We observe that the classical notion of numerical radius gives rise to a notion of smoothness in the space of bounded linear operators on certain Banach spaces, whenever the numerical radius is a norm. We demonstrate an important class of \emph{real} Banach space $ \mathbb{X} $ for which the numerical radius defines a norm on $ \mathbb{L}(\mathbb{X}), $ the space of all bounded linear operators on $ \mathbb{X} $. We characterize Birkhoff-James orthogonality in the space of bounded linear operators on a finite-dimensional Banach space, endowed with the numerical radius norm. Some examples are also discussed to illustrate the geometric differences between the numerical radius norm and the usual operator norm, from the viewpoint of operator smoothness.
We introduce the notion of approximate smoothness in a normed linear space. We characterize this property and show the connections between smoothness and approximate smoothness for some spaces. As an application, we consider in particular the Birkhoff-James orthogonality and its right-additivity under the assumption of approximate smoothness.
In 2000 Kadets et al. introduced the notions of acs, luacs and uacs spaces, which form common generalisations of well-known rotundity and smoothness properties of Banach spaces. In a recent preprint the author introduced some further related notions and investigated the behaviour of these geometric properties under the formation of absolute sums. This paper is in a sense a continuation of the previous work. Here we will study the behaviour of said properties under the formation of Köthe-Bochner spaces, thereby generalising some results of Sirotkin on the acs, luacs and uacs properties of $L^p$…
We introduce a suitable notion of asymptotic smoothness on infinite dimensional Banach spaces, and we prove that, under some structural restrictions on the space, the convex envelope of an asymptotically smooth function is asymptotically smooth. Furthermore, we study convexity and smoothness properties of polynomial norms, and we obtain that a polynomial norm of degree N has modulus of convexity of power type N.
We study the notions of acs, luacs and uacs Banach spaces which were introduced by V. Kadets et al. in 2000 and form common generalisations of the usual rotundity and smoothness properties of Banach spaces. In particular, we are interested in (mainly infinite) absolute sums of such spaces. We also introduce some new classes of spaces that lie inbetween those of acs and uacs spaces and study their behaviour under the formation of absolute sums as well.
We introduce the notions of strong asymptotic uniform smoothness and convexity. We show that the injective tensor product of strongly asymptotically uniformly smooth spaces is asymptotically uniformly smooth. This applies in particular to uniformly smooth spaces admitting a monotone FDD, extending a result by Dilworth, Kutzarova, Randrianarivony, Revalski and Zhivkov. Our techniques also provide a characterisation of Orlicz functions $M, N$ such that the space of compact operators $\mathcal K(h_M,h_N)$ is asymptotically uniformly smooth. Finally we show that $\mathcal K(X, Y)$ is not strictly …
We introduce stronger versions of the usual notions of martingale type p <= 2 and cotype q >= 2 of a Banach space X and show that these concepts are equivalent to uniform p-smoothness and q-convexity, respectively. All these are metric concepts, so they depend on the particular norm in X. These concepts allow us to get some more insight into the fine line between X being isomorphic to a uniformly p-smooth space or being uniformly p-smooth itself. Instead of looking at Banach spaces, we consider linear operators between Banach spaces right away. The situation of a Banach space X can be rediscov…
Generalizing the notion of numerical range and numerical radius of an operator on a Banach space, we introduce the notion of joint numerical range and joint numerical radius of tuple of operators on a Banach space. We study the convexity of the joint numerical range. We show that the joint numerical radius defines a norm if and only if the numerical radius defines a norm on the corresponding space. Then we prove that on a finite-dimensional Banach space, the joint numerical radius can be retrieved from the extreme points. Furthermore, we introduce a notion of joint numerical index of a Banach …
A new construction of decomposition smoothness spaces of homogeneous type is considered. The smoothness spaces are based on structured and flexible decompositions of the frequency space $\mathbb{R}^d\backslash\{0\}$. We construct simple adapted tight frames for $L_2(\mathbb{R}^d)$ that can be used to fully characterise the smoothness norm in terms of a sparseness condition imposed on the frame coefficients. Moreover, it is proved that the frames provide a universal decomposition of tempered distributions with convergence in the tempered distributions modulo polynomials. As an application of th…
We consider the difference $f(H_1)-f(H_0)$ for self-adjoint operators $H_0$ and $H_1$ acting in a Hilbert space. We establish a new class of estimates for the operator norm and the Schatten class norms of this difference. Our estimates utilise ideas of scattering theory and involve conditions on $H_0$ and $H_1$ in terms of the Kato smoothness. They allow for a much wider class of functions $f$ (including some unbounded ones) than previously available results do. As an important technical tool, we propose a new notion of Schatten class valued smoothness and develop a new framework for double op…
There are two notions of sparsity associated to a frame $Ψ=(ψ_i)_{i\in I}$: Analysis sparsity of $f$ means that the analysis coefficients $(\langle f,ψ_i\rangle)_i$ are sparse, while synthesis sparsity means that $f=\sum_i c_iψ_i$ with sparse coefficients $(c_i)_i$. Here, sparsity of $c=(c_i)_i$ means $c\in\ell^p(I)$ for a given $p<2$. We show that both notions of sparsity coincide if $Ψ={\rm SH}(φ,ψ;δ)$ is a discrete (cone-adapted) shearlet frame with 'nice' generators $φ,ψ$ and fine enough sampling density $δ>0$. The required 'niceness' is explicitly quantified in terms of Fourier-decay and …
We define and discuss transfinite asymptotic notions of smoothability, type, and equal norm type. We prove distinctness of these notions for a proper class of ordinals and that each class is an ideal. We also extend some results of Godefroy, Kalton, and Lancien to operators and ordinals greater than zero regarding the equivalence of equal norm asymptotic type and uniform renormings with power type smoothness. Finally, we discuss an extension of a non-linear result for quasi-reflexive, asymptotically $p$-smoothable Banach spaces to quasi-reflexive Banach spaces with asymptotic equal norm type $…
We study the geometry of $\mathcal{L}(X)_w,$ the space of all bounded linear operators on a Banach space $X,$ endowed with the numerical radius norm, whenever the numerical radius defines a norm. We obtain the form of the extreme points of the unit ball of the dual space of $\mathcal{L}(X)_w.$ Using this structure, we explore Birkhoff-James orthogonality, best approximation and deduce distance formula in $\mathcal{L}(X)_w.$ A special attention is given to the case of operators satisfying a notion of smoothness. Finally, we obtain an equivalence between Birkhoff-James orthogonality in $\mathcal…
In this paper we develop the theory of Fourier multiplier operators $T_{m}:L^{p}(\mathbb{R}^{d};X)\to L^{q}(\mathbb{R}^{d};Y)$, for Banach spaces $X$ and $Y$, $1\leq p\leq q\leq \infty$ and $m:\mathbb{R}^d\to \mathcal{L}(X,Y)$ an operator-valued symbol. The case $p=q$ has been studied extensively since the 1980's, but far less is known for $p<q$. In the scalar setting one can deduce results for $p<q$ from the case $p=q$. However, in the vector-valued setting this leads to restrictions both on the smoothness of the multiplier and on the class of Banach spaces. For example, one often needs that …
In this paper, we introduce the notion of the Bishop-Phelps-Bollobás property for numerical radius (BPBp-$ν$) for a subclass of the space of bounded linear operators. Then, we show that certain subspaces of $\mathcal{L}(L_1(μ))$ have the BPBp-$ν$ for every finite measure $μ$. As a consequence we deduce that the subspaces of finite-rank operators, compact operators and weakly compact operators on $L_1(μ)$ have the BPBp-$ν$.
In this paper, we aim to introduce the notion of the spectral radius of bounded linear operators acting on a complex Hilbert space $\mathcal{H}$, which are bounded with respect to the seminorm induced by a positive operator $A$ on $\mathcal{H}$. Mainly, we show that $r_A(T)\leq ω_A(T)$ for every $A$-bounded operator $T$, where $r_A(T)$ and $ω_A(T)$ denote respectively the $A$-spectral radius and the $A$-numerical radius of $T$. This allows to establish that $r_A(T)=ω_A(T)=\|T\|_A$ for every $A$-normaloid operator $T$, where $\|T\|_A$ is denoted to be the $A$-operator seminorm of $T$. Moreover,…
From the interpretation of Linear Logic multiplicative disjunction as the $\varepsilon$-product defined by Laurent Schwartz, we construct several models of Differential Linear Logic based on usual mathematical notions of smooth maps. This improves on previous results, by R. Blute, T. Ehrhard and C. Tasson, based on convenient smoothness where only intuitionist models were built. We isolate a completeness condition, called k-quasi-completeness, and an associated notion stable by duality called k-reflexivity, allowing for a $*$-autonomous category of k-reflexive spaces in which the dual of the t…
Several notions of spectral radius arise in the study of nonlinear order-preserving positively homogeneous self-maps of cones in Banach spaces. We give conditions that guarantee that all these notions lead to the same value. In particular, we give a Collatz-Wielandt type formula, which characterizes the growth rate of the orbits in terms of eigenvectors in the closed cone or super-eigenvectors in the interior of the cone. This characterization holds when the cone is normal and when a quasi-compactness condition, involving an essential spectral radius defined in terms of $k$-set-contractions, i…
