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Nigel J. Kalton

Publications and source records attributed to Nigel J. Kalton.

32 records · Page 2Linked to original sources
arXiv1994

Unconditional bases and unconditional finite-dimensional decompositions in Banach spaces

Let $X$ 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 for $X$ by forming an unconditional basis for each $E_n.$ For example, we show that if $\sup \dim E_n<\infty$ and $X$ has Gordon-Lewis local unconditional structure then $X$ has an unconditional basis of this type. We also give an example of a non-Hilbertian space $X$ with the property that whenever $Y$ is a closed subspace of $X$ with a UFDD $(E_n)$ such that $\sup\dim E_n<\infty$ then $Y$ has an unconditional basis, showing that a recent result of Komorowski and Tomczak-Jaegermann cannot be improved.

math.FA↗
arXiv1994

Complex interpolation of Hardy-type subspaces

We consider the problem of complex interpolation of certain Hardy-type subspaces of Köthe function spaces. For example, suppose $X_0$ and $X_1$ are Köthe function spaces on the unit circle $\bold T,$ and let $H_{X_0}$ and $H_{X_1}$ be the corresponding Hardy spaces. Under mild conditions on $X_0,X_1$ we give a necessary and sufficient condition for the complex interpolation space $[H_{X_0},H_{X_1}]_θ$ to coincide with $H_{X_θ}$ where $X_θ=[X_0,X_1]_θ.$ We develop a very general framework for such results and our methods apply to many more general sitauations including the vector-valued case.

math.FA↗
arXiv1993

Property $(M)$, $M$-ideals, and almost isometric structure of Banach spaces

We study $M$-ideals of compact operators by means of the property~$(M)$ introduced in \cite{Kal-M}. Our main result states for a separable Banach space $X$ that the space of compact operators on $X$ is an $M$-ideal in the space of bounded operators if (and only if) $X$ does not contain a copy of $\ell_{1}$, has the metric compact approximation property, and has property~$(M)$. The investigation of special versions of property~$(M)$ leads to results on almost isometric structure of some classes of Banach spaces. For instance, we give a simple necessary and sufficient condition for a Banach space to embed almost isometrically into an $\ell_{p}$-sum of finite-dimensional spaces resp.\ into $c_{0}$, and for $2<p<\iy$ we prove that a subspace of $L_{p}$ embeds almost isometrically into $\ell_{p}$ if and only if it does not contain a subspace isomorphic to $\ell_{2}$.

math.FA↗
arXiv1992

Surjective isometries on rearrangement-invariant spaces

We prove that if $X$ is a real rearrangement-invariant function space on $[0,1]$, which is not isometrically isomorphic to $L_2,$ then every surjective isometry $T:X\to X$ is of the form $Tf(s)=a(s)f(σ(s))$ for a Borel function $a$ and an invertible Borel map $σ:[0,1] \to [0,1].$ If $X$ is not equal to $L_p$, up to renorming, for some $1\le p\le \infty$ then in addition $|a|=1$ a.e. and $σ$ must be measure-preserving.

math.FA↗
arXiv1992

Factorization theorems for quasi-normed spaces

We extend Pisier's abstract version of Grothendieck's theorem to general non-locally convex quasi-Banach spaces. We also prove a related result on factoring operators through a Banach space and apply our results to the study of vector-valued inequalities for Sidon sets. We also develop the local theory of (non-locally convex) spaces with duals of weak cotype 2.

math.FA↗
arXiv1992

Interpolation of compact operators by the methods of Calderón and Gustavsson-Peetre

Let $ X=(X_0,X_1)$ and $ Y=(Y_0,Y_1)$ be Banach couples and suppose $T: X\to Y$ is a linear operator such that $T:X_0\to Y_0$ is compact. We consider the question whether the operator $T:[X_0,X_1]_θ\to [Y_0,Y_1]_θ$ is compact and show a positive answer under a variety of conditions. For example it suffices that $X_0$ be a UMD-space or that $X_0$ is reflexive and there is a Banach space so that $X_0=[W,X_1]_α$ for some $0<α<1.$

math.FA↗
arXiv1992

On vector-valued inequalities for Sidon sets and sets of interpolation

Let $E$ be a Sidon subset of the integers and suppose $X$ is a Banach space. Then Pisier has shown that $E$-spectral polynomials with values in $X$ behave like Rademacher sums with respect to $L_p-$norms. We consider the situation when $X$ is a quasi-Banach space. For general quasi-Banach spaces we show that a similar result holds if and only if $E$ is a set of interpolation ($I_0$-set). However for certain special classes of quasi-Banach spaces we are able to prove such a result for larger sets. Thus if $X$ is restricted to be ``natural'' then the result holds for all Sidon sets. We also consider spaces with plurisubharmonic norms and introduce the class of analytic Sidon sets.

math.FA↗
arXiv1992

Calderón couples of re-arrangement invariant spaces

We examine conditions under which a pair of re-arrangement invariant function spaces on $[0,1]$ or $[0,\infty)$ form a Calderón couple. A very general criterion is developed to determine whether such a pair is a Calderón couple, with numerous applications. We give, for example, a complete classification of those spaces $X$ which form a Calderón couple with $L_{\infty}.$ We specialize our results to Orlicz spaces and are able to give necessary and sufficient conditions on an Orlicz function $F$ so that the pair $(L_F,L_{\infty})$ forms a Calderón pair.

math.FA↗
arXiv1992

On nonatomic Banach lattices and Hardy spaces

We are interested in the question when a Banach space $X$ with an unconditional basis is isomorphic (as a Banach space) to an order-continuous nonatomic Banach lattice. We show that this is the case if and only if $X$ is isomorphic as a Banach space with $X(\ell_2)$. This and results of J. Bourgain are used to show that spaces $H_1(\bold T^n)$ are not isomorphic to nonatomic Banach lattices. We also show that tent spaces introduced in \cite{4} are isomorphic to Rad $H_1$.

math.FA↗