In this article, we concentrate on the Berezin transform of the absolute value of a bounded linear operator $T$ defined on the Bergman space $L_a^2\left(\mathbb{D}\right)$ of the open unit disk. We establish some sufficient conditions on $T$ which guarantee that the Berezin transform of $\left|T\right|$ majorizes the Berezin transform of $\left|T^{\ast }\right|$. We have shown that $T$ is self-adjoint and $T^2=T^3$ if and only if there exists a normal idempotent operator $S$ on $L_a^2\left(\mathbb{D}\right)$ such that $\rho \left(T\right)=\rho \left(\right|S{|}^2)=\rho (\left|S^{\ast }{|}^2\right)$, where $\rho \left(T\right)$ is the Berezin transform of $T$. We also establish that if $T$ is compact and $\left|T^n\right|=|T{|}^n$ for some $n\in \mathbb{N}$, $n\ne 1$, then $\rho \left(\right|T^n\left|\right)=\rho \left(\right|T{|}^n)$ for all $n\in \mathbb{N}$. Further, if $T=U\left|T\right|$ is the polar decomposition of $T$, then we present necessary and sufficient conditions on $T$ such that $|T{|}^{1/2}$ intertwines with $U$ and a contraction $X$ belonging to $\mathcal{L}\left(L_a^2\right(\mathbb{D}\left)\right)$.

We study bilinear operators acting on a product of Hilbert spaces of integrable functions—zero-valued for couples of functions whose convolution equals zero—that we call convolution-continuous bilinear maps. We prove a factorization theorem for them, showing that they factor through ${\ell }^1$. We also present some applications for the case when the range space has some relevant properties, such as the Orlicz or Schur properties. We prove that ${\ell }^1$ is the only Banach space for which there is a norming bilinear map which equals zero exactly in those couples of functions whose convolution is zero. We also show some examples and applications to generalized convolutions.

We obtain estimates for the essential norm of the composition operators acting on the general spaces $H_{\omega ,p}$ of Hardy spaces. Our characterization is given in terms of generalized Nevanlinna counting functions.

In this article, we prove that if $K$ is a nonempty weakly compact convex set in a Banach space such that $K$ has the hereditary fixed-point property (FPP) and $\mathfrak{F}$ is a commuting family of isometry mappings on $K$, then there exists a point in $C\left(K\right)$ which is fixed by every member in $\mathfrak{F}$ whenever $C\left(K\right)$ is a compact set. Also, we give an example to show that $C\left(K\right)$, the Chebyshev center of $K$, need not be invariant under isometry maps. This example answers the question as to whether the Chebyshev center is invariant under isometry maps. Furthermore, we give a simple example to illustrate that Lim’s center, as introduced by Lim, is different from the Chebyshev center.

We study an interplay between operator algebras and the geometry of rational elliptic curves. Namely, let ${\mathcal{O}}_B$ be the Cuntz–Krieger algebra given by a square matrix $B=(b-1,1,b-2,1)$, where $b$ is an integer greater than or equal to $2$. We prove that there exists a dense, self-adjoint subalgebra of ${\mathcal{O}}_B$ which is isomorphic (modulo an ideal) to a twisted homogeneous coordinate ring of the rational elliptic curve $\mathcal{E}\left(\mathbb{Q}\right)=\left\{\right(x,y,z)\in {\mathbb{P}}^2(\mathbb{C})\mid y^{2}z=x(x-z\left)\right(x-\frac{b-2}{b+2}z\left)\right\}$.

In this short note, we further Ng’s work by extending Bekka amenability and weak Bekka amenability to general locally compact quantum groups, and we generalize some of Ng’s results to the general case. In particular, we show that a locally compact quantum group $\mathbb{G}$ is coamenable if and only if the contra-corepresentation of its fundamental multiplicative unitary $W_{\mathbb{G}}$ is Bekka-amenable, and that $\mathbb{G}$ is amenable if and only if its dual quantum group’s fundamental multiplicative unitary $W_{\hat{\mathbb{G}}}$ is weakly Bekka-amenable.

In this article, we introduce a new class of noncyclic mappings called generalized pointwise noncyclic contractions, and we prove a best proximity pair theorem for this class of noncyclic mappings in the setting of strictly convex Banach spaces. Our conclusions generalize a result due to Kirk and Royalty. We also study convergence of iterates of noncyclic contraction mappings in uniformly convex Banach spaces.

In this article we investigate the numerical ranges of several classes of operators. It is shown that, if we let $T$ be a hyponormal operator and let $\epsilon >0$, then there exists a compact operator $K$ with norm less than $\epsilon$ such that $T+K$ is hyponormal and has a closed numerical range. Moreover we prove that the statement of the above type holds for other operator classes, including weighted shifts, normaloid operators, triangular operators, and block-diagonal operators.

Let $\mathcal{A}$ and $\mathcal{B}$ be commutative Banach algebras. Then a multiplicative linear functional $\theta$ on $\mathcal{B}$ induces a multiplication on the Cartesian product space $\mathcal{A}\times \mathcal{B}$ given by $(a,b)(c,d)=(ac+\theta (d)a+\theta (b)c,bd)$ for all $(a,b),(c,d)\in \mathcal{A}\times \mathcal{B}$. We show that this Lau product is stable with respect to the spectral properties like the unique uniform norm property, the spectral extension property, the multiplicative Hahn–Banach property, and the unique semisimple norm property under certain conditions on $\theta$.

We construct a topology on the standard Hilbert module $l^2\left(\mathcal{A}\right)$ over a unital $W^{\ast }$-algebra $\mathcal{A}$ such that any “compact” operator (i.e., any operator in the norm closure of the linear span of the operators of the form $x\mapsto z\langle y,x\rangle$) maps bounded sets into totally bounded sets.

It is known that if $M$ is a finite-dimensional Banach space, or a strictly convex space, or the space ${\ell }_1$, then every nonexpansive bijection $F:B_M\to B_M$ of its unit ball $B_M$ is an isometry. We extend these results to nonexpansive bijections $F:B_E\to B_M$ between unit balls of two different Banach spaces. Namely, if $E$ is an arbitrary Banach space and $M$ is finite-dimensional or strictly convex, or the space ${\ell }_1$, then every nonexpansive bijection $F:B_E\to B_M$ is an isometry.

By the use of methods of real analysis and weight functions, we study the equivalent properties of a Hilbert-type integral inequality with the nonhomogeneous kernel. The constant factor related to the Hurwitz zeta function is proved to be the best possible. As a corollary, a few equivalent conditions of a Hilbert-type integral inequality with the homogeneous kernel are deduced. We also consider their operator expressions.