In this article, we first give an essential characterization of Toeplitz operators with quasihomogeneous symbols on the weighted pluriharmonic Bergman space of the unit polydisk. Then we completely characterize when the product of two Toeplitz operators with monomial-type symbols is a Toeplitz operator. As a result, some interesting higher-dimensional phenomena appear on the unit polydisk.

Let $K\mathcal{(H)}$ and $\mathcal{B(H)}$ be the sets of all compact operators and all bounded linear operators, respectively, on the Hilbert space $\mathcal{H}$. In this article, we mainly show that if $\Phi \in CB(K{\mathcal{(H)}}^{\ast },\mathcal{B}\mathcal{(K)})$, then there exist ${\Phi }_i\in CP(K{\mathcal{(H)}}^{\ast },\mathcal{B(K)})$, for $i=1,2,3,4$, such that $\Phi =({\Phi }_1-{\Phi }_2)+\sqrt{-1}({\Phi }_3-{\Phi }_4)$. However, $CP(K{\mathcal{(H)}}^{\ast },\mathcal{B(K)})⊈CB(K{\mathcal{(H)}}^{\ast },\mathcal{B}\mathcal{(K)})$, where $CB(V,W)$ and $CP(V,W)$ are the sets of all completely bounded maps and all completely positive maps from $V$ into $W$, respectively.

We investigate the convergence rate of the generalized Bochner–Riesz means $S_R^{\delta ,\gamma }$ on $L^p$-Sobolev spaces in the sharp range of $\delta$ and $p$ ($p\ge 2$). We give the relation between the smoothness imposed on functions and the rate of almost-everywhere convergence of $S_R^{\delta ,\gamma }$. As an application, the corresponding results can be extended to the $n$-torus ${\mathbb{T}}^n$ by using some transference theorems. Also, we consider the following two generalized Bochner–Riesz multipliers, $(1-|\xi {|}^{{\gamma }_1}{)}_{+}^{\delta }$ and $(1-|\xi {|}^{{\gamma }_2}{)}_{+}^{\delta }$, where ${\gamma }_1$, ${\gamma }_2$, $\delta$ are positive real numbers. We prove that, as the maximal operators of the multiplier operators with respect to the two functions, their $L^2\left(\right|x{|}^{-\beta })$-boundedness is equivalent for any ${\gamma }_1$, ${\gamma }_2$ and fixed $\delta$.

It is known that if a Banach space $Y$ is a $u$-ideal in its bidual $Y^{**}$ with respect to the canonical projection on the third dual $Y^{***}$ , then $Y^{*}$ contains “many” functionals admitting a unique norm-preserving extension to $Y^{**}$—the dual unit ball $B_{Y^{*}}$ is the norm-closed convex hull of its weak${}^{*}$ strongly exposed points by a result of Å. Lima from 1995. We show that if $Y$ is a strict $u$-ideal in a Banach space $X$ with respect to an ideal projection $P$ on $X^{*}$ , and $X/Y$ is separable, then $B_{Y^{*}}$ is the ${\tau }_P$-closed convex hull of functionals admitting a unique norm-preserving extension to $X$, where ${\tau }_P$ is a certain weak topology on $Y^{*}$ defined by the ideal projection $P$.

Let $G$ be a discrete group acting on a unital $C^{\ast }$-algebra $\mathcal{A}$ by $\ast$-automorphisms. We characterize (in terms of the dynamics) when the inclusion $\mathcal{A}\subseteq \mathcal{A}{⋊}_{r}G$ has a unique conditional expectation, and when it has a unique pseudoexpectation in the sense of Pitts; we do likewise for the inclusion $\mathcal{A}\subseteq \mathcal{A}⋊G$. As an application, we re-prove (and potentially extend) some known $C^{\ast }$-simplicity results for $\mathcal{A}{⋊}_{r}G$.

We characterize a class of matrices that is unitarily similar to a complex symmetric matrix via the discrete Fourier transform.

We study I-convexity and Q-convexity, two geometric properties introduced by Amir and Franchetti. We point out that a Banach space $X$ has the weak fixed-point property when $X$ is I-convex (or Q-convex) with a strongly bimonotone basis. By means of some characterizations of I-convexity and Q-convexity in Banach spaces, we obtain criteria for these two convexities in the Orlicz–Bochner function space $L_{(M)}(\mu ,X)$: that $L_{(M)}(\mu ,X)$ is I-convex (or Q-convex) if and only if $L_{(M)}(\mu )$ is reflexive and $X$ is I-convex (or Q-convex).

Using $C^{\ast }$-algebraic techniques and especially AF-algebras, we present a complete classification of the continuous unitary representations for a class of infinite wreath product groups. These nonlocally compact groups are realized by a topological completion of the semidirect product of the countably infinite symmetric group acting on the countable direct product of a finite Abelian group.

We propose a definition of frames in Krein spaces which generalizes the concept of $J$-frames defined relatively recently by Giribet, Maestripieri, Martínez-Pería, and Massey. The difference consists in the fact that a $J$-frame is related to maximal definite subspaces ${\mathcal{M}}_{\pm }$ which are not assumed to be uniformly definite. The latter allows us to extend the set of $J$-frames. In particular, some $J$-orthogonal Schauder bases can be interpreted as $J$-frames.

In this article, given some positive Borel measure $\mu$, we define two integration operators to be

$$I_{\mu }(f)(z)={\int }_{\mathbf{D}}f(w)K(z,w)e^{-2\phi (w)}d\mu (w)$$ and

$$J_{\mu }(f)(z)={\int }_{\mathbf{D}}|f(w)K(z,w)|e^{-2\phi (w)}d\mu (w).$$ We characterize the boundedness and compactness of these operators from the Bergman space $A_{\phi }^p$ to $L_{\phi }^q$ for $1<p,q<\infty$, where $\phi$ belongs to a large class ${\mathcal{W}}_0$, which covers those defined by Borichev, Dhuez, and Kellay in 2007. We also completely describe those $\mu$’s such that the embedding operator is bounded or compact from $A_{\phi }^p$ to $L_{\phi }^q(d\mu )$, $0<p,q<\infty$.

In this paper we completely characterize the norm attainment set of a bounded linear operator between Hilbert spaces. In fact, we obtain two different characterizations of the norm attainment set of a bounded linear operator between Hilbert spaces. We further study the extreme contractions on various types of finite-dimensional Banach spaces, namely Euclidean spaces, and strictly convex spaces. In particular, we give an elementary alternative proof of the well-known characterization of extreme contractions on a Euclidean space, which works equally well for both the real and the complex case. As an application of our exploration, we prove that it is possible to characterize real Hilbert spaces among real Banach spaces, in terms of extreme contractions on their $2$-dimensional subspaces.

We study doubly stochastic operators with zero entropy. We generalize three famous theorems: Rokhlin’s theorem on genericity of zero entropy, Kushnirenko’s theorem on equivalence of discrete spectrum and nullity, and Halmos–von Neumann’s theorem on representation of maps with discrete spectrum as group rotations.