For a fixed Banach operator ideal $\mathcal A$, we use the notion of $\mathcal A$-compact sets of Carl and Stephani to study $\mathcal A$-compact polynomials and $\mathcal A$-compact holomorphic mappings. Namely, those mappings $g\colon X\rightarrow Y$ such that every $x \in X$ has a neighborhood $V_x$ such that $g(V_x)$ is relatively $\mathcal A$-compact. We show that the behavior of $\mathcal A$-compact polynomials is determined by its behavior in any neighborhood of any point. We transfer some known properties of $\mathcal A$-compact operators to $\mathcal A$-compact polynomials. In order to study $\mathcal A$-compact holomorphic functions, we appeal to the $\mathcal A$-compact radius of convergence which allows us to characterize the functions in this class. Under certain hypothesis on the ideal $\mathcal A$, we give examples showing that our characterization is sharp.
A sufficient condition is established for the existence of a solution to the equation $\mathcal{T}(u,\mathcal{C}(u))=u$, by considering a class of Kannan type equicontraction mappings $\mathcal{T}:\mathcal{A}\times \overline{\mathcal{C}(\mathcal{A})}\to Ξ$, where $\mathcal{A}$ is a convex, closed and bounded subset of a Banach space $Ξ$ and $\mathcal{C}$ is a compact mapping. To fulfil the desired purpose, we engage the Sadovskii's theorem, involving the measure of noncompactness. The relevance of the acquired results has been illustrated by considering a certain class of initial value problem…
Let $E$ and $F$ be complex Banach spaces, $U$ be an open subset of $E$ and $1\leq p\leq\infty$. We introduce and study the notion of a Cohen strongly $p$-summing holomorphic mapping from $U$ to $F$, a holomorphic version of a strongly $p$-summing linear operator. For such mappings, we establish both Pietsch domination/factorization theorems and analyse their linearizations from $\mathcal{G}^\infty(U)$ (the canonical predual of $\mathcal{H}^\infty(U)$) and their transpositions on $\mathcal{H}^\infty(U)$. Concerning the space $\mathcal{D}_p^{\mathcal{H}^\infty}(U,F)$ formed by such mappings and …
Let $\mathcal{A}$ and $\mathcal{U}$ be Banach algebras and $θ$ be a nonzero character on $\mathcal{A}$. Then the \textit{Lau product Banach algebra} $\mathcal{A}\times_θ\mathcal{U}$ associated with the Banach algebras $\mathcal{A}$ and $\mathcal{U}$ is the $l^1$-direct sum $\mathcal{A}\oplus\mathcal{U}$ equipped with the algebra multiplication $(a,u)(a',u')=(ab,θ(a)u'+θ(a')u+uu')\, (a,a'\in\mathcal{A},u,u'\in\mathcal{U})$ and $l^1$-norm. In this paper we shall investigate the derivations and multipliers from this Banach algebras and study the automatic continuity of these mappings. We also stu…
Let ${\mathfrak M}=({\mathcal M},ρ)$ be a metric space and let $X$ be a Banach space. Let $F$ be a set-valued mapping from ${\mathcal M}$ into the family ${\mathcal K}_m(X)$ of all compact convex subsets of $X$ of dimension at most $m$. The main result in our recent joint paper with Charles Fefferman (which is referred to as a "Finiteness Principle for Lipschitz selections") provides efficient conditions for the existence of a Lipschitz selection of $F$, i.e., a Lipschitz mapping $f:{\mathcal M}\to X$ such that $f(x)\in F(x)$ for every $x\in{\mathcal M}$. We give new alternative proofs of this…
Let ${\mathfrak M}=({\mathcal M},ρ)$ be a metric space and let $X$ be a Banach space. Let $F$ be a set-valued mapping from ${\mathcal M}$ into the family ${\mathcal K}_m(X)$ of all compact convex subsets of $X$ of dimension at most $m$. The main result in our recent joint paper with Charles Fefferman (which is referred to as a "Finiteness Principle for Lipschitz selections") provides efficient conditions for the existence of a Lipschitz selection of $F$, i.e., a Lipschitz mapping $f:{\mathcal M}\to X$ such that $f(x)\in F(x)$ for every $x\in{\mathcal M}$. We give new alternative proofs of this…
Related to the concept of $p$-compact operator with $p\in [1,\infty]$ introduced by Sinha and Karn, this paper deals with the space $\mathcal{H}^\infty_{\mathcal{K}_p}(U,F)$ of all Banach-valued holomorphic mappings on an open subset $U$ of a complex Banach space $E$ whose ranges are relatively $p$-compact subsets of $F$. We characterize such holomorphic mappings as those whose Mujica's linearisations on the canonical predual of $\mathcal{H}^\infty(U)$ are $p$-compact operators. This fact allows us to make a complete study of them. We show that $\mathcal{H}^\infty_{\mathcal{K}_p}$ is a surject…
We study a mapping $τ_G$ of the cone ${\mathbf B}^+({\mathcal H})$ of bounded nonnegative self-adjoint operators in a complex Hilbert space ${\mathcal H}$ into itself. This mapping is defined as a strong limit of iterates of the mapping ${\mathbf B}^+({\mathcal H})\ni X\mapstoμ_G(X)=X-X:G\in{\mathbf B}^+({\mathcal H})$, where $G\in{\mathbf B}^+({\mathcal H})$ and $X:G$ is the parallel sum. We find explicit expressions for $τ_G$ and establish its properties. In particular, it is shown that $τ_G$ is sub-additive, homogeneous of degree one, and its image coincides with set of its fixed points whi…
Regarding the geometry of a real normed space ${\mathcal X}$, we mainly introduce a notion of approximate bisectrix-orthogonality on vectors $x, y \in {\mathcal X}$ as follows: $${x\np{\varepsilon}}_W y \mbox{if and only if} \sqrt{2}\frac{1-\varepsilon}{1+\varepsilon}\|x\|\,\|y\|\leq \Big\|\,\|y\|x+\|x\|y\,\Big\|\leq\sqrt{2}\frac{1+\varepsilon}{1-\varepsilon}\|x\|\,\|y\|.$$ We study class of linear mappings preserving the approximately bisectrix-orthogonality ${\np{\varepsilon}}_W$. In particular, we show that if $T: {\mathcal X}\to {\mathcal Y}$ is an approximate linear similarity, then $${x\…
Let $\mathcal{L}$ be a subspace lattice on a Banach space $X$ and let $δ:\mathrm{Alg}\mathcal{L}\rightarrow B(X)$ be a linear mapping. If $\vee\{L\in \mathcal{L}: L_-\nsupseteq L\}=X$ or $\wedge\{L_-:L\in \mathcal{L}, L_-\nsupseteq L\}=(0)$, we show that the following three conditions are equivalent: (1) $δ(AB)=δ(A)B+Aδ(B)$ whenever $AB=0$; (2) $δ(AB+BA)=δ(A)B+Aδ(B)+δ(B)A+Bδ(A)$ whenever $AB+BA=0$; (3) $δ$ is a generalized derivation and $δ(I)\in (\mathrm{Alg}\mathcal{L})^\prime$. If $\vee\{L\in \mathcal{L}: L_-\nsupseteq L\}=X$ or $\wedge\{L_-:L\in \mathcal{L}, L_-\nsupseteq L\}=(0)$ and $δ$ …
The purpose of this article is to prove a generalisation of the Besicovitch-Federer projection theorem about a characterisation of rectifiable and unrectifiable sets in terms of their projections. For an $m$-unrectifiable set $Σ\subset\mathbb{R}^n$ having finite Hausdorff measure and $\varepsilon>0$, we prove that for a mapping $f\in\mathcal{C}^1(U,\mathbb{R}^n)$ having constant, equal to $m$, rank of the Jacobian matrix there exists a mapping $f_\varepsilon$ whose rank of the Jacobian matrix is also constant, equal to $m$, such that $\|f_\varepsilon-f\|_{\mathcal{C}^1}<\varepsilon$ and $\math…
We study the generic behavior of the method of successive approximations for set-valued mappings in separable Banach spaces. We consider the case of nonexpansive mappings with convex and compact point images and show that for the typical such mapping and typical points of its domain the sequence of successive approximations is unique and converges to a fixed point of the mapping.
Applying a linearization theorem due to J. Mujica, we study the ideals of bounded holomorphic mappings $\mathcal{H}^\infty\circ\mathcal{I}$ generated by composition with an operator ideal $\mathcal{I}$. The bounded-holomorphic dual ideal of $\mathcal{I}$ is introduced and its elements are characterized as those that admit a factorization through $\mathcal{I}^\mathrm{dual}$. For complex Banach spaces $E$ and $F$, we also analyze new ideals of bounded holomorphic mappings from an open subset $U\subseteq E$ to $F$ such as $p$-integral holomorphic mappings and $p$-nuclear holomorphic mappings with…
Let $\mathcal H$ be a complex infinite-dimensional separable Hilbert space, and let $\mathcal K(\mathcal H)$ be the $C^*$-algebra of compact linear operators in $\mathcal H$. Let $(E,\|\cdot\|_E)$ be a symmetric sequence space. If $\{μ(n,x)\}$ are the singular values of $x\in\mathcal K(\mathcal H)$, let $\mathcal C_E=\{x\in\mathcal K(\mathcal H): \{μ(n,x)\}\in E\}$ with $\|x\|_{\mathcal C_E}=\|\{μ(n,x)\}\|_E$, $x\in\mathcal C_E$, be the Banach ideal of compact operators generated by $E$. Let $\mathcal C_E^h=\{x\in\mathcal C_E : x=x^*\}$ be the real Banach subspace of self-adjoint operators in …
Let $\mathcal H$ be an infinite-dimensional Hilbert space, and let $\mathcal B(\mathcal H)$ ($\mathcal K(\mathcal H)$) be the $C^*$-algebra of bounded (respectively, compact) linear operators in $\mathcal H$. Let $(E,\|\cdot\|_E)$ be a fully symmetric sequence space. If $\{s_n(x)\}_{n=1}^\infty$ are the singular values of $x\in\mathcal K(\mathcal H)$, let $\mathcal C_E=\{x\in\mathcal K(\mathcal H): \{s_n(x)\}\in E\}$ with $\|x\|_{\mathcal C_E}=\|\{s_n(x)\}\|_E$, $x\in\mathcal C_E$, be the Banach ideal of compact operators generated by $E$. We show that the averages $A_n(T)(x)=\frac1{n+1}\sum\l…
In this paper we study absolutely summing mappings on Banach spaces by exploring the cotype of their domains and ranges. It is proved that every $n$% -linear mapping from $\mathcal{L}_{\infty}$-spaces into $\mathbb{K}$ is $% (2;2,...,2,\infty)$-summing and also shown that every $n$-linear mapping from $\mathcal{L}_{\infty}$-spaces into $F$ is $(q;2,...,2)$-summing whenever $F$ has cotype $q.$ We also give new examples of analytic summing mappings and polynomial and multilinear versions of a linear Extrapolation Theorem.
A multiplicative Hankel operator is an operator with matrix representation $M(α) = \{α(nm)\}_{n,m=1}^\infty$, where $α$ is the generating sequence of $M(α)$. Let $\mathcal{M}$ and $\mathcal{M}_0$ denote the spaces of bounded and compact multiplicative Hankel operators, respectively. In this note it is shown that the distance from an operator $M(α) \in \mathcal{M}$ to the compact operators is minimized by a nonunique compact multiplicative Hankel operator $N(β) \in \mathcal{M}_0$, $$\|M(α) - N(β)\|_{\mathcal{B}(\ell^2(\mathbb{N}))} = \inf \left \{\|M(α) - K \|_{\mathcal{B}(\ell^2(\mathbb{N}))} …
In this manuscript, we claim that the newly introduced $\mathcal{F}$-metric spaces are Hausdorff and also first countable. Moreover, we assert that every separable $\mathcal{F}$-metric space is second countable. Additionally, we acquire some interesting fixed point results concerning altering distance functions for contractive-type mappings and Kannan-type contractive mappings in this exciting context. However, most of the findings are well-furnished by several non-trivial numerical examples. Finally, we raise an open problem regarding the metrizability of such kind of spaces.
Let $\mathcal{A}$ and $\mathcal{B}$ be two unital complex $\ast $-algebras such that $\mathcal{A}$ has a nontrivial projection. In this paper, we study the structure of bijective mappings $Φ:\mathcal{A}\rightarrow \mathcal{B}$ preserving sum of triple products $α_{1} ab^{*}c+α_{2} acb^{*}+α_{3} b^{*}ac +α_{4} cab^{*}+α_{5} b^{*}ca+α_{6} cb^{*}a,$ where the scalars $\{α_{k}\}_{k=1}^{6}$ are complex numbers satisfying some conditions. Applications of obtained results are given.
We prove in a uniform way that all Denjoy--Carleman differentiable function classes of Beurling type $C^{(M)}$ and of Roumieu type $C^{\{M\}}$, admit a convenient setting if the weight sequence $M=(M_k)$ is log-convex and of moderate growth: For $\mathcal C$ denoting either $C^{(M)}$ or $C^{\{M\}}$, the category of $\mathcal C$-mappings is cartesian closed in the sense that $\mathcal C(E,\mathcal C(F,G))\cong \mathcal C(E\times F, G)$ for convenient vector spaces. Applications to manifolds of mappings are given: The group of $\mathcal C$-diffeomorphisms is a regular $\mathcal C$-Lie group if $…
Let $E$ be a locally convex space, $U\subseteq\mathbb{R}^n$ as well as $V\subseteq\mathbb{R}^m$ be open and $k,l\in\mathbb{N}_0\cup\left\{\infty\right\}$. Locally convex spaces $C^{k,l}(U\times V,E)$ of functions with different degrees of differentiability in the $U$- and $V$-variable were recently studied by H.Alzaareer, who established an exponential law of the form $C^{k,l}(U\times V,E)\cong C^k(U,C^l(V,E))$. We establish an analogous exponential law $C^{k,l}_{\mathcal{W}_1\otimes\mathcal{W}_2}(U\times V,E)\cong C^k_{\mathcal{W}_1}(U,C^l_{\mathcal{W}_2}(V,E))$ for suitable spaces of weighte…
Let $(E,\mathcal E,μ)$ be a measure space and $G\colon E\times E\to [0,\infty]$ be measurable. Moreover, let $\mathcal F\!_{ui}$ denote the set of all $q\in\mathcal E^+$ (measurable numerical functions $q\ge 0$ on $E$) such that $\{G(x,\cdot)q\colon x\in E\}$ is uniformly integrable, and let $\mathcal F\!_{co}$ denote the set of all $q\in\mathcal E^+$ such that the mapping $f\mapsto G(fq) :=\int G(\cdot,y) f(y) q(y)\,dμ(y)$ is a compact operator on the space $\mathcal E_b$ of bounded measurable functions on $E$ (equipped with the sup-norm). It is shown that $\mathcal F\!_{ui}=\mathcal F\!_{co}…
Using Mujica's linearization theorem, we extend to the holomorphic setting some classical characterizations of compact (weakly compact, Rosenthal, Asplund) linear operators between Banach spaces such as the Schauder, Gantmacher and Gantmacher-Nakamura theorems and the Davis-Figiel-Johnson-Pelczynski, Rosenthal and Asplund factorization theorems.
For an operator ideal $\mathcal A$, we study the composition operator ideals ${\mathcal A}\circ{\mathcal K}$, ${\mathcal K}\circ{\mathcal A}$ and ${\mathcal K}\circ{\mathcal A}\circ{\mathcal K}$, where $\mathcal K$ is the ideal of compact operators. We introduce a notion of an $\mathcal A$-approximation property on a Banach space and characterise it in terms of the density of finite rank operators in ${\mathcal A}\circ{\mathcal K}$ and ${\mathcal K}\circ{\mathcal A}$. We propose the notions of $\ell_{\infty}$-extension and $\ell_{1}$-lifting properties for an operator ideal $\mathcal A$ and st…
We prove the existence of a non-trivial hyperinvariant subspace for several sets of polynomially compact operators. The main results of the paper are: (i) a non-trivial norm closed algebra $\mathcal A\subseteq \mathcal B(\mathscr X)$ which consists of polynomially compact quasinilpotent operators has a non-trivial hyperinvariant subspace; (ii) if there exists a non-zero compact operator in the norm closure of the algebra generated by an operator band $\mathcal S$, then $\mathcal S$ has a non-trivial hyperinvariant subspace.
We introduce a concept of bilinear ideal of jointly completely bounded mappings between operator spaces. In particular, we study the bilinear ideals $\mathcal{N}$ of completely nuclear, $\mathcal{I }$ of completely integral, $\mathcal{E}$ of completely extendible bilinear mappings, $\mathcal{MB}$ multiplicatively bounded and its symmetrization $\mathcal{SMB}$. We prove some basic properties of them, one of which is the fact that $\mathcal{I}$ is naturally identified with the ideal of (linear) completely integral mappings on the injective operator space tensor product.
Let $π: \widetilde{\mathcal{N}} \to \mathcal{N}$ be a Riemannian covering, with $\mathcal{N}$, $\widetilde{\mathcal{N}}$ smooth compact connected Riemannian manifolds. If $\mathcal{M}$ is an $m$-dimensional compact simply-connected Riemannian manifold, $0<s<1$ and $2 \le sp< m$, we prove that every mapping $u \in W^{s, p} (\mathcal{M}, \mathcal{N})$ has a lifting in $W^{s,p}$, i.e., we have $u = π\, \circ \, \widetilde{u}$ for some mapping $\widetilde{u} \in W^{s, p} (\mathcal{M}, \widetilde{\mathcal{N}})$. Combined with previous contributions of Bourgain, Brezis and Mironescu and Bethuel and …
We review the current state of the art concerning the characterization of traces of the spaces $W^{1, p} (\mathbb{B}^{m-1}\times (0,1), \mathcal{N})$ of Sobolev mappings with values into a compact manifold $\mathcal{N}$. In particular, we exhibit a new analytical obstruction to the extension, which occurs when $p < m$ is an integer and the homotopy group $π_p (\mathcal{N})$ is non trivial. On the positive side, we prove the surjectivity of the trace operator when the fundamental group $π_1 (\mathcal{N})$ is finite and $π_2 (\mathcal{N}) \simeq \dotsb \simeq π_{\lfloor p - 1 \rfloor} (\mathcal{…
In this brief article we characterize the relatively compact subsets of $\mathcal{A}^p$ for the topology $σ(\mathcal{A}^p,\mathcal{R}^q)$ (see below), by the weak compact subsets of $L^p$ . The spaces $\mathcal{R}^q$ endowed with the weak topology induced by $\mathcal{A}^p$, was recently employed to create the convex risk theory of random processes. The weak compact sets of $\mathcal{A}^p$ are important to characterize the so-called Lebesgue property of convex risk measures, to give a complete description of the Makcey topology on $\mathcal{R}^q$ and for their use in the optimization theory.
We define the notion of $φ$-perturbation of a densely defined adjointable mapping and prove that any such mapping $f$ between Hilbert ${\mathcal A}$-modules over a fixed $C^*$-algebra ${\mathcal A}$ with densely defined corresponding mapping $g$ is ${\mathcal A}$-linear and adjointable in the classical sense with adjoint $g$. If both $f$ and $g$ are everywhere defined then they are bounded. Our work concerns with the concept of Hyers--Ulam--Rassias stability originated from the Th.M. Rassias' stability theorem that appeared in his paper [On the stability of the linear mapping in Banach spaces,…
We introduce a uniform structure on any Hilbert $C^*$-module $\mathcal N$ and prove the following theorem: suppose, $F:{\mathcal M}\to {\mathcal N}$ is a bounded adjointable morphism of Hilbert $C^*$-modules over $\mathcal A$ and $\mathcal N$ is countably generated. Then $F$ belongs to the Banach space generated by operators $θ_{x,y}$, $θ_{x,y}(z):=x\langle y,z\rangle$, $x\in {\mathcal N}$, $y,z\in {\mathcal M}$ (i.e. $F$ is ${\mathcal A}$-compact, or "compact") if and only if $F$ maps the unit ball of ${\mathcal M}$ to a totally bounded set with respect to this uniform structure (i.e. $F$ is …
We define Hardy spaces $\mathcal{H}^p$ for quasiregular mappings in the plane, and show that for a particular class of these mappings many of the classical properties that hold in the classical setting of analytic mappings still hold. This particular class of quasiregular mappings can be characterised in terms of composition operators when the symbol is quasiconformal. Relations between Carleson measures and Hardy spaces play an important role in the discussion. This program was initiated and developed for Hardy spaces of quasiconformal mappings by Astala and Koskela in 2011 in their paper $\m…
A mapping $f: {\mathcal M} \to {\mathcal N}$ between Hilbert $C^*$-modules approximately preserves the inner product if \[\|< f(x), f(y)> - < x, y> \| \leq ϕ(x, y),\] for an appropriate control function $ϕ(x,y)$ and all $x, y \in {\mathcal M}$. In this paper, we extend some results concerning the stability of the orthogonality equation to the framework of Hilbert $C^*$-modules on more general restricted domains. In particular, we investigate some asymptotic behavior and the Hyers--Ulam--Rassias stability of the orthogonality equation.
Let $(Ω, \mathcal{F}, \mathbf{P})$ be a probability space, $ξ$ be a random variable on $(Ω, \mathcal{F}, \mathbf{P})$, $\mathcal{G}$ be a sub-$σ$-algebra of $\mathcal{F}$, and let $\mathbf{E}^\mathcal{G} = \mathbf{ E}(\cdot | \mathcal{G})$ be the corresponding conditional expectation operator. We obtain sharp estimates for the moments of $ξ- \mathbf{E}^\mathcal{G}ξ$ in terms of the moments of $ξ$. This allows us to find the optimal constant in the bounded compact approximation property of $L^p([0, 1])$, $1 < p < \infty$.
We introduce the class of operator $p$-compact mappings and completely right $p$-nuclear operators, which are natural extensions to the operator space framework of their corresponding Banach operator ideals. We relate these two classes, define natural operator space structures and study several properties of these ideals. We show that the class of operator $\infty$-compact mappings in fact coincides with a notion already introduced by Webster in the nineties (in a very different language). This allows us to provide an operator space structure to Webster's class.
We construct an essential extension of $\mathcal K(\ell_2({\mathfrak{c}}))$ by $\mathcal K(\ell_2)$, where ${\mathfrak{c}}$ denotes the cardinality of continuum, i.e., a $C^*$-algebra $\mathcal A\subseteq \mathcal B(\ell_2)$ satisfying the short exact sequence $$0\rightarrow \mathcal K(\ell_2)\xrightarrowι \mathcal A \rightarrow\mathcal K(\ell_2({\mathfrak{c}}))\rightarrow 0,$$ where $ι[\mathcal K(\ell_2)]$ is an essential ideal of $\mathcal A$ such that the algebra of multipliers $\mathcal M(\mathcal A)$ of $\mathcal A$ is equal to the unitization of $\mathcal A$. In particular $\mathcal A$ i…
A common approach for describing classes of functions and probability measures on a topological space $\mathcal{X}$ is to construct a suitable map $Φ$ from $\mathcal{X}$ into a vector space, where linear methods can be applied to address both problems. The case where $\mathcal{X}$ is a space of paths $[0,1] \to \mathbb{R}^n$ and $Φ$ is the path signature map has received much attention in stochastic analysis and related fields. In this article we develop a generalized $Φ$ for the case where $\mathcal{X}$ is a space of maps $[0,1]^d \to \mathbb{R}^n$ for any $d \in \mathbb{N}$, and show that th…
We prove two results about Wigner distributions. Firstly, that the Wigner transform is the only sesquilinear map ${\mathcal S}(\mathbb{R}^n) \times {\mathcal S}(\mathbb{R}^n) \to {\mathcal S}(\mathbb{R}^{2n})$ which is bounded and covariant under phase-space translations and linear symplectomorphisms. Consequently, the Wigner distributions form the only set of quasidistributions which is invariant under linear symplectic transformations. Secondly, we prove that the maximal group of (linear or non-linear) coordinate transformations that preserves the set of (pure or mixed) Wigner distributions …
