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We study the metric projection onto the closed convex cone in a real Hilbert space $\mathscr {H}$ generated by a sequence $\mathcal {V} = \{v_n\}_{n=0}^\infty $. The first main result of this article provides a sufficient condition under which the closed convex cone generated by $\mathcal {V}$ coincides with the following set:
$$ \begin{align*} \mathcal{C}[[\mathcal{V}]]: = \bigg\{\sum_{n=0}^\infty a_n v_n\Big|a_n\geq 0,\text{ the series }\sum_{n=0}^\infty a_n v_n\text{ converges in } \mathscr{H}\bigg\}. \end{align*} $$
Then, by adapting classical results on general convex cones, we give a useful description of the metric projection onto $\mathcal {C}[[\mathcal {V}]]$. As an application, we obtain the best approximations of many concrete functions in $L^2([-1,1])$ by polynomials with nonnegative coefficients.
In this paper we consider the following problem: let Xk, be a Banach space with a normalised basis (e(k, j))j, whose biorthogonals are denoted by
${(e_{(k,j)}^*)_j}$
, for
$k\in\N$
, let
$Z=\ell^\infty(X_k:k\kin\N)$
be their l∞-sum, and let
$T:Z\to Z$
be a bounded linear operator with a large diagonal, i.e.,
Under which condition does the identity on Z factor through T? The purpose of this paper is to formulate general conditions for which the answer is positive.
Li et al. [‘Weak 2-local isometries on uniform algebras and Lipschitz algebras’, Publ. Mat.63 (2019), 241–264] generalized the Kowalski–Słodkowski theorem by establishing the following spherical variant: let A be a unital complex Banach algebra and let
$\Delta : A \to \mathbb {C}$ be a mapping satisfying the following properties:
(a)
$\Delta $
is 1-homogeneous (that is,
$\Delta (\lambda x)=\lambda \Delta (x)$
for all
$x \in A$
,
$\lambda \in \mathbb C$
);
Then
$\Delta $
is linear and there exists
$\lambda _{0} \in \mathbb {T}$
such that
$\lambda _{0}\Delta $
is multiplicative. In this note we prove that if (a) is relaxed to
$\Delta (0)=0$
, then
$\Delta $
is complex-linear or conjugate-linear and
$\overline {\Delta (\mathbf {1})}\Delta $
is multiplicative. We extend the Kowalski–Słodkowski theorem as a conclusion. As a corollary, we prove that every 2-local map in the set of all surjective isometries (without assuming linearity) on a certain function space is in fact a surjective isometry. This gives an affirmative answer to a problem on 2-local isometries posed by Molnár [‘On 2-local *-automorphisms and 2-local isometries of B(H)', J. Math. Anal. Appl.479(1) (2019), 569–580] and also in a private communication between Molnár and O. Hatori, 2018.
Non-amenability of ${\mathcal {B}}(E)$ has been surprisingly difficult to prove for the classical Banach spaces, but is now known for E = ℓp and E = Lp for all 1 ⩽ p < ∞. However, the arguments are rather indirect: the proof for L1 goes via non-amenability of $\ell ^\infty ({\mathcal {K}}(\ell _1))$ and a transference principle developed by Daws and Runde (Studia Math., 2010).
In this note, we provide a short proof that ${\mathcal {B}}(L_1)$ and some of its subalgebras are non-amenable, which completely bypasses all of this machinery. Our approach is based on classical properties of the ideal of representable operators on L1, and shows that ${\mathcal {B}}(L_1)$ is not even approximately amenable.
Bożejko and Speicher associated a finite von Neumann algebra MT to a self-adjoint operator T on a complex Hilbert space of the form $\mathcal {H}\otimes \mathcal {H}$ which satisfies the Yang–Baxter relation and $ \left\| T \right\| < 1$. We show that if dim$(\mathcal {H})$ ⩾ 2, then MT is a factor when T admits an eigenvector of some special form.
We study the stability of the differential process of Rochberg and Weiss associated with an analytic family of Banach spaces obtained using the complex interpolation method for families. In the context of Köthe function spaces, we complete earlier results of Kalton (who showed that there is global bounded stability for pairs of Köthe spaces) by showing that there is global (bounded) stability for families of up to three Köthe spaces distributed in arcs on the unit circle while there is no (bounded) stability for families of four or more Köthe spaces. In the context of arbitrary pairs of Banach spaces, we present some local stability results and some global isometric stability results.
We consider the Cauchy problem for a general class of parabolic partial differential equations in the Euclidean space ℝN. We show that given a weighted Lp-space $L_w^p({\mathbb {R}}^N)$ with 1 ⩽ p < ∞ and a fast growing weight w, there is a Schauder basis $(e_n)_{n=1}^\infty$ in $L_w^p({\mathbb {R}}^N)$ with the following property: given an arbitrary positive integer m there exists nm > 0 such that, if the initial data f belongs to the closed linear span of en with n ⩾ nm, then the decay rate of the solution of the problem is at least t−m for large times t.
The result generalizes the recent study of the authors concerning the classical linear heat equation. We present variants of the result having different methods of proofs and also consider finite polynomial decay rates instead of unlimited m.
We prove an extension of Pisier’s inequality (1986) with a dimension-independent constant for vector-valued functions whose target spaces satisfy a relaxation of the UMD property.
For integers $p,b\geq 2$, let $D=\{0,1,\ldots ,b-1\}$ be a set of consecutive digits. It is known that the Cantor measure $\unicode[STIX]{x1D707}_{pb,D}$ generated by the iterated function system $\{(pb)^{-1}(x+d)\}_{x\in \mathbb{R},d\in D}$ is a spectral measure with spectrum
where $S=pD$. We give conditions on $\unicode[STIX]{x1D70F}\in \mathbb{Z}$ under which the scaling set $\unicode[STIX]{x1D70F}\unicode[STIX]{x1D6EC}(pb,S)$ is also a spectrum of $\unicode[STIX]{x1D707}_{pb,D}$. These investigations link number theory and spectral measures.
A Δ-point x of a Banach space is a norm-one element that is arbitrarily close to convex combinations of elements in the unit ball that are almost at distance 2 from x. If, in addition, every point in the unit ball is arbitrarily close to such convex combinations, x is a Daugavet point. A Banach space X has the Daugavet property if and only if every norm-one element is a Daugavet point. We show that Δ- and Daugavet points are the same in L1-spaces, in L1-preduals, as well as in a big class of Müntz spaces. We also provide an example of a Banach space where all points on the unit sphere are Δ-points, but none of them are Daugavet points. We also study the property that the unit ball is the closed convex hull of its Δ-points. This gives rise to a new diameter-two property that we call the convex diametral diameter-two property. We show that all C(K) spaces, K infinite compact Hausdorff, as well as all Müntz spaces have this property. Moreover, we show that this property is stable under absolute sums.
Let
$\{M_{n}\}_{n=1}^{\infty }$
be a sequence of expanding matrices with
$M_{n}=\operatorname{diag}(p_{n},q_{n})$
, and let
$\{{\mathcal{D}}_{n}\}_{n=1}^{\infty }$
be a sequence of digit sets with
${\mathcal{D}}_{n}=\{(0,0)^{t},(a_{n},0)^{t},(0,b_{n})^{t},\pm (a_{n},b_{n})^{t}\}$
, where
$p_{n}$
,
$q_{n}$
,
$a_{n}$
and
$b_{n}$
are positive integers for all
$n\geqslant 1$
. If
$\sup _{n\geqslant 1}\{\frac{a_{n}}{p_{n}},\frac{b_{n}}{q_{n}}\}<\infty$
, then the infinite convolution
$\unicode[STIX]{x1D707}_{\{M_{n}\},\{{\mathcal{D}}_{n}\}}=\unicode[STIX]{x1D6FF}_{M_{1}^{-1}{\mathcal{D}}_{1}}\ast \unicode[STIX]{x1D6FF}_{(M_{1}M_{2})^{-1}{\mathcal{D}}_{2}}\ast \cdots \,$
is a Borel probability measure (Cantor–Dust–Moran measure). In this paper, we investigate whenever there exists a discrete set
$\unicode[STIX]{x1D6EC}$
such that
$\{e^{2\unicode[STIX]{x1D70B}i\langle \unicode[STIX]{x1D706},x\rangle }:\unicode[STIX]{x1D706}\in \unicode[STIX]{x1D6EC}\}$
is an orthonormal basis for
$L^{2}(\unicode[STIX]{x1D707}_{\{M_{n}\},\{{\mathcal{D}}_{n}\}})$
.
We prove that the class of reflexive asymptotic-
$c_{0}$
Banach spaces is coarsely rigid, meaning that if a Banach space
$X$
coarsely embeds into a reflexive asymptotic-
$c_{0}$
space
$Y$
, then
$X$
is also reflexive and asymptotic-
$c_{0}$
. In order to achieve this result, we provide a purely metric characterization of this class of Banach spaces. This metric characterization takes the form of a concentration inequality for Lipschitz maps on the Hamming graphs, which is rigid under coarse embeddings. Using an example of a quasi-reflexive asymptotic-
$c_{0}$
space, we show that this concentration inequality is not equivalent to the non-equi-coarse embeddability of the Hamming graphs.
A basic problem in the theory of partially ordered vector spaces is to characterise those cones on which every order-isomorphism is linear. We show that this is the case for every Archimedean cone that equals the inf-sup hull of the sum of its engaged extreme rays. This condition is milder than existing ones and is satisfied by, for example, the cone of positive operators in the space of bounded self-adjoint operators on a Hilbert space. We also give a general form of order-isomorphisms on the inf-sup hull of the sum of all extreme rays of the cone, which extends results of Artstein–Avidan and Slomka to infinite-dimensional partially ordered vector spaces, and prove the linearity of homogeneous order-isomorphisms in a variety of new settings.
In this note we prove that the Kalton interlaced graphs do not equi-coarsely embed into the James space
${\mathcal{J}}$
nor into its dual
${\mathcal{J}}^{\ast }$
. It is a particular case of a more general result on the non-equi-coarse embeddability of the Kalton graphs into quasi-reflexive spaces with a special asymptotic structure. This allows us to exhibit a coarse invariant for Banach spaces, namely the non-equi-coarse embeddability of this family of graphs, which is very close to but different from the celebrated property
${\mathcal{Q}}$
of Kalton. We conclude with a remark on the coarse geometry of the James tree space
${\mathcal{J}}{\mathcal{T}}$
and of its predual.
The Chebyshev conjecture posits that Chebyshev subsets of a real Hilbert space
$X$
are convex. Works by Asplund, Ficken and Klee have uncovered an equivalent formulation of the Chebyshev conjecture in terms of uniquely remotal subsets of
$X$
. In this tradition, we develop another equivalent formulation in terms of Chebyshev subsets of the unit sphere of
$X\times \mathbb{R}$
. We characterise such sets in terms of the image under stereographic projection. Such sets have superior structure to Chebyshev sets and uniquely remotal sets.
We study the Daugavet property in tensor products of Banach spaces. We show that
$L_{1}(\unicode[STIX]{x1D707})\widehat{\otimes }_{\unicode[STIX]{x1D700}}L_{1}(\unicode[STIX]{x1D708})$
has the Daugavet property when
$\unicode[STIX]{x1D707}$
and
$\unicode[STIX]{x1D708}$
are purely non-atomic measures. Also, we show that
$X\widehat{\otimes }_{\unicode[STIX]{x1D70B}}Y$
has the Daugavet property provided
$X$
and
$Y$
are
$L_{1}$
-preduals with the Daugavet property, in particular, spaces of continuous functions with this property. With the same techniques, we also obtain consequences about roughness in projective tensor products as well as the Daugavet property of projective symmetric tensor products.
In this paper, we investigate the weighted multilinear boundedness properties of the maximal higher order Calderón commutator for the dimensions larger than two. We establish all weighted multilinear estimates on the product of the
$L^{p}(\mathbb{R}^{d},w)$
space, including some peculiar endpoint estimates of the higher dimensional Calderón commutator.
Suppose that
$0<|\unicode[STIX]{x1D70C}|<1$
and
$m\geqslant 2$
is an integer. Let
$\unicode[STIX]{x1D707}_{\unicode[STIX]{x1D70C},m}$
be the self-similar measure defined by
$\unicode[STIX]{x1D707}_{\unicode[STIX]{x1D70C},m}(\cdot )=\frac{1}{m}\sum _{j=0}^{m-1}\unicode[STIX]{x1D707}_{\unicode[STIX]{x1D70C},m}(\unicode[STIX]{x1D70C}^{-1}(\cdot )-j)$
. Assume that
$\unicode[STIX]{x1D70C}=\pm (q/p)^{1/r}$
for some
$p,q,r\in \mathbb{N}^{+}$
with
$(p,q)=1$
and
$(p,m)=1$
. We prove that if
$(q,m)=1$
, then there are at most
$m$
mutually orthogonal exponential functions in
$L^{2}(\unicode[STIX]{x1D707}_{\unicode[STIX]{x1D70C},m})$
and
$m$
is the best possible. If
$(q,m)>1$
, then there are any number of orthogonal exponential functions in
$L^{2}(\unicode[STIX]{x1D707}_{\unicode[STIX]{x1D70C},m})$
.
We show the existence of a measurable selector in Carpenter’s Theorem due to Kadison. This solves a problem posed by Jasper and the first author in an earlier work. As an application we obtain a characterization of all possible spectral functions of shift-invariant subspaces of
$L^{2}(\mathbb{R}^{d})$
and Carpenter’s Theorem for type
$\text{I}_{\infty }$
von Neumann algebras.
We introduce a notion of modulated topological vector spaces, that generalises, among others, Banach and modular function spaces. As applications, we prove some results which extend Kirk’s and Browder’s fixed point theorems. The theory of modulated topological vector spaces provides a very minimalist framework, where powerful fixed point theorems are valid under a bare minimum of assumptions.