Let $A$ and $B$ be Banach function algebras on compact Hausdorff spaces $X$ and $Y$, respectively. Given a non-zero scalar $\alpha$and $s,t\in \Bbb N$ we characterize the general form of suitable powers of surjective maps $T, T': A \longrightarrow B$ satisfying $\|(Tf)^s (T'g)^t-\alpha\|_Y=\|f^s g^t-\alpha \|_X$, for all $f,g \in A$, where $\|\cdot \|_X$ and $\|\cdot \|_Y$ denote the supremum norms on $X$ and $Y$, respectively. A similar result is given for the case where $T=T'$ and $T$ is defined between certain subsets of $A$ and $B$. We also show that if $T: A\longrightarrow B$ is a surjective map satisfying the stronger condition$R_\pi((Tf)^{s}(Tg)^{t}-\alpha)\cap R_\pi(f^{s}g^{t}-\alpha)\neq\varnothing $ for all $f,g \in A$, where $R_\pi(\cdot)$ denotes the peripheral range of the algebra elements, then there exists a homeomorphism $\varphi$ from the Choquet boundary $c(B)$ of $B$ onto the Choquet boundary $c(A)$ of $A$ such that $(Tf)^{d}(y)=(T1)^{d}(y)\,(f \circ \varphi(y))^{d}$ for all $f\in A$ and $y\in c(B)$,where $d$ is the greatest common divisor of $s$ and $t$.

Let $\mathcal{A}$ be a unital algebra over a number field $\mathbb{K}$. A linear mapping $\delta$ from $\mathcal{A}$ into itself is called a generalized ($m, n, l$)-Jordan centralizer if it satisfies $(m+n+l)\delta(A^2)-m\delta(A)A-nA\delta(A)-lA\delta(I)A\in \mathbb{K}I$ for every $A\in \mathcal{A}$, where $m\geq0, n\geq0, l\geq0$ are fixed integers with $m+n+l\neq 0$. In this paper, we study generalized ($m, n, l$)-Jordan centralizers on generalized matrix algebras and some reflexive algebras alg$\mathcal{L}$, where $\mathcal{L}$ is a CSL or satisfies $\vee\{L: L\in \mathcal{J}(\mathcal{L})\}=X$ or $\wedge\{L_-: L\in \mathcal{J}(\mathcal{L})\}=(0)$, and prove that each generalized ($m, n, l$)-Jordan centralizer of these algebras is a centralizer when $m+l\geq1$ and $n+l\geq1$.

Higher rank semigraph algebras are introduced by mixing concepts of ultragraph algebras and higher rank graph algebras. This yields a kind of higher rank generalisation of ultragraph algebras. We prove Cuntz--Krieger uniqueness theorems for cancelling semigraph algebras and aperiodic saturated semigraph algebras.

We consider the perturbation and expression for the generalized inverse and Moore--Penrose inverse of closed linear operator under a weaker perturbation condition. As a application, we also investigate the perturbation for the Moore--Penrose inverse of closed $EP$ operator. Some new and interesting perturbation results and examples are obtained in this paper.

The object of this paper is to investigate some inclusion relationships and a number of other useful properties among certain subclasses of analytic and $p$-valent functions, which are defined here by certain integral operator.

In this paper, we prove that if a finite disjoint union of translates $\bigcup_{k=1}^n\{f_k(x-\gamma)\}_{\gamma\in\Gamma_k}$ in $L^p(\mathbb{R}^d)$ $(p \in (1, \infty))$ is a $p'$-Bessel sequence for some $p' \in (1, \infty)$, then the disjoint union $\Gamma=\bigcup_{k=1}^n \Gamma_k$ has finite upper Beurling density, and that if $\bigcup_{k=1}^n\{f_k(x-\gamma)\}_{\gamma\in\Gamma_k}$ is a $(C_q)$-system with $1/p+1/q=1$, then $\Gamma$ has infinite upper Beurling density. Thus, no finite disjoint union of translates in $L^p(\mathbb{R}^d)$ can form a $p'$-Bessel $(C_q)$-system for any $p'\in (1,\infty)$. Furthermore, by using techniques from the geometry of Banach spaces, we obtain that, for $p \in (1, \le2)$, no finite disjoint union of translates in $L^p(\mathbb{R}^d)$ can form an unconditional basis.

We investigate the traceability of positive integral operators on $L^2(X,\mu)$ when $X$ is a Hausdorff locally compact second countable space and $\mu$ is a non-degenerate, $\sigma$-finite and locally finite Borel measure. This setting includes other cases proved in the literature, for instance the one in which $X$ is a compact metric space and $\mu$ is a special finite measure. The results apply to spheres, tori and other relevant subsets of the usual space $\mathbb{R}^m$.

In the present work we deal with the approximation properties of certain linear polynomial operators in rearrangement invariant quasi Banach function spaces. We obtain some Jackson type direct theorem and sharp converse theorem of trigonometric approximation with respect to fractional positive order moduli of smoothness in these spaces.

In the survey paper, along one of several main lines of bounding the ratio of two gamma functions, the authors retrospect and analyse Wendel's double inequality, Kazarinoff's refinement of Wallis' formula, Watson's monotonicity, Gautschi's double inequality, Kershaw's first double inequality, and the (logarithmically) complete monotonicity results of functions involving ratios of two gamma or $q$-gamma functions obtained by Bustoz, Ismail, Lorch, Muldoon, and other mathematicians.

We obtain some Hermite--Hadamard type inequalities for convex functions in a global non-positive curvature space.

Let $({\mathcal X}, d, \mu)$ be a metric measure space and satisfy the so-called upper doubling condition and the geometrically doubling condition. In this paper, the authors establish an interpolation result that a sublinear operator which is bounded from the Hardy space $H^1(\mu)$ to $L^{1,\,\infty}(\mu)$ and from $L^\infty(\mu)$ to the BMO-type space RBMO($\mu$) is also bounded on $L^p(\mu)$ for all $p\in(1,\,\infty)$. This extension is not completely straightforward and improves the existing result

In this paper, we define the classes $F(p,q,s)$ of quaternion-valued functions, then we characterize quaternion Bloch functions by quaternion $F(p,q,s)$ functions in the unit ball of $\mathbb{R}^3$. Further, some important basic properties of these functions are also considered.

As is well-known, each positive operator $T$ acting on a Hilbert space has a positive square root which is realized by means of functional calculus. However, it is not always true that an operator have a square root. In this paper, by means of Schauder basis theory we obtain that if a backward operator weighted shift $T$ with multiplicity $2$ is not strongly irreducible, then there exists a backward shift operator $B$ (maybe unbounded) such that $T=B^2$. Furthermore, the backward operator weighted shifts in the sense of Cowen-Douglas are also considered.

In the present paper, taking some advantages offered by the context of finite dimensional Hilbert spaces, we shall give a complete characterizations of certain distinguished classes of operators (self-adjoint, unitary reflection, normal) in terms of operator inequalities. These results extend previous characterizations obtained by the second author.

The paper gives a detailed survey of recent results on elliptic problems in Hilbert spaces of generalized smoothness. The latter are the isotropic Hörmander spaces $H^{s,\varphi}:=B_{2,\mu}$, with $\mu(\xi)=\langle\xi\rangle^{s}\varphi(\langle\xi\rangle)$ for $\xi\in\mathbb{R}^{n}$. They are parametrized by both the real number $s$ and the positive function $\varphi$ varying slowly at $+\infty$ in the Karamata sense. These spaces form the refined Sobolev scale, which is much finer than the Sobolev scale $\{H^{s}\}\equiv\{H^{s,1}\}$ and is closed with respect to the interpolation with a function parameter. The Fredholm property of elliptic operators and elliptic boundary-value problems is preserved for this new scale. Theorems of various type about a solvability of elliptic problems are given. A~local refined smoothness is investigated for solutions to elliptic equations. New sufficient conditions for the solutions to have continuous derivatives are found. Some applications to the spectral theory of elliptic operators are given.