In this article, we study the perturbation analysis for the (skew) Hermitian matrix least squares problem (LSP). Suppose that $\mathcal{S}$ and $\hat{\mathcal{S}}$ are two sets of solutions to the (skew) Hermitian matrix least squares problem $AX{A}^H=B$ and the perturbed Hermitian matrix least squares problem $\hat{A}\hat{X}{\hat{A}}^H=\hat{B}$, respectively. For any given $X\in \mathcal{S}$, we derive general expressions of the least squares solutions $\hat{X}\in \hat{\mathcal{S}}$ that are closest to $X$, and we present the corresponding distances between them under appropriate norms. Perturbation bounds for the nearest least squares solutions are further derived.

In this note, we formulate recent stability results for Hardy inequalities in the language of Folland and Stein’s homogeneous groups. Consequently, we obtain remainder estimates for Rellich-type inequalities on homogeneous groups. Main differences from the Euclidean results are that the obtained stability estimates hold for any homogeneous quasinorm.

We prove that for any two elements $A$, $B$ in a factor $\mathcal{M}$, if $B$ commutes with all the unitary conjugates of $A$, then either $A$ or $B$ is in $\mathbb{C}I$. Then we obtain an equivalent condition for the situation that the $C$-numerical radius ${\omega }_C(\cdot )$ is a weakly unitarily invariant norm on finite factors, and we also prove some inequalities on the $C$-numerical radius on finite factors. As an application, we show that for an invertible operator $T$ in a finite factor $\mathcal{M}$, $f\left({△}_{\lambda }\right(T\left)\right)$ is in the weak operator closure of the set $\left\{{\sum }_{i=1}^n{z}_i{U}_{i}f\right(T)U_i^{\ast }\mid n\in \mathbb{N},(U_i{)}_{1\le i\le n}\in \mathcal{U}\left(\mathcal{M}\right),{\sum }_{i=1}^n\left|z_i\right|\le 1\}$, where $f$ is a polynomial, ${△}_{\lambda }\left(T\right)$ is the $\lambda$-Aluthge transform of $T$, and $0\le \lambda \le 1$.

For $\epsilon >0$ and a bounded linear operator $T$ acting on some Hilbert space, the $\epsilon$-pseudospectrum of $T$ is ${\sigma }_{\epsilon }\left(T\right)=\{z\in \mathbb{C}:\Vert (zI-T{)}^{-1}\Vert >1/\epsilon \}$ and the $\epsilon$-pseudospectral radius of $T$ is $r_{\epsilon }\left(T\right)=sup\hspace{0.17em}\left\{\right|z|:z\in {\sigma }_{\epsilon }(T\left)\right\}$. In this article, we provide a characterization of those operators $T$ satisfying $r_{\epsilon }\left(T\right)=r\left(T\right)+\epsilon$ for all $\epsilon >0$. Here $r\left(T\right)$ denotes the spectral radius of $T$.

Let $A$ be the AF algebra whose scaled ordered group $K_0\left(A\right)$ is $(G\oplus H,(G_{+}\setminus \left\{0\right\})\oplus H\cup \{(0,0)\},\tilde{g}\oplus 0)$, where $(G,G_{+},\tilde{g})$ is the scaled ordered group $K_0\left(B\right)$ of a unital simple AF algebra $B$, and $H$ is a countable torsion-free Abelian group. Let $\sigma$ be an order 2 scaled ordered automorphism of $K_0\left(A\right)$, defined by $\sigma (g,h)=(g,-h)$, where $(g,h)\in G\oplus H$. We show that there is an order $2$ automorphism $\alpha$ of $A$ such that ${\alpha }_{\ast }=\sigma$. This gives a partial answer to a lifting question posed by Blackadar. Incidentally, the lift $\alpha$ we construct has the tracial Rokhlin property. Consequently, the crossed product $C^{\ast }({\mathbb{Z}}_2,A,\alpha )$ is a unital simple AH algebra with no dimension growth.

We present some results on the nuclearity (or trace class) of integral operators acting on $L^2(X,\nu )$ under specific conditions. These results improve and adapt a number of methods found in references on this subject. Our discussions take place within the context of special subsets (and manifolds) of the Euclidean space (endowed with weighted Lebesgue measure), second-countable spaces, and Lusin and Souslin spaces (endowed with $\sigma$-finite Borel measure).

We initiate a study of operator approximate biprojectivity for quantum group algebra $L^1\left(\mathbb{G}\right)$, where $\mathbb{G}$ is a locally compact quantum group. We show that if $L^1\left(\mathbb{G}\right)$ is operator approximately biprojective, then $\mathbb{G}$ is compact. We prove that if $\mathbb{G}$ is a compact quantum group and $\mathbb{H}$ is a non-Kac-type compact quantum group such that both $L^1\left(\mathbb{G}\right)$ and $L^1\left(\mathbb{H}\right)$ are operator approximately biprojective, then $L^1\left(\mathbb{G}\right)\hat{\otimes }{L}^1\left(\mathbb{H}\right)$ is operator approximately biprojective, but not operator biprojective.

We present an alternative solution to nonsingular cubic moment problems, using techniques that are expected to be useful for higher-degree truncated moment problems. In particular, we apply the theory of recursively determinate moment matrices to deal with a case of rank-increasing moment matrix extensions.

Let $\mathcal{L}=-\Delta +\mu$ be the generalized Schrödinger operator on ${\mathbb{R}}^n$, $n\ge 3$, where $\Delta$ is the Laplacian and $\mu \not\equiv 0$ is a nonnegative Radon measure on ${\mathbb{R}}^n$. In this article, we give a characterization of ${\mathrm{BMO}}_{\mathcal{L}}$ in terms of Carleson measures, where ${\mathrm{BMO}}_{\mathcal{L}}$ is the $\mathrm{BMO}$-type space associated with the generalized Schrödinger operator.

Let $K$ be a hypergroup. The purpose of this article is to study the notions of amenability of the hypergroup algebras $L\left(K\right)$, $M\left(K\right)$, and $L(K{)}^{\ast \ast }$. Among other results, we obtain a characterization of approximate amenability of $L(K{)}^{\ast \ast }$. Moreover, we introduce the Banach space $L_{\infty }(K,L(K\left)\right)$ and prove that the dual of a Banach hypergroup algebra $L\left(K\right)$ can be identified with $L_{\infty }(K,L(K\left)\right)$. In particular, $L\left(K\right)$ is an $F$-algebra. By using this fact, we give necessary and sufficient conditions for $K$ to be left-amenable.

In this article, we generalize a well-known operator version of Jensen’s inequality to normal operators. The main techniques employed here are the spectral theory for bounded normal operators on a Hilbert space, and different Jensen-type inequalities. We emphasize the application of a vector version of Jensen’s inequality. By applying our results, some classical inequalities obtained for self-adjoint operators can also be extended.

This article has two clear motivations, one technical and one practical. The technical motivation unifies in a single formulation a huge family of inequalities that have been produced separately over the last ninety years in different contexts. But we do not just join inequalities; our method also creates a family of inequalities that were invisible by previous approaches. The practical motivation is to show that our new approach has the strength to attack various problems. We provide new applications of our family of inequalities, continuing recent work by Maia, Nogueira, and Pellegrino.