We first study the Volterra operator $V$ acting on spaces of Dirichlet series. We prove that $V$ is bounded on the Hardy space $h^p_0$ for any $0< p \leq \infty$, and is compact on $h^p_0$ for $1< p \leq \infty$. Furthermore, we show that $V$ is cyclic but not supercyclic on $h^p_0$ for any $0 < p < \infty$. Corresponding results are also given for $V$ acting on Bergman spaces $h^p_{w,0}$. We then study the Volterra type integration operators $T_g$. We prove that if $T_g$ is bounded on the Hardy space $h^p$, then it is bounded on the Bergman space $h^p_w$.

In the setting of Fock–Sobolev spaces of positive orders over the complex plane, Choe and Yang showed that if the one of the symbols of two commuting Toeplitz operators with bounded symbols is non-trivially radial, then the other must also be radial. In this paper, we extend this result to the Fock–Sobolev space of negative order using the Fock-type space with a confluent hypergeometric function.

In this paper, we study the traces and the extensions for weighted Sobolev spaces on upper half spaces when the weights reach to the borderline cases. We first give a full characterization of the existence of trace spaces for these weighted Sobolev spaces, and then study the trace parts and the extension parts between the weighted Sobolev spaces and a new kind of Besov-type spaces (on hyperplanes) which are defined by using integral averages over selected layers of dyadic cubes.

We present new bounds for the Berezin number inequalities which improve on the existing bounds. We also obtain bounds for the Berezin norm of operators as well as the sum of two operators.

In this paper, we mainly study the hyponormality of dual Toeplitz operators on the orthogonal complement of the harmonic Bergman space. First we show that the dual Toeplitz operator with the bounded symbol is hyponormal if and only if it is normal. Then we obtain a necessary and sufficient condition for the dual Toeplitz operator $S_\varphi$ with the symbol $\varphi(z)=az^{n_1}\overline{z}^{m_1}+bz^{n_2}\overline{z}^{m_2}$ ($n_1,n_2,m_1,m_2\in \mathbb N\ \text{and}\ a,b\in \mathbb C)$ to be hyponormal. Finally, we show that the rank of the commutator of two dual Toeplitz operators must be an even number if the commutator has a finite rank.

We show that the following properties of the C^*-algebras in a class P are inherited by simple unital C^*-algebras in the class of asymptotically tracially in P: (1) n-comparison, (2) α-comparison (1 ≤ α < ∞).

We prove that a surjective map (on the positive cones of unital C^*-algebras) preserves the minimum spectrum values of harmonic means if and only if it has a Jordan *-isomorphism extension to the whole algebra. We represent weighted geometric mean preserving bijective maps on the positive cones of prime C^*-algebras in terms of Jordan *-isomorphisms of the algebras. We also characterize order isomorphisms and orthoisomorphisms of the projection lattice of the von Neumann algebra of all bounded linear operators on a Hilbert space, answering an open question arisen by Dye. Finally, we give a description for Fuglede-Kadison determinant preserving maps on the positive cone of a finite von Neumann algebra and improve Gaal and Nayak's work on this topic.

In this paper, we consider an abstract third-order differential equation and deduce some results on the maximal regularity of its strict solution. We assume that the inhomogeneity appearing in the right-hand term of this equation belongs to some anistropic Hölder spaces. We illustrate our results by a BVP involving a 3D Laplacian posed in a cusp domain of $\mathbb{R}^4$.

For bounded linear operators $A,B,C$ and $D$ on a Banach space $X$, we show that if $B A C=B D B$ and $C D B=C A C$ then $I-AC$ is generalized Drazin--Riesz invertible if and only if $I-BD$ is generalized Drazin--Riesz invertible, which gives a positive answer to Question 4.9 in Yan, Zeng and Zhu [Complex Anal. Oper. Theory 14, Paper No. 12 (2020)]. In particular, we show that Jacobson's lemma holds for generalized Drazin--Riesz inverses.

In this paper, we study some properties of weighted composition operators on a class of weighted Bergman spaces \begin{document}$ A_\varphi^p $\end{document} with 0 < p ≤ ∞ and \begin{document}$ \varphi \in \mathcal{W}_0 $\end{document}. Also, we completely characterize the q-Carleson measure for \begin{document}$ A_\varphi^p $\end{document} in terms of the averaging function and the generalized Berezin transform with 0 < q < ∞. As applications, the boundedness and compactness of weighted composition operators acting from one Bergman space \begin{document}$ A_\varphi^p $\end{document} to another \begin{document}$ A_\varphi^q $\end{document} are equivalently described and the Schatten class property of the weighted composition operator acting on \begin{document}$ A_\varphi^2 $\end{document} are given. Our main results are expressed in terms of certain Berezin type integral transforms.

Quantum uncertainty relations are mathematical inequalities that describe the lower bound of products of standard deviations of observables (i.e., bounded or unbounded self-adjoint operators). By revealing a connection between standard deviations of quantum observables and numerical radius of operators, we establish a universal uncertainty relation for $k$ observables, of which the formulation depends on the even or odd quality of $k$. This universal uncertainty relation is tight at least for the cases $k=2$ and $k=3$. For two observables, the uncertainty relation is a simpler reformulation of Schrödinger's uncertainty principle, which is also tighter than Heisenberg's and Robertson's uncertainty relations.

The main purpose of this paper is to define prime and introduce non-commutative arithmetics based on Thompson's group F. Defining primes in a non-abelian monoid is highly non-trivial, which relies on a concept called "castling". Three types of castlings are essential to grasp the arithmetics. The divisor function τ on Thompson's monoid S satisfies τ(uv) ≤ τ(u)τ(v) for any u, v ∈ S. Then the limit τ₀(u)=lim_n→∞(τ(u_n))^1/n exists. The quantity Ç(S)=sup_1≠u∈S τ₀(u)/τ(u) describes the complexity for castlings in S. We show that Ç(S)=1. Moreover, the Möbius function on S is calculated. And the Liouville function Ω on S is studied.

In this paper, we show that every super weakly compact convex subset of a Banach space is an absolute uniform retract, and that it also admits the uniform compact approximation property. These can be regarded as extensions of Lindenstrauss and Kalton’s corresponding results.

Recently, Gehér and Šemrl have characterized the general form of surjective isometries of the set of all projections on an infinite-dimensional separable Hilbert space using unitaries and antiunitaries. In this paper, we study the surjective L²-isometries of the projection lattice of an infinite dimensional Hilbert space and show that every such isometry can also be described by unitaries and antiunitaries.