We introduce Banach spaces of vector-valued random variables motivated from mathematical finance. So-called risk functionals are defined in a natural way on these Banach spaces, and it is shown that these functionals are Lipschitz continuous. Since the risk functionals cannot be defined on strictly larger spaces of random variables, this creates an area of particular interest with regard to the spaces presented. We elaborate key properties of these Banach spaces and give representations of their dual spaces in terms of vector measures with values in the dual space of the state space.

In this article, we study Toeplitz operators with nonnegative symbols on the ${\mathcal{A}}_2$-weighted harmonic Bergman space. We characterize the boundedness, compactness, and invertibility of Toeplitz operators with nonnegative symbols on this space.

Partial actions of groups on $C^{\ast }$-algebras and the closely related actions and coactions of Hopf algebras have received much attention in recent decades. They arise naturally as restrictions of their global counterparts to noninvariant subalgebras, and the ambient enveloping global (co)actions have proven useful for the study of associated crossed products. In this article, we introduce the partial coactions of $C^{\ast }$-bialgebras, focusing on $C^{\ast }$-quantum groups, and we prove the existence of an enveloping global coaction under mild technical assumptions. We also show that partial coactions of the function algebra of a discrete group correspond to partial actions on direct summands of a $C^{\ast }$-algebra, and we relate partial coactions of a compact or its dual discrete $C^{\ast }$-quantum group to partial coactions or partial actions of the dense Hopf subalgebra. As a fundamental example, we associate to every discrete $C^{\ast }$-quantum group a quantum Bernoulli shift.

We study properties under which the domain of a closed derivation $\delta :D\left(\delta \right)\to A$ of a generalized B${}^{\ast }$-algebra $A$ remains invariant under analytic functional calculus. For a complete, generalized B${}^{\ast }$-algebra with jointly continuous multiplication, two sufficient conditions are assumed: that the unit of $A$ belongs to the domain of the derivation, along with a condition related to the coincidence ${\sigma }_A\left(x\right)={\sigma }_{D\left(\delta \right)}\left(x\right)$ of the (Allan) spectra for every element $x\in D\left(\delta \right)$. Certain results are derived concerning the spectra for a general element of the domain, in the realm of a domain which is advertibly complete or enjoys the Q-property. For a closed $\ast$-derivation $\delta$ of a complete GB${}^{\ast }$-algebra with jointly continuous multiplication such that $1\in D\left(\delta \right)$ and $x$ a normal element of the domain, $f\left(x\right)\in D\left(\delta \right)$ for every analytic function on a neighborhood of the spectrum of $x$. We also give an example of a closed derivation of a GB${}^{\ast }$-algebra which does not contain the identity element. A condition for a closed derivation of a GB${}^{\ast }$-algebra $A$ to be the generator of a one-parameter group of automorphisms of $A$ is provided along with a generalization of the Lumer–Phillips theorem for complete locally convex spaces.

In the setting of the Heisenberg group, we define weighted Hardy spaces by means of their atomic characterization, and we establish the (sharp) boundedness of Hausdorff operators on power-weighted Hardy spaces. Moreover, we obtain sufficient and necessary conditions for the boundedness of Hausdorff operators on local Hardy spaces in the Heisenberg group.

Let $(X_n\Vert \cdot {\Vert }_n)$ denote a sequence of real Banach spaces. Let

$$X={\oplus }_1{X}_n=\left\{\right(x_n):x_n\in X_n\text{for any}n\in \mathbb{N},{\sum }_{n=1}^{\infty }\Vert x_n{\Vert }_n<\infty \}.$$ In this article, we investigate some properties of best approximation operators associated with finite-dimensional subspaces of $X$. In particular, under a number of additional assumptions on $\left(X_n\right)$, we characterize finite-dimensional Chebyshev subspaces $Y$ of $X$. Likewise, we show that the set

$$\mathrm{Nuniq}=\{x\in X:card(P_Y\left(x\right))>1\}$$ is nowhere dense in $Y$, where $P_Y$ denotes the best approximation operator onto $Y$. Finally, we demonstrate various (mainly negative) results on the existence of continuous selection for metric projection and we provide examples illustrating possible applications of our results.

Let $A$, $B$ be positive semidefinite $n\times n$ matrices, and let $\alpha \in (0,1)$. We show that if $f$ is an increasing submultiplicative function on $[0,\infty )$ with $f\left(0\right)=0$ such that $f\left(t\right)$ and $f^2\left(t^{1/2}\right)$ are convex, then for every unitarily invariant norm. Moreover, if $\alpha \in [0,1]$ and $X$ is an $n\times n$ matrix with $X\ne 0$, then for every unitarily invariant norm. These inequalities present generalizations of recent results of Zou and Jiang and of Audenaert.

A sufficient condition for higher-order compact embeddings on bounded domains in Carnot–Carathéodory spaces is established for the class of rearrangement-invariant function spaces. The condition is expressed in terms of compactness of a suitable $1$-dimensional integral operator depending on the isoperimetric function relative to the Carnot–Carathéodory structure of the relevant sets. The general result is then applied to particular Sobolev spaces built upon Lebesgue and Lorentz spaces.

This article is devoted to the investigation of self-adjoint (and, more generally, proper) extensions of Phillips symmetric operators (PSO). A closed densely defined symmetric operator with equal defect numbers is considered a Phillips symmetric operator if its characteristic function is a constant on ${\mathbb{C}}_{+}$. We present equivalent definitions of PSO and prove that proper extensions with real spectra of a given PSO are similar to each other. Our results imply that one-point interaction of the momentum operator $i\frac{d}{dx}+\alpha \delta (x-y)$ leads to unitarily equivalent self-adjoint operators with Lebesgue spectra. Self-adjoint operators with nontrivial spectral properties can be obtained as a result of more complicated perturbations of the momentum operator. In this way, we study special classes of perturbations which can be characterized as one-point interactions defined by the nonlocal potential $\gamma \in L_2\left(\mathbb{R}\right)$.

In this article, via establishing a new atomic characterization of the Musielak–Orlicz Hardy space $H^{\phi }\left({\mathbb{R}}^n\right)$ [which is essentially deduced from the known molecular characterization of $H^{\phi }\left({\mathbb{R}}^n\right)$] and some estimates on a new discrete Littlewood–Paley $g$-function and a Peetre-type maximal function, together with using the known intrinsic $g$-function characterization of $H^{\phi }\left({\mathbb{R}}^n\right)$, the authors obtain several equivalent characterizations of $H^{\phi }\left({\mathbb{R}}^n\right)$ in terms of wavelets, which extend the wavelet characterizations of both Orlicz–Hardy spaces and the weighted Hardy spaces, and are available to the typical and useful Musielak–Orlicz Hardy space $H^{log}\left({\mathbb{R}}^n\right)$. The novelty of this approach is that the new adapted atomic characterization of $H^{\phi }\left({\mathbb{R}}^n\right)$ compensates the inconvenience in applications of the supremum appearing in the original definition of atoms, which play crucial roles in the proof of the main theorem of this article and may have further potential applications.

In this article, we investigate the properties of the sharp partial order in unital rings, and we study additive maps preserving the minus partial order in both directions in the setting of unital semiprime Banach algebras with essential socle.

We consider operator-valued Bochner–Riesz means with weight function $h_{\kappa }^2$ under a finite reflection group for the Dunkl transform. We establish the maximal inequality of the weighted Hardy–Littlewood maximal function, and we apply it to the maximal inequality of operator-valued Bochner–Riesz means $B_R^{\delta }(h_{\kappa }^2;f)\left(x\right)$ for $\delta >{\lambda }_{\kappa }:=\frac{d-1}{2}+{\sum }_{j=1}^d{\kappa }_j$. Furthermore, we also obtain the corresponding pointwise convergence theorem.