Green’s theorem gives a Morita equivalence $C_0(G/H,A)⋊G\sim A⋊H$ for a closed subgroup $H$ of a locally compact group $G$ acting on a $C^{\ast }$-algebra $A$. We prove an analogue of Green’s theorem in the case $G=\mathbb{Z}$, where the automorphism generating the action is replaced by a Hilbert $C^{\ast }$-bimodule.

In this article, we give some characterizations of Lipschitz space via commutators of bilinear singular integral operators and bilinear fractional integral operators, respectively.

We obtain global rates of approximation by $q$-Durrmeyer operators $D_{n,q}(f;x)$ for the functions in the class $L_p\left(\right[0,1\left]\right),1\le p\le \infty$. First, rates of approximation in terms of the norms of $f$ and $f\text{'}$ and in terms of the ordinary modulus of smoothness are obtained. Subsequently, we obtain rates of approximation in terms of the generalized modulus of smoothness ${\omega }_{\phi }(f,\delta )$.

For $0<ϵ\le 1$ and an element $a$ of a complex unital Banach algebra $\mathcal{A}$, we prove the following two topological properties about the level sets of the condition spectrum. (1) If $ϵ=1$, then the $1$-level set of the condition spectrum of $a$ has an empty interior unless $a$ is a scalar multiple of the unity. (2) If $0<ϵ<1$, then the $ϵ$-level set of the condition spectrum of $a$ has an empty interior in the unbounded component of the resolvent set of $a$. Further, we show that, if the Banach space $X$ is complex uniformly convex or if $X^{\ast }$ is complex uniformly convex, then, for any operator $T$ acting on $X$, the level set of the $ϵ$-condition spectrum of $T$ has an empty interior.

In this paper, we consider the Ulam–Hyers stability of the functional equations $$f(ux-vy,uy-vx)=f(x,y)f(u,v),$$ $$f(ux+vy,uy-vx)=f(x,y)f(u,v),$$ $$f(ux+vy,uy+vx)=f(x,y)f(u,v),$$ $$f(ux-vy,uy+vx)=f(x,y)f(u,v)$$ for all $x,y,u,v\in \mathbb{R}$, where $f:{\mathbb{R}}^2\to \mathbb{R}$, which arise from number theory and are connected with the characterizations of the determinant and permanent of two-by-two matrices.

Assume that $K$ is a closed convex subset of a uniformly convex Banach space $E$, and assume that $\left\{T\right(s){\}}_{s>0}$ is a nonexpansive semigroup on $K$. By using the following implicit iteration sequence $\left\{x_n\right\}$ defined by $$x_n=(1-{\alpha }_n)x_{n-1}+{\alpha }_n\cdot \frac{1}{t_n}{\int }_0^{t_n}T\left(s\right)x_{n}ds,\forall n\ge 1,$$ the main purpose of this paper is to establish a weak convergence theorem for the nonexpansive semigroup $\left\{T\right(s){\}}_{s>0}$ in uniformly convex Banach spaces without the Opial property. Our results are different from some recently announced results.

Modifying the moduli of supporting convexity and supporting smoothness, we introduce new moduli for Banach spaces which occur, for example, as lengths of catheti of right-angled triangles (defined via so-called quasiorthogonality). These triangles have two boundary points of the unit ball of a Banach space as endpoints of their hypotenuse, and their third vertex lies in a supporting hyperplane of one of the two other vertices. Among other things, it is our goal to quantify via such triangles the local deviation of the unit sphere from its supporting hyperplanes. We prove respective Day–Nordlander-type results involving generalizations of the modulus of convexity and the modulus of Banaś.

Let $R\left(\mathbb{D}\right)$ be the algebra generated in the Sobolev space $W^{22}\left(\mathbb{D}\right)$ by the rational functions with poles outside the unit disk $\overline{\mathbb{D}}$. This is called the Sobolev disk algebra. In this article, the commutant of the multiplication operator $M_{B\left(z\right)}$ on $R\left(\mathbb{D}\right)$ is studied, where $B\left(z\right)$ is an n-Blaschke product. We prove that an operator $A\in \mathcal{L}\left(R\right(\mathbb{D}\left)\right)$ is in $\mathcal{A}\text{'}\left(M_{B\left(z\right)}\right)$ if and only if $A={\sum }_{i=1}^n{M}_{h_i}{M}_{\Delta \left(z\right)}^{-1}{T}_i$, where $\{h_i{\}}_{i=1}^n\subset R(\mathbb{D})$, and $T_i\in \mathcal{L}\left(R\right(\mathbb{D}\left)\right)$ is given by $\left(T_{i}g\right)\left(z\right)={\sum }_{j=1}^n(-1{)}^{i+j}{\Delta }_{ij}(z\left)g\right(G_{j-1}\left(z\right))$, $i=1,2,\dots ,n$, $G_0\left(z\right)\equiv z$.

Let $A$ be a positive invertible matrix, and let $B$ be a normal matrix. Following the investigation of Ando, we show that $\Vert A♯\left(B^{\ast }{A}^{-1}B\right)\Vert \ge \Vert B\Vert$, where $♯$ denotes the geometric mean, fails in general.

This note presents a further study of the class of Cuntz–Krieger-type algebras. A necessary and sufficient condition is identified that ensures that the algebra is purely infinite, the ideal structure is studied, and nuclearity is proved by presenting the algebra as a crossed product of an AF-algebra by an abelian group. The results are applied to examples of Cuntz–Krieger-type algebras, such as higher-rank semigraph $C^{\ast }$-algebras and higher-rank Exel–Laca algebras.

For a linear-delay equation on an arbitrary Banach space, we describe a condition so that the Lyapunov exponents of the equation persist under sufficiently small linear as well as nonlinear perturbations. We consider both cases of discrete and continuous time with the study of delay-difference equations and delay equations, respectively. The delay can be any number from zero to infinity.

In this paper we describe the complex-valued solutions defined on a double coset hypergroup of the exponential, additive, and quadratic functional equations. Moreover, the $m$-sine functions on a double coset hypergroup are discussed. The double coset hypergroup we consider is closely related to affine groups and spherical functions on them.