The numerical range of a matrix, according to Kippenhahn, is determined by a hyperbolic determinantal form of linear Hermitian matrices associated to the matrix. On the other hand, using Riemann theta functions, Helton and Vinnikov confirmed that a hyperbolic form always admits a determinantal representation of linear real symmetric matrices. The Riemann matrix of the hyperbolic curve plays the main role in the existence of real symmetric matrices. In this article, we implement computations of the Riemann matrix and the Abel–Jacobi variety of the hyperbolic curve associated to a determinantal polynomial of a matrix. Further, we prove that the lattice of the Abel–Jacobi variety is decomposed into the direct sum of two orthogonal lattices for some $4\times 4$ Toeplitz matrices.

We study the filtering of the perspective of a regular operator map of several variables through a completely positive linear map. By this method we are able to extend known operator inequalities of two variables to several variables, with applications in the theory of operator means of several variables. We also extend Lieb and Ruskai’s convexity theorem from two to $n+1$ operator variables for any natural number $n$.

Dual pairs of frames yield a procedure for obtaining perfect reconstruction of elements in the underlying Hilbert space in terms of superpositions of the frame elements. However, practical constraints often force us to apply sequences that do not exactly form dual frames. In this article, we consider the important case of generalized shift-invariant systems and provide various ways of estimating the deviation from perfect reconstruction that occur when the systems do not form dual frames. The deviation from being dual frames will be measured either in terms of a perturbation condition or in terms of the deviation from equality in the duality conditions.

Let $X$ be a Banach space, let $T\in L\left(X\right)$ be a bounded linear operator, and let $T_n$ be a restriction of $T$ on $R\left(T^n\right)$. This article should be viewed as a note on the research work of Carpintero et al. We give here several different proofs for completeness, and we show the relations of $T$ and $T_n$ to a much greater extent. Moreover, we give sufficient conditions for which Weyl-type theorems for $T$ are equivalent to Weyl-type theorems for $T_n$.

We obtain a sufficient condition to ensure that weighted backward shift operators on Köthe sequence spaces ${\lambda }_p\left(A\right)$ admit an invariant distributionally $\epsilon$-scrambled subset for any $0<\epsilon <diam{\lambda }_p\left(A\right)$. In particular, every Devaney chaotic weighted backward shift operator on ${\lambda }_p\left(A\right)$ supports such a subset.

Given $n\ge 2$, $z_{ij}\in \mathbb{T}$ such that $z_{ij}={\overline{z}}_{ji}$ for $1\le i,j\le n$ and $z_{ii}=1$ for $1\le i\le n$, and integers $p_1,\dots ,p_n\ge 1$, we show that the universal ${\mathrm{C}}^{*}$-algebra generated by unitaries $u_1,\dots ,u_n$ such that $u_i^{p_i}{u}_j^{p_j}=z_{ij}{u}_j^{p_j}{u}_i^{p_i}$ for $1\le i,j\le n$ is not simple if at least one exponent $p_i$ is at least two. We indicate how the method of proof by “working with various quotients” can be used to establish nonsimplicity of universal ${\mathrm{C}}^{*}$-algebras in other cases.

In this article, we study the (modified) von Neumann–Jordan constant and Zbăganu constant of $\pi /2$-rotation invariant norms on ${\mathbb{R}}^2$. Some estimations of these geometric constants are given. As an application, we construct various examples consisting of $\pi /2$-rotation invariant norms.

In this article we study new $L^p$-boundedness properties for the Mehler–Fock transform of general order on the spaces $L^p\left(\right(0,\infty ),e^{\alpha x}dx)$ and $L^p\left(\right(0,\infty ),(1+x{)}^{\gamma }dx)$, $1\le p\le \infty$, and $\alpha ,\gamma \in \mathbb{R}$. We also obtain Parseval-type relations over these spaces.

This note aims to present some operator inequalities for unitarily invariant norms. First, a Zhan-type inequality for unitarily invariant norms is given. Moreover, some operator inequalities for the Cauchy–Schwarz type are also established.

In this paper, we establish the existence of three possibly nontrivial solutions for a Dirichlet problem on the real line without assuming on the nonlinearity asymptotic conditions at infinity. As a particular case, when the nonlinearity is superlinear at zero and sublinear at infinity, the existence of two nontrivial solutions is obtained. This approach is based on variational methods and, more precisely, a critical points theorem, which assumes a more general condition than the classical Palais–Smale condition, is exploited.

We give a sufficient and necessary condition for an analytic function $f\left(z\right)$ on the unit ball $\mathbb{B}$ in ${\mathbb{C}}^n$ with Hadamard gaps, that is, for $f\left(z\right)={\sum }_{k=1}^{\infty }{P}_{n_k}\left(z\right)$ where $P_{n_k}\left(z\right)$ is a homogeneous polynomial of degree $n_k$ and $n_{k+1}/n_k\ge c>1$ for all $k\in \mathbb{N}$, to belong to the weighted-type space $H_{\mu }^{\infty }$ and the corresponding little weighted-type space $H_{\mu ,0}^{\infty }$ under some condition posed on the weighted funtion $\mu$. We also study the growth rate of those functions in $H_{\mu }^{\infty }$.

In this paper, we investigate local Lie derivations of a certain class of operator algebras and show that, under certain conditions, every local Lie derivation of such an algebra is a Lie derivation.