We combine dyadic analysis through Haar-type wavelets (defined on Christ’s families of generalized cubes) and the Lax–Milgram theorem in order to prove the existence of Green’s functions for fractional Laplacians on some function spaces of vanishing small resolution in spaces of homogeneous type.

We consider continuous gradient operators $F$ acting in a real Hilbert space $H$, and we study their surjectivity under the basic assumption that the corresponding functional $\langle F\left(x\right),x\rangle$—where $\langle \cdot \rangle$ is the scalar product in $H$—is coercive. While this condition is sufficient in the case of a linear operator (where one in fact deals with a bounded self-adjoint operator), in the general case we supplement it with a compactness condition involving the number $\omega \left(F\right)$ introduced by Furi, Martelli, and Vignoli, whose positivity indeed guarantees that $F$ is proper on closed bounded sets of $H$. We then use Ekeland’s variational principle to reach the desired conclusion. In the second part of this article, we apply the surjectivity result to give a perspective on the spectrum of these kinds of operators—ones not considered by Feng or the above authors—when they are further assumed to be sublinear and positively homogeneous.

In this article, we introduce the concept of generalized multipliers for g-frames. In fact, we show that every generalized multiplier for g-Bessel sequences is a generalized multiplier for the induced sequences, in a special sense. We provide some sufficient and/or necessary conditions for the invertibility of generalized multipliers. In particular, we characterize g-Riesz bases by invertible multipliers. We look at which perturbations of g-Bessel sequences preserve the invertibility of generalized multipliers. Finally, we investigate how to find a matrix representation of operators on a Hilbert space using g-frames, and then we characterize g-Riesz bases and g-orthonormal bases by applying such matrices.

We characterize orthogonally complemented submodules in Hilbert $C^{\ast }$-modules by their orthogonal closures. Applying Magajna’s characterization of Hilbert $C^{\ast }$-modules over $C^{\ast }$-algebras of compact operators by the complementing property of submodules, we give an elementary proof of Schweizer’s characterization of Hilbert $C^{\ast }$-modules over $C^{\ast }$-algebras of compact operators. Also, we prove analogous characterization theorems for $C^{\ast }$-algebras of compact operators related to topological properties of submodules of strict completions of Hilbert modules over a nonunital $C^{\ast }$-algebra.

We produce and study a family of representations of relative graph algebras on Hilbert spaces that arise from the orbits of points of $1$-dimensional dynamical systems, where the underlying Markov interval maps $f$ have escape sets. We identify when such representations are faithful in terms of the transitions to the escape subintervals.

Let $A$ and $B$ be complex Banach algebras, and let $\varphi ,{\varphi }_1$, and ${\varphi }_2$ be surjective maps from $A$ onto $B$. Denote by $\partial \sigma (x)$ the boundary of the spectrum of $x$. If $A$ is semisimple, $B$ has an essential socle, and $\partial \sigma (xy)=\partial \sigma ({\varphi }_1(x){\varphi }_2(y))$ for each $x,y\in A$, then we prove that the maps $x\mapsto {\varphi }_1(\mathbf{1}){\varphi }_2(x)$ and $x\mapsto {\varphi }_1(x){\varphi }_2(\mathbf{1})$ coincide and are continuous Jordan isomorphisms. Moreover, if $A$ is prime with nonzero socle and ${\varphi }_1$ and ${\varphi }_2$ satisfy the aforementioned condition, then we show once again that the maps $x\mapsto {\varphi }_1(\mathbf{1}){\varphi }_2(x)$ and $x\mapsto {\varphi }_1(x){\varphi }_2(\mathbf{1})$ coincide and are continuous. However, in this case we conclude that the maps are either isomorphisms or anti-isomorphisms. Finally, if $A$ is prime with nonzero socle and $\varphi$ is a peripherally multiplicative map, then we prove that $\varphi$ is continuous and either $\varphi$ or $-\varphi$ is an isomorphism or an anti-isomorphism.

Let $\mathcal{H}$ and $\mathcal{K}$ be complex separable Hilbert spaces. Given the operators $A\in \mathcal{B}(\mathcal{H})$ and $B\in \mathcal{B}(\mathcal{K},\mathcal{H})$, we define , where $X\in \mathcal{B}(\mathcal{H},\mathcal{K})$ and $Y\in \mathcal{B}(\mathcal{K})$ are unknown elements. In this article, we give a necessary and sufficient condition for $M_{X,Y}$ to be a (right) Weyl operator for some $X\in \mathcal{B}(\mathcal{H},\mathcal{K})$ and $Y\in \mathcal{B}(\mathcal{K})$. Moreover, we show that if $dim\mathcal{K}<\infty$, then $M_{X,Y}$ is a left Weyl operator for some $X\in \mathcal{B}(\mathcal{H},\mathcal{K})$ and $Y\in \mathcal{B}(\mathcal{K})$ if and only if $[AB]$ is a left Fredholm operator and $ind([AB])\le dim\mathcal{K}$; if $dim\mathcal{K}=\infty$, then $M_{X,Y}$ is a left Weyl operator for some $X\in \mathcal{B}(\mathcal{H},\mathcal{K})$ and $Y\in \mathcal{B}(\mathcal{K})$.

In this article we characterize the form of each 2-local Lie derivation on a von Neumann algebra without central summands of type $I_1$. We deduce that every 2-local Lie derivation $\delta$ on a finite von Neumann algebra $\mathcal{M}$ without central summands of type $I_1$ can be written in the form $\delta (A)=AE-EA+h(A)$ for all $A$ in $\mathcal{M}$, where $E$ is an element in $\mathcal{M}$ and $h$ is a center-valued homogenous mapping which annihilates each commutator of $\mathcal{M}$. In particular, every linear 2-local Lie derivation is a Lie derivation on a finite von Neumann algebra without central summands of type $I_1$. We also show that every 2-local Lie derivation on a properly infinite von Neumann algebra is a Lie derivation.

Our main goal in this article is to give some functional inequalities involving a (convex) functional and its Fenchel conjugate. As a consequence, we obtain some refinements of the so-called Fenchel inequality as well as its reverse. Inequalities of interest illustrating the previous theoretical results are provided as well.

In this article we give sufficient conditions for the hypoellipticity in the first level of the abstract complex generated by the differential operators $L_j=\frac{\partial }{\partial t_j}+\frac{\partial \varphi }{\partial t_j}(t,A)A$, $j=1,2,\dots ,n$, where $A:D(A)\subset X\to X$ is a sectorial operator in a Banach space $X$, with $\Re \sigma (A)>0$, and $\varphi =\varphi (t,A)$ is a series of nonnegative powers of $A^{-1}$ with coefficients in $C^{\infty }(\Omega )$, $\Omega$ being an open set of ${\mathbb{R}}^n$ with $n\in \mathbb{N}$ arbitrary. Analogous complexes have been studied by several authors in this field, but only in the case $n=1$ and with $X$ a Hilbert space. Therefore, in this article, we provide an improvement of these results by treating the question in a more general setup. First, we provide sufficient conditions to get the partial hypoellipticity for that complex in the elliptic region. Second, we study the particular operator $A=1-\Delta :W^{2,p}({\mathbb{R}}^N)\subset L^p({\mathbb{R}}^N)\to L^p({\mathbb{R}}^N)$, for $1\le p\le 2$, which will allow us to solve the problem of points which do not belong to the elliptic region.

Our principal aim in this article is to show the equivalence of two concepts used recently in the theory of Banach algebras. The result we present here solves an open problem raised by Jeribi and Krichen in their 2015 book.

In this article we introduce a method of constructing functions with claimed properties by using the Tychonoff theorem. As an application of this method we show that the Carathéodory distance $c_D$ of convex domains $D$ in a complex, locally convex, Hausdorff, and infinite-dimensional topological vector space is approximated by the Carathéodory distances $c_{D\cap Y}$ in finite-dimensional linear subspaces $Y$. Originally this result is due to Dineen, Timoney, and Vigué who apply ultrafilters in their proof.