Lipschitz image of a measure-null set can have a null complement

We give two examples which show that in infinite dimensional Banach spaces the measure-null sets are not preserved by Lipschitz homeomorphisms. There exists a closed setD ? ?₂ which contains a translate of any compact set in the unit ball of ?₂ and a Lipschitz isomorphismF of ?₂ onto ?₂ so thatF(D) is contained in a hyperplane. LetX be a Banach space with an unconditional basis. There exists a Borel setA?X and a Lipschitz isomorphismF ofX onto itself so that the setsX/A andF(A) are both Haar null.     

Operators whose dual has non-separable range

Let X and Y be separable Banach spaces and T:X→Y be a bounded linear operator. We characterize the non-separability of T^?(Y^?) by means of fixing properties of the operator T.     

Semigroups and abstract Gevrey spaces

Conditions are found under which a closed linear operator A in a Banach space X generates a continuous semigroup in a linear topological space Y which is dense in X. The space Y is an abstract Gevrey space associated with the operator A. This is an abstract setting for some results for hyperbolic systems with data in spaces of Gevrey functions.     

Affine Approximation of Lipschitz Functions and Nonlinear Quotients

New concepts related to approximating a Lipschitz function between Banach spaces by affine functions are introduced. Results which clarify when such approximations are possible are proved and in some cases a complete characterization of the spaces X, Y for which any Lipschitz function from X to Y can be so approximated is obtained. This is applied to the study of Lipschitz and uniform quotient mappings between Banach spaces. It is proved, in particular, that any Banach space which is a uniform quotient of L _p , 1 < p < , is already isomorphic to a linear quotient of L _p . Submitted: June 1998, revised: December 1998.     

Preserving affine baire classes by perfect affine maps

Let ? : X → Y be an affine continuous surjection between compact convex sets. Suppose that the canonical copy of the space of real-valued affine continuous functions on Y in the space of real-valued affine continuous functions on X is complemented. We show that if F is a topological vector space, then f : Y → F is of affine Baire class α whenever the composition f ? ? is of affine Baire class α. This abstract result is applied to extend known results on affine Baire classes of strongly affine Baire mappings.     

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.     

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.     

Absolute Lipschitz extendability

A metric space X is said to be absolutely Lipschitz extendable if every Lipschitz function f from X into any Banach space Z can be extended to any containing space Y?X, where the loss in the Lipschitz constant in the extension is independent of $\text{Y,}\text{Z}$, and f. We show that various classes of natural metric spaces are absolutely Lipschitz extendable. To cite this article: J.R. Lee, A. Naor, C. R. Acad. Sci. Paris, Ser. I 338 (2004).     

On bordisms of real algebraic M-varieties

Any morphism of nonsingular complete real algebraic varieties F: Y → X determines a holomorphic mapping of the sets of complex points F _?: Y (?) → X(?) as well as a differentiable mapping of the sets of real points F _?: Y(?) → X(?). These two mappings determine classes of nonoriented bordisms [F _?] ∈ MO_2m (X((?)), [F _?] ∈ M O_m (X(?)), where m = dim Y. The paper describes relationship between these two classes of bordisms.     

On local convexity of nonlinear mappings between Banach spaces

We find conditions for a smooth nonlinear map f: U → V between open subsets of Hilbert or Banach spaces to be locally convex in the sense that for some c and each positive ? < c the image f(B _?(x)) of each ?-ball B _?(x) ? U is convex. We give a lower bound on c via the second order Lipschitz constant Lip₂(f), the Lipschitz-open constant Lip_o(f) of f, and the 2-convexity number conv₂(X) of the Banach space X.     

Minimal distance upper bounds for the perturbation of least squares problems in Hilbert spaces

Let X and Y be Hilbert spaces, and let T : X → Y be a bounded linear operator with closed range. In this paper, we present an optimal perturbation result on the least squares solutions to the operator equation Tx = y under the most general condition.     

Approximating solutions of Hammerstein type equations in Banach spaces

Abstract

Let X be a real Banach space and X^? be its dual. Let F: X → X^? and K: X^? → X be Lipschitz monotone mappings. In this paper an explicit iterative scheme is constructed for approximating solutions of the Hammerstein type equation, 0 = u + KF u, when they exist. Strong convergence of the scheme is obtained under appropriate conditions. Our results improve and unify many of the results in the literature.     

Bernstein–Doetsch-type criteria for the continuity and Lipschitz continuity of convex set-valued mappings

The Bernstein–Doetsch criterion (for convex and midconvex functionals) has been repeatedly generalized to convex and midconvex set-valued mappings F: X → 2 ^Y ; continuity and local Lipschitz continuity were understood in the sense of the Hausdorff distance. However, all such results imposed restrictive additional boundedness-type conditions on the images F(x). In this paper, the Bernstein–Doetsch criterion is generalized to arbitrary convex and midconvex set-valued mappings acting on normed linear spaces X,Y.     

Some mapping properties of p-summing adjoint operators

Let X and Y be Banach spaces and u be a continuous linear operator from X to Y. We prove that if u^*, the adjoint operator of u, is p-summing for some p?1, then for any q?2, u takes (almost) unconditionally summable sequences in X into members of , the projective tensor product of ?q and Y.     

Hyperreflexivity and operator ideals

Suppose (B,β) is an operator ideal, and A is a linear space of operators between Banach spaces X and Y. Modifying the classical notion of hyperreflexivity, we say that A is called B-hyperreflexive if there exists a constant C such that, for any T∈B(X,Y) with α=supβ(qTi)<∞ (the supremum runs over all isometric embeddings i into X, and all quotient maps of Y, satisfying qAi=0), there exists a∈A, for which β(T−a)?Cα. In this paper, we give examples of B-hyperreflexive spaces, as well as of spaces failing this property. In the last section, we apply SE-hyperreflexivity of operator algebras (SE is a regular symmetrically normed operator ideal) to constructing operator spaces with prescribed families of completely bounded maps.     

Characterizations of new Cohen summing bilinear operators

We prove two characterizations of new Cohen summing bilinear operators. The first one is: Let X, Y and Z be Banach spaces, 1 < p < ∞, V : X × Y → Z a bounded linear operator and n ≥ 2 a natural number. Then V is new Cohen p-summing if and only if for all Banach spaces X₁,?…?, X_n and all p-summing operators U : X₁ × · · · × X_n → X, the operator V ? (U, I_Y) : X₁ × · · · × X_n × Y → Z is -summing. The second result is: Let H be a Hilbert space,, Y, Z Banach spaces and V : H × Y → Z a bounded bilinear operator and 1 < p < ∞. Then V is new Cohen p-summing if and only if for all Banach spaces E and all p-summing operators U : E → H, the operator V ? (U, I_Y) is (p, p*)-dominated.     

The weak metric approximation property and ideals of operators

We study the weak metric approximation property introduced by Lima and Oja. We show that a Banach space X has the weak metric approximation property if and only if F(Y,X), the space of finite rank operators, is an ideal in W(Y,X∗∗), the space of weakly compact operators for all Banach spaces Y.     

Weighted composition operators between weighted spaces of vector-valued holomorphic functions on Banach spaces

Let B(E, F) be the Banach Space of all continuous linear operators from a Banach Space E into a Banach space F. Let U_X and U_Y be balanced open subsets of Banach spaces X and Y, respectively. Let V and W be two Nachbin families of continuous weights on U_X and U_Y, respectively. Let HV(U_X, E) (or HV₀(U_X, E)) and HW(U_Y, F) (or HW₀(U_Y, F)) be the weighted spaces of vector-valued holomorphic functions. In this paper, we investigate the holomorphic mappings ? : U_Y → U_X and ψ : U_Y → B(E, F) which generate weighted composition operators between these weighted spaces.     

Perturbing the boundary conditions of the generator of a cosine family

Let A be a densely defined, closed linear operator (which we shall call maximal operator) with domain D(A) on a Banach space X and consider closed linear operators L:D(A)???X and ??:D(A)???X (where ?X is another Banach space called boundary space). Putting conditions on L and ??, we show that the second order abstract Cauchy problem for the operator A _?? with A _?? u=Au and domain D(A _??):={u??D(A):Lu=??u} is well-posed and thus it generates a cosine operator function on the Banach space X.     

Unbounded strictly singular operators

Let D(T)⊂X→Y be an unbounded linear operator where X and Y are normed spaces. It is shown that if Y is complete then T is strictly singular if and only if T is the sum of a continuous strictly singular operator and an unbounded finite rank operator. A counterexample is constructed for the case in which Y is not complete.