It is well known that every function in Hardy space can be factorized into an inner function and outer function. Since the factorization is unique, if we fix a function in Hardy space, inner and outer factors must be control by each other. In this note, we give an inner-outer factorization on $\mathcal{Q}_p$ spaces and some subspace of $\mathcal{Q}_p$ spaces, where $p \in (0,1)$.

This paper mainly divided into two parts. The first part gives same facts about topological properties of certain linear topological spaces, inclusion relations and matrix mappings. The second part establishes some identities or estimates for the matrix operator norms and the Hausdorff measures of noncompactness of certain matrix operators, characterize some classes of compact operators on these spaces.

Let $X$ be a locally compact space and let $(Y,d)$ be a nontrivial metric space such that $d$ has the Heine--Borel property. We prove Ascoli-type theorem for locally bounded quasicontinuous functions from $X$ to $Y$. Using the above result we also prove an Ascoli-type theorems for the spaces of minimal usco and minimal cusco maps from $X$ to $\Bbb R$.

The closed range and Fredholm properties of the upper-triangular operator matrix $M=(A, C; 0, B) \in\mathcal{B}({\mathcal H}_1\oplus {\mathcal H}_2)$ are studied, where ${\mathcal H}_1$ and ${\mathcal H}_2$ are Hilbert spaces. It is shown that the range $\mathcal{R}(M)$ of $M$ is closed if and only if the following statements hold: (i) $\mathcal{R}(P_{\mathcal{R} (A)^\perp}C|_{\mathcal{N}(B)})$ is closed, (ii) $\mathcal{R}(A)+\mathcal{R}(P_{\overline{\mathcal{R}(A)}} C|_{\mathcal{N}(P_{\mathcal{R}(A)^\perp}C|_{\mathcal{N}(B)})})={\overline{\mathcal{R}(A)}}$, (iii) $\mathcal{R}(B^{*})+\mathcal{R}(P_{\mathcal{N}(B)^\perp}C^* |_{\mathcal{R}(P_{\mathcal{R} (A)^\perp}C|_{\mathcal{N}(B)})^\perp})={\overline{\mathcal{R}(B^*)}}$,\\ where $P_{\mathcal G}$ denotes the orthogonal projection onto ${\mathcal G}$ along ${\mathcal G}^\perp$. Moreover, the analogues for the Fredholmness of $M$ are further presented.

We prove that any Banach algebra that is geometrically close to a Ball Algebra must automatically consist of analytic functions and must have a very similar algebraic structure to the original algebra.

In this paper, we introduce a type of generalized Hausdorff operators and characterize the boundedness of these operators on Lebesgue spaces and central Morrey spaces. Moreover, we obtain the operator norms on these spaces. We also obtain sufficient and necessary conditions which ensure the boundedness of their commutators on Lebesgue spaces and central Morrey spaces with symbols in central BMO spaces. As applications, we give a new method to obtain sharp bounds for weighted Hardy operators and weighted Cesàro operators on Lebesgue spaces and central Morrey spaces.

It is shown that, in a non-necessarily Hausdorff real topological vector space, if a subset is a countable disjoint union of convex sets closed in the subset, then those convex sets must be its convex components. On the other hand, by means of convex components we extend the notion of extreme point to non-convex sets, which entails a new equivalent reformulation of the Krein--Milman property (involving drops among other objects). Finally, we study the nature of convex functions and provide some results on their support in order to introduce the concept of multi-slice, that is, slices determined by convex functions (instead of by linear functions). Among other things, we prove that the boundary of a closed convex set with non-empty interior can be obtained as the set of support points of a certain lower semi-continuous convex function on that convex set.

In this paper, we study the problem of isometric extension on the unit sphere of the space $l\cap l^p(H)$ for $0 < p < 1$. We obtain that an isometric mapping of the unit sphere $S(l\cap l^p(H))$ onto itself can be extended to an isometry on the whole space $l\cap l^p(H)$.

We study a concept of weak external hyperconvexity in quasi-metric setting. In this article, we generalize further well-known results about weak external hyperconvexity from metric to quasi-metric point of view. In particular, we show the necessary condition of a subset of $q$-hyperconvex $T_0$-quasi-metric space to be weakly externally $q$-hypeconvex.

Given a normed space $X$ we consider the hyperspace $k(X)$ of all non-empty compact convex subsets of $X$ endowed with the Hausdorff distance. We prove that if $T: X \longrightarrow X$ is an $(m,q)$-isometry, then it is possible that the map $k(T) : k(X) \longrightarrow k(X)$, $k(T) C := TC$, is not an $(m,q)$-isometry. Moreover, if $\widehat{k(X)}$ is the R{\aa}dstr\"{o}m space associated to the hyperspace $k(X)$, then $\mathcal{T}: k(X) \longrightarrow k(X)$ is an $(m,q)$-isometry if and only if $\widehat{\mathcal{T}}: \widehat{k(X)} \longrightarrow \widehat{k(X)}$ is an $(m,q)$-isometry.

Let $L:=-\mathrm{div}(A\nabla)+V$ be a Schrödinger type operator with the nonnegative potential $V$ belonging to the reverse H\"older class $RH_{q}(\mathbb{R}^n)$ for some $q\in(n/2,\infty]$ and $n\ge3$, where $A$ satisfies the uniformly elliptic condition. Assume that $\varphi:\,\mathbb{R}^n\times[0,\infty)\to[0,\infty)$ is a function such that $\varphi(x,\cdot)$ is an Orlicz function and $\varphi(\cdot,t)\in {\mathbb A}_{\infty}(\mathbb{R}^n)$ (the class of uniformly Muckenhoupt weights). In this article, the author proves that the operators $VL^{-1}$, $V^{1/2}\nabla L^{-1}$ and $\nabla^2L^{-1}$ are bounded from the Musielak--Orlicz--Hardy space associated with $L$, $H_{\varphi,\,L}(\mathbb{R}^n)$, to the Musielak-Orlicz space $L^{\varphi}(\mathbb{R}^n)$ or $H_{\varphi,\,L}(\mathbb{R}^n)$ under some further assumptions for $\varphi$ and $A$, which further implies a maximal inequality for $L$ in the scale of $H_{\varphi,\,L}(\mathbb{R}^n)$. All these results improve the known results by weakening the assumption for $\varphi$ and $L$.

In this paper we generalize the max algebra system of nonnegative matrices to the class of nonnegative tensors and derive its fundamental properties. If $\mathbb{A} \in \Re_ + ^{\left[ {m,n} \right]}$ is a nonnegative essentially positive tensor such that satisfies the condition class NC, we prove that there exist $\mu \left( \mathbb{A} \right)$ and a corresponding positive vector $x$ such that $\mathop {\max }\limits_{1 \le{i_2}\cdots {i_m} \le n} \left\{ {{a_{i{i_2}\cdots {i_m}}}{x_{{i_2}}}\cdots {x_{{i_m}}}} \right\}=\mu \left( \mathbb{A} \right) x_i^{m - 1},\,\,\,\,i = 1,2,\cdots ,n.$ This theorem, is well known as the max algebra version of Perron--Frobenius theorem for this new system.

In this paper, strong convergence of Halpern iteration is shown for a quasi-strongly nonexpansive sequence of multivalued mappings in complete CAT(0) spaces.

We construct Hopf algebra isomorphisms of discrete (multiplier) Hopf $C^*$-bialgebras from $K$-theoretical data, without assuming that the Haar weight is tracial.

Two applications of nets are given. The first is an extension of the Bochner integral to arbitrary locally convex spaces, leading to an integration theory of more general vector valued functions then in the classical approach by Gelfand and Pettis. The second application starts with the observation that an operator on a Hilbert space is trace class if and only if the net of "principal trace minors" converges. The notion of a "determinant class operator" then is defined as one for which the net of determinantal principal minors converges. It is shown that for a normal operator $A$ this condition coincides with $1-A$ being trace class.

Motivated by the refinements and reverses of arithmetic-geometric mean and arithmetic-harmonic mean inequalities for scalars and matrices, in this article, we generalize the scalar and matrix inequalities for the difference between arithmetic mean and harmonic mean. In addition, relevant inequalities for the Hilbert-Schmidt norm and determinant are established.

Let $A=(a_{n,k})_{n,k\geq 0}$ be a non-negative matrix. We denote by $L_{\ell_p(w),C_{q}^r(w)}(A)$ the supremum of those $\ell,$ satisfying the following inequality: \[{\left( {\sum\limits_{n = 0}^\infty {{w_n}{{\left( {\frac{1}{{{{\left( {1 + r} \right)}^n}}}\sum\limits_{k = n}^\infty {\frac{{{{\left( {1 + r} \right)}^k}}}{{1 + k}}\sum\limits_{j = 0}^\infty {{a_{k,j}}{x_j}} } } \right)}^q}} } \right)^{1/q}} \ge \ell {\left( {\sum\limits_{n = 0}^\infty {{w_n}x_n^p} } \right)^{1/p}},\] where $x\geq 0$, $x\in \ell_p(w)$, $r\in (0,1)$, $q\le p$ are numbers in $(0,1)$ and $\left(w_n \right)_{n=0}^\infty$ is a non-negative and non-increasing sequence of real numbers. In this paper, first we introduce the weighted sequence space $C_{p}^r(w)~(p \in (0,1))$ of non-absolute type which is a $p$-normed space and is isometrically isomorphic to the space $\ell_p(w)$. Then we focus on the evaluation of $L_{\ell_p(w),C_{q}^r(w)}(A^t)$ for a lower triangular matrix $A$, where $q\le p$ are in $(0,1)$. A lower estimate is obtained. Moreover, in this paper a Hardy type formula is obtained for $L_{\ell_p,C_{q}^r}(H_\mu ^\alpha )$ where $H^\alpha_\mu$ is the generalized Hausdorff matrix, $q\le p\le1$ are positive numbers and $\alpha\geq 0.$ A similar result is also established for the case in which $H^\alpha_\mu$ is replaced by ${(H_\mu ^\alpha )^t}$.

This work creates two categories of "array-weighted sets" for the purposes of constructing universal matrix-normed spaces and algebras. These universal objects have the analogous universal property to the free vector space, lifting maps completely bounded on a generation set to a completely bounded linear map of the matrix-normed space. Moreover, the universal matrix-normed algebra is used to prove the existence of a free product for matrix-normed algebras using algebraic methods.

In this paper, by introducing the class of quasi hereditarily normaloid polaroid operators, we obtain a theoretical and general framework from which Weyl type theorems may be promptly established for many of these classes of operators. This framework also entails Weyl type theorems for perturbations $f(T + A)$, where $A$ is algebraic and commutes with $T,$ and $f$ is an analytic function, defined on an open neighborhood of the spectrum of $T +A$, such that $f$ is non constant on each of the components of its domain.

The Cauchy-Schwarz (C-S) inequality is one of the most famous inequalities in mathematics. In this survey article, we first give a brief history of the inequality. Afterward, we present the C-S inequality for inner product spaces. Focusing on operator inequalities, we then review some significant recent developments of the C-S inequality and its reverses for Hilbert space operators and elements of Hilbert $C^*$-modules. In particular, we pay special attention to an operator Wielandt inequality.