We are concerned with fractional Sobolev-type integro-differential equations in Banach spaces with operator pairs and impulsive conditions, where the operator pairs generate propagation families. With the help of the theory of propagation family and Laplace transforms, along with an estimate for a special sequence improved in this article, we introduce a definition of mild solutions to the impulsive problem for these abstract fractional Sobolev-type integro-differential equations and we establish general existence theorems and a continuous dependence theorem, which essentially extend some previous conclusions. In our results, the operator $B$ could be unbounded, and the existence of an operator $B^{-1}$ is not necessarily needed. Moreover, we give some examples to illustrate our main results.

Let $L$ be a one-to-one operator of type $w$ with $w\in [0,\pi /2)$, which satisfies the Davies–Gaffney estimates and has a bounded holomorphic calculus, and let $p(\cdot )$ be a measurable function on ${\mathbb{R}}^n$ with $0<p_{-}:=ess{inf\hspace{0.17em}}_{x\in {\mathbb{R}}^n}p(x)\le ess{sup\hspace{0.17em}}_{x\in {\mathbb{R}}^n}p(x)=:p_{+}<\infty$. Under the assumption that $p(\cdot )$ satisfies the global log-Hölder condition, we introduce the variable Hardy–Lorentz space $H_L^{p(\cdot ),q}({\mathbb{R}}^n)$ for $0<q<\infty$ and construct its molecular decomposition. Furthermore, we investigate the dual spaces of the variable Hardy–Lorentz space $H_L^{p(\cdot ),q}({\mathbb{R}}^n)$ with $0<p_{-}\le p_{+}\le 1$ and $0<q<\infty$. These results are new even when $p(\cdot )$ is a constant.

Let $\mathcal{M}$ be a $\sigma$-finite von Neumann algebra equipped with a normal faithful state $\phi$, and let $\mathcal{A}$ be a maximal subdiagonal algebra of $\mathcal{M}$. We prove a Stein–Weiss-type interpolation theorem of Haagerup noncommutative $H^p$-spaces associated with $\mathcal{A}$.

Let $I$ be a countably infinite index set, and let $X$ be a Banach sequence space over $I$. In this article, we characterize the disjoint hypercyclic and supercyclic weighted pseudoshift operators on $X$ in terms of the weights, the OP-basis, and the shift mappings on $I$. Also, the shifts on weighted $L^p$ spaces of a directed tree and the operator weighted shifts on ${\ell }^2(\mathbb{Z},\mathcal{K})$ are investigated as special cases.

The theory of algebraic extensions of Banach algebras is well established, and there are many constructions which yield interesting extensions. In particular, Cole’s method for extending uniform algebras by adding square roots of functions to a given uniform algebra has been used to solve many problems within uniform algebra theory. However, there are numerous other examples in the theory of uniform algebras that can be realized as extensions of a uniform algebra, and these more general extensions have received little attention in the literature. In this paper, we investigate more general classes of uniform algebra extensions. We introduce a new class of extensions of uniform algebras, and show that several important properties of the original uniform algebra are preserved in these extensions. We also show that several well-known examples from the theory of uniform algebras belong to these more general classes of uniform algebra extensions.

We study general variants of spaces of holomorphic functions on circular domains on the complex plane. We define Hardy-type spaces generated by Banach function lattices, for which we prove the Carleson theorem. We also analyze canonical embeddings of such spaces into appropriate function lattices. Finally, we study composition operators on Hardy-type spaces on circular domains and characterize order-boundedness of such maps.

Let $(\Omega ,\mathcal{F},\mathbb{P})$ be a probability space, and let $\phi :\Omega \times [0,\infty )\to [0,\infty )$ be a Musielak–Orlicz function. In this article, we establish the atomic characterizations of weak martingale Musielak–Orlicz Hardy spaces ${\mathit{WH}}_{\phi }^s(\Omega )$, ${\mathit{WH}}_{\phi }^M(\Omega )$, ${\mathit{WH}}_{\phi }^S(\Omega )$, $W{P}_{\phi }(\Omega )$, and $W{Q}_{\phi }(\Omega )$. We then use these atomic characterizations to obtain the boundedness of $\sigma$-sublinear operators from weak martingale Musielak–Orlicz Hardy spaces to weak Musielak–Orlicz spaces, as well as some martingale inequalities which further clarify the relationships among these weak martingale Musielak–Orlicz Hardy spaces. All these results improve and generalize the corresponding results on weak martingale Orlicz–Hardy spaces. Moreover, we improve all the known results on weak martingale Musielak–Orlicz Hardy spaces. In particular, both the boundedness of $\sigma$-sublinear operators and the martingale inequalities, for weak weighted martingale Hardy spaces as well as for weak weighted martingale Orlicz–Hardy spaces, are new.

We study the uniform path connectivity of sets of matrix tuples that satisfy some additional constraints, and more specifically, given $\epsilon >0$, a fixed metric $ð$ in $M_n(\mathbb{C}{)}^m$ induced by the operator norm $\Vert \cdot \Vert$, any collection of $r$ nonconstant polynomials $p_1(x_1,\dots ,x_m),\dots ,p_r(x_1,\dots ,x_m)$ over $\mathbb{C}$ with finite zero set $\mathbf{Z}(p_1,\dots ,p_r)\subset {\mathbb{C}}^m$ and any $m$-tuple $\mathbf{X}=(X_1,\dots ,X_m)$ in the set ${\mathbb{ZD}}_n^m(p_1,\dots ,p_r)\subseteq M_n^m\left(\mathbb{C}\right)$ of commuting normal matrix contractions such that $\Vert p_j(Y_1,\dots ,Y_m)\Vert =0$ for each $(Y_1,\dots ,Y_m)\in {\mathbb{ZD}}_n^m(p_1,\dots ,p_r)$ and each $1\le j\le r$. The author proves the existence of paths between arbitrary $m$-tuples that belong to the intersection of ${\mathbb{ZD}}_n^m(p_1,\dots ,p_r)$ and the open $\delta$-ball $B_{ð}(\mathbf{X},\delta )$ centered at $\mathbf{X}$ for some $\delta >0$ that can be chosen independently of $n$. In addition, the author proves that the aforementioned paths are contained in the intersection of $B_{ð}(\mathbf{X},\epsilon )$ and ${\mathbb{ZD}}_n^m(p_1,\dots ,p_r)$. Some connections of the main results with structure-preserving perturbation theory and preconditioning techniques are outlined.

We investigate the existence of solutions of an infinite system of integral equations of Volterra–Hammerstein type on the real half-axis. The method applied in our study is connected with the construction of a suitable measure of noncompactness in the space of continuous and bounded functions defined on the real half-axis with values in the space $c_0$ consisting of real sequences converging to zero and equipped with the classical supremum norm. We use the mentioned measure of noncompactness together with a fixed point theorem of Darbo type. Our investigations are illustrated by an example.

Let $\phi :{\mathbb{R}}^n\times [0,\infty )\to [0,\infty )$ satisfy that, for any given $x\in {\mathbb{R}}^n$, $\phi (x,\cdot )$ is an Orlicz function and $\phi (\cdot ,t)$ is a Muckenhoupt $A_{\infty }({\mathbb{R}}^n)$ weight uniformly in $t\in (0,\infty )$. In this article, via using the atomic and Littlewood–Paley function characterizations of the weak Musielak–Orlicz Hardy space ${\mathit{WH}}^{\phi }({\mathbb{R}}^n)$, for any $\alpha \in (0,1]$ and $s\in {\mathbb{Z}}_{+}$, we first establish its $s$-order intrinsic square function characterizations in terms of the intrinsic Lusin area function $S_{\alpha ,s}$, the intrinsic $g$-function $g_{\alpha ,s}$ and the intrinsic $g_{\lambda }^{\ast }$-function $g_{\lambda ,\alpha ,s}^{\ast }$ with the best known range $\lambda \in (2+2(\alpha +s)/n,\infty )$.

We establish the semi-Fredholm theory on Hilbert $C^{\ast }$-modules as a continuation of Fredholm theory on Hilbert $C^{\ast }$-modules established by Mishchenko and Fomenko. We give a definition of a semi-Fredholm operator on Hilbert $C^{\ast }$-module, and we prove that these semi-Fredholm operators are those that are one-sided invertible modulo compact operators, that the set of proper semi-Fredholm operators is open, and many other results that generalize their classical counterparts.

In the first part, we describe the dual ${\ell }^2(\mathsf{A}{)}^{\prime }$ of the standard Hilbert $C^{\ast }$-module ${\ell }^2(\mathsf{A})$ over an arbitrary (not necessarily unital) $C^{\ast }$-algebra $\mathsf{A}$. When $\mathsf{A}$ is a von Neumann algebra, this enables us to construct explicitly a self-dual Hilbert $\mathsf{A}$-module ${\ell }_{\mathrm{strong}}^2(\mathsf{A})$ that is isometrically isomorphic to ${\ell }^2(\mathsf{A}{)}^{\prime }$, which contains ${\ell }^2(\mathsf{A})$, and whose $\mathsf{A}$-valued inner product extends the original inner product on ${\ell }^2(\mathsf{A})$. This serves as a concrete realization of a general construction for Hilbert $C^{\ast }$-modules over von Neumann algebras introduced by Paschke. Then we introduce a concept of a weak Bessel sequence and a weak frame in Hilbert $C^{\ast }$-modules over von Neumann algebras. The dual ${\ell }^2(\mathsf{A}{)}^{\prime }$ is recognized as a suitable target space for the analysis operator. We describe fundamental properties of weak frames such as the correspondence with surjective adjointable operators, the canonical dual, the reconstruction formula, and so on, first for self-dual modules and then, working in the dual, for general modules. The last part describes a class of Hilbert $C^{\ast }$-modules over $L^{\infty }(I)$, where $I$ is a bounded interval on the real line, that appears naturally in connection with Gabor (i.e., Weyl–Heisenberg) systems. We then demonstrate that Gabor Bessel systems and Gabor frames in $L^2(\mathbb{R})$ are in a bijective correspondence with weak Bessel systems and weak frames of translates by $a$ in these modules over $L^{\infty }[0,\frac{1}{b}]$, where $a,b>0$ are the lattice parameters. In this setting new proofs of several classical results on Gabor frames are demonstrated and some new ones are obtained.