In this paper, we show that, for the rational inner function $\theta (z,w)=\frac{zw+\overline{b}w+\overline{c}z+\overline{d}}{1+bz+cw+dzw}$, $S_z$ is reducible on the quotient module ${\mathcal{K}}_{\theta }=H^2\ominus \theta H^2$ over the bidisk if and only if $\theta$ is the product of two one-variable inner functions.

We study Banach function spaces (BFS) on locally compact topological groups. We focus on totally bounded sets in BFS. We obtain the Riesz–Kolmogorov compactness theorem as well as the Sudakov theorem.

In this paper we introduce a subclass of Cowen–Douglas operators of high-rank case denoted by ${\mathcal{FB}}_{n,1}(\Omega )$. Each operator $T\in {\mathcal{FB}}_{n,1}(\Omega )$ is induced by one Cowen–Douglas operator with rank $n$, another Cowen–Douglas operator with rank $1$, and an intertwining operator between them. By using this definition, we can construct plenty of Cowen–Douglas operators with high rank. By discussing the curvature of line bundle and second fundamental form of some rank $2$ bundle and its subbundle, we give the unitary classification of operators in ${\mathcal{FB}}_{n,1}(\Omega )$ and we reduce the number of unitary invariants of this kind of operators from $(n+1{)}^2$ to two.

We introduce and study a generalized approximate orthogonality relation in real normed linear spaces, namely, approximate ${\rho }_{\lambda }^v$-orthogonality. We investigate the relation between this generalized approximate orthogonality and approximate Birkhoff–James orthogonality. In particular, we show that every approximately ${\rho }_{\lambda }^v$-orthogonality-preserving linear mapping is necessarily a scalar multiple of an almost isometry.

We give a natural Ky Fan minimax inequality version of set-valued maps, and we deal with a type of vector equilibrium problem for set-valued mappings on a special dense set not on the whole domain. We use these results as applications to study the solutions of a generalized set-valued vector variational inequality.

In this short article, we mainly give some examples to prove that the main results of Section 4 of the paper [W. S. Liao and J. Wu, Ann. Func. Anal. 6 (2015), no. 3, 191–202] are not true. Then we give the corrected matrix inequalities.

We say that a map $f:X\to Y$ between two real normed spaces is a phase-isometry if $\{\Vert f(x)+f(y)\Vert ,\Vert f(x)-f(y)\Vert \}=\{\Vert x+y\Vert ,\Vert x-y\Vert \}$ holds for all $x,y\in X$. Two maps $f,g:X\to Y$ are called phase-equivalent if there is a phase function $\epsilon :X\to \{-1,1\}$ such that $\epsilon f=g$. By studying the properties of surjective phase-isometries on the Tsirelson space $T$, we show that such maps are phase-equivalent to linear isometries. This gives a real version of Wigner’s theorem for the Tsirelson space.

In the following we show that, under some conditions, $\phi$-morphisms preserve duals and approximate duals of frames in Hilbert $C^{*}$-modules. Moreover, using $\phi$-morphisms and some concepts related to frame theory such as modular Riesz bases, canonical duals, tensor products, and Bessel multipliers, we construct new approximate duals, considering in particular the approximate duals constructed by compact operators and morphisms.

Controlled frames and g-frames were considered recently as generalizations of frames in Hilbert spaces. In this paper we generalize some of the known results in frame theory to controlled g-frames. We obtain some new properties of controlled g-frames and obtain new controlled g-frames by considering controlled g-frames for its components. And we also find some new resolutions of the identity. Furthermore, we study the stabilities of controlled g-frames under small perturbations.

Let $X(\mathbb{R})$ be a separable Banach function space such that the Hardy–Littlewood maximal operator $M$ is bounded on $X(\mathbb{R})$ and on its associate space $X\text{'}(\mathbb{R})$. Suppose that $a$ is a Fourier multiplier on the space $X(\mathbb{R})$. We show that the Fourier convolution operator $W^0(a)$ with symbol $a$ is compact on the space $X(\mathbb{R})$ if and only if $a=0$. This result implies that nontrivial Fourier convolution operators on Lebesgue spaces with Muckenhoupt weights are never compact.

Let $T_1$ and $T_2$ be two piecewise smooth circle homeomorphisms with countably many break points and identical irrational rotation number. We provide a sufficient condition for $C^1$-smoothness of the conjugation between $T_1$ and $T_2$.

In the present note, following a new approach recently described by Khosravian-Arab, Dehghan, and Eslahchi, we construct a new kind of $\alpha$-Bernstein operator and study a uniform convergence estimate for these operators. We also prove some direct results involving the asymptotic theorems. Finally, we illustrate the convergence of the operators to a certain function with the help of Maple software.