitz matrices generalized the Gohberg-Semencul, Ben-Artzi-Shalom and Heinig-Rost formulas and are useful for complexity analysis.