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.     

On factors ofC([0, 1]) with non-separable dual

LetC denote the Banach space of scalar-valued continuous functions defined on the closed unit interval. It is proved that ifX is a Banach space andT:C→X is a bounded linear operator withT ^* X ^* non-separable, then there is a subspaceY ofC, isometric toC, such thatT|Y is an isomorphism. An immediate consequence of this and a result of A. Pelczynski, is that every complemented subspace ofC with non-separable dual is isomorphic (linearly homeomorphic) toC. The research for this paper was partially supported by NSF-GP-30798X. An erratum to this article is available at .     

Operators on <Emphasis Type="Italic">C</Emphasis>[0,1] preserving copies of asymptotic ℓ<Subscript>1</Subscript> spaces

Given separable Banach spaces X, Y, Z and a bounded linear operator T:X→Y, then T is said to preserve a copy of Z provided that there exists a closed linear subspace E of X isomorphic to Z and such that the restriction of T to E is an into isomorphism. It is proved that every operator on C([0,1]) which preserves a copy of an asymptotic ℓ₁ space also preserves a copy of C([0,1]).     

Characteristic classes and obstruction theory for infinite loop spaces

The classical extension problem is to determine whether or not a given mapg:AY, defined on a given subspaceA of a spaceX, has an extensionXY. The present paper examines this question in the special case where thek-invariants ofY are cohomology classes of finite order (for instance ifY is an infinite loop space).     

Banach spacesX withX** separable

LetX be a Banach space and letZ be a closed subspace ofX** which containsX. It is proved in this paper that, in the caseX** separable, there exists a closed subspaceY ofX such thatX+ =Z, the closure ofY inX** for the weak-star topology.     

Universal non-compact operators between super-reflexive Banach spaces and the existence of a complemented copy of Hilbert space

Suppose that 1<p≦2, 2≦q<∞. The formal identity operatorI:l _p→l _qfactorizes through any given non-compact operator from ap-smooth Banach space into aq-convex Banach space. It follows that ifX is a 2-convex space andY is an infinite dimensional subspace ofX which is isomorphic to a Hilbert space, thenY contains an isomorphic copy ofl ₂ which is complemented inX.     

G-narrow operators and G-rich subspaces

Let X and Y be Banach spaces. An operator G: X → Y is a Daugavet center if ‖G +T‖ = ‖G‖+‖T‖ for every rank-1 operator T. For every Daugavet center G we consider a certain set of operators acting from X, so-called G-narrow operators. We prove that if J is the natural embedding of Y into a Banach space E, then E can be equivalently renormed so that an operator T is (J ○ G)-narrow if and only if T is G-narrow. We study G-rich subspaces of X: Z ? X is called G-rich if the quotient map q: X → X/Z is G-narrow.     

Topological direct sum decompositions of banach spaces

LetY andZ be two closed subspaces of a Banach spaceX such thatY≠lcub;0rcub; andY+Z=X. Then, ifZ is weakly countably determined, there exists a continuous projectionT inX such that ∥T∥=1,T(X)⊃Y, T ⁻¹(0)⊂Z and densT(X)=densY. It follows that every Banach spaceX is the topological direct sum of two subspacesX ₁ andX ₂ such thatX ₁ is reflexive and densX ₂**=densX**/X.     

Almost Invariant Half-spaces of Algebras of Operators

Given a Banach space X and a bounded linear operator T on X, a subspace Y of X is almost invariant under T if TY í Y+F{TY\subseteq Y+F} for some finite-dimensional “error” F. In this paper, we study subspaces that are almost invariant under every operator in an algebra \mathfrak A{\mathfrak A} of operators acting on X. We show that if \mathfrak A{\mathfrak A} is norm closed then the dimensions of “errors” corresponding to operators in \mathfrak A{\mathfrak A} must be uniformly bounded. Also, if \mathfrak A{\mathfrak A} is generated by a finite number of commuting operators and has an almost invariant half-space (that is, a subspace with both infinite dimension and infinite codimension) then \mathfrak A{\mathfrak A} has an invariant half-space.     

Automorphisms inc 0,l 1 andm

It is proved that ifX=c ₀ orm and ifY andZ are subspaces ofX withX/Y andX/Z non-reflexive, then any isomorphism ofY ontoZ has an extension to an automorphism ofX. A dual result is obtained forX=l ₁. This research was partially supported by NSF-GP-8964.     

Some topics on the continuation and smoothing of vector functions

We consider problems of continuation of vector functions from a subspace to the entire space and of smoothing problems for these functions. It is shown that there exists a reflexive separable spaceX and a subspaceY such that even a very smooth mapping ofY does not extend to a uniformly continuous mapping of a neighborhood ofY.Translated fromMatematicheskie Zametki, Vol. 58, No. 6, pp. 906–916, December, 1995.     

Building and solving matrix spectral problems

When an infinite dimensional operator T: X → X is approximated with (a slight perturbation of) an operator T_n : X → X of finite rank less than or equal to n, the spectral elements of an auxiliary matrix Z ∈ ℂ^{n ×n} , lead to those of Tn, if they are computed exactly. This contribution covers a general theoretical framework for matrix problems issued from finite rank discretizations and perturbed variants, the stop criterion of the QR method for eigenvalues, the possibility of using the Newton method to compute a Schur form, and the use of Newton method to refine coarse approximate bases of spectral subspaces. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

EmbeddingL 1 in a Banach lattice

We show that ifX is a Banach lattice containing no copy ofc ₀ and ifZ is a subspace ofX isomorphic toL ₁[0, 1] then (a)Z contains a subspaceZ ₀ isomorphic toL ₁ and complemented inX and (b)X contains a complemented sublattice isomorphic and lattice-isomorphic toL ₁.     

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.     

Invariant measures for certain expansive ℤ2-actions

Letp>1 be prime, and letY ⊂X=(ℤ/pℤ)^{ℤ 2}) be an infinite, closed, shift-invariant subgroup with the following properties: the restriction toY of the shift-actionσ of ℤ² onX is mixing with respect to the Haar measureλ _Y ofY, and every closed, shift-invariant subgroupZ ⊂Y is finite. We prove that every sufficiently mixing, non-atomic, shift-invariant probability measureμ onY is equal toλ _Y . The author would like to thank the Department of Mathematics, University of Vienna, for hospitality while this work was done.     

Extending an operator from a Hilbert space to a larger Hilbert space,so as to reduce its spectrum

In our earlier paper [1] we showed that given any elementx of a commutative unital Banach algebraA, there is an extensionA′ ofA such that the spectrum ofx inA′ is precisely the essential spectrum ofx inA. In [2], we showed further that ifT is a continuous linear operator on a Banach spaceX, then there is an extensionY ofX such thatT extends continuously to an operatorT ⁻ onY, and the spectrum ofT ⁻ is precisely the approximate point spectrum ofT. In this paper we take the second of these results, and show further that ifX is a Hilbert space then we can ensure thatY is also a Hilbert space; so any operatorT on a Hilbert spaceX is the restriction to one copy ofX of an operatorT ⁻ onX ⊕X, whose spectrum is precisely the approximate point spectrum ofT. This result is “best possible” in the sense that if isany extension to a larger Banach space of an operatorT, it is a standard exercise that the approximate point spectrum ofT is contained in the spectrum of .     

Nonamenable products are not treeable

LetX andY be infinite graphs such that the automorphism group ofX is nonamenable and the automorphism group ofY has an infinite orbit. We prove that there is no automorphism-invariant measure on the set of spanning trees in the direct productX×Y. This implies that the minimal spanning forest corresponding to i.i.d. edge-weights in such a product has infinitely many connected components almost surely. Research partially supported by NSF grant DMS-9803597.     

On Optimal Feedback Control for Stationary Linear Systems

We study linear-quadratic optimal control problems for finite dimensional stationary linear systems A X+B U=Z with output Y=C X+D U from the viewpoint of linear feedback solution. We interpret solutions in relation to system robustness with respect to disturbances Z and relate them to nonlinear matrix equations of Riccati type and eigenvalue-eigenvector problems for the corresponding Hamiltonian system. Examples are included along with an indication of extensions to continuous, i.e., infinite dimensional, systems, primarily of elliptic type.     

On Certain Densely Invariant Quantities of Linear Operators

Let L(X,Y) denote the class of linear transformations T:D(T) ? X → Y where X and Y are normed spaces. A quantity f is called densely invariant if for every system L(X, Y) and every operator T ? L(X,Y) we have f(T/E)= f(T) whenever E is a core of T. The density invariance of certain well known quantities is established. In case Y is complete and T is closable, a stronger property is shown to hold for some of these quantitites, namely invariance under restriction to dense subspaces.