We will show that if $\mathcal{M}$ is a factor, then for any pair $\varphi ,p\in\mathcal{M}_*^+$ of normal positive linear functionals on $\mathcal{M}$, the inequality: $$ \|\varphi\|\leq \|\psi\| $$ is equivalent to the fact that there exist a countable family $\{\varphi_i : i\in I\}\subset \mathcal{M}_*^+$ in $\mathcal{M}_*^+$ and a family $\{u_i : i\in I\}\subset\mathcal{M}$ of partial isometries in $\mathcal{M}$ such that $$ \varphi=\sum_{ i\in I} \varphi_i,\quad \sum_{ i\in I} u_i{\varphi_i}u_i^*\leq \psi, \quad \text{and} \quad u_i^*u_i=s(\varphi_i), i\in I, $$ where $s(\omega), \omega\in \mathcal{M}_*^+$, means the support projection of $\omega$. Furthermore, if $\|\varphi\|=\|\psi\|$, then the equality replaces the inequality in the second statement. In the case that $\mathcal{M}$ is not of type III$_1$, the family of partial isometries can be replaced by a family of unitaries in $\mathcal{M}$. One cannot expect to have this result in the usual integration theory. To have a similar result, one needs to bring in some kind of non-commutativity. Let $\{X, \mu\}$ be a $\sigma$-finite semifinite measure space and $G$ be an ergodic group of automorphisms of $L^\infty(X,\mu)$, then for a pair $f$ and $g$ of $\mu$-integrable positive functions on $X$, the inequality: $$ \int_X f(x)\text{d} \mu(x)\leq \int_X g(x)\text{d} \mu(x) $$ is equivalent to the existence of a countable families $\{f_i: i\in I\}\subset L^1(X, \mu)$ of positive integrable functions and $\{\gamma_i: i\in I\}$ in $G$ such that $$ f=\sum_{ i\in I} f_i\quad\text{and}\quad \sum_{ i\in I} \gamma_i(f_i)\leq g, $$ where the summation and inequality are all taken in the ordered Banach space $L^1(X, \mu)$ and the action of $G$ on $L^1(X, \mu)$ is defined through the duality between $L^\infty(X, \mu)$ and $L^1(X, \mu)$, i.e., $$ (\gamma(f))(x)=f(\gamma^{-1} x)\frac{d\mu\circ \gamma^{-1}}{d\mu}(x), \quad f\in L^1(X, \mu). $$

In 1987, we established an operator inequality as follows; $A \ge B \ge 0 $ $\Longrightarrow (A^{\frac {r}{2}} A^p A^{\frac {r}{2}})^{\frac{1}{q}} \ge (A^{\frac {r}{2}} B^p A^{\frac {r}{2}})^{\frac{1}{q}}$ holds for (*) $ p \ge 0$, $q \ge 1$, $r \ge 0$ with $(1+r)q \ge p+r.$ It is an extension of Löwner-Heinz inequality. The purpose of this paper is to explain geometrical background of the domain by (*), and to give brief survey of recent results of its applications.

Let $B(X, Y)$ denote the set of all bounded linear operators from Banach space $X$ to Banach space $Y$. In this paper, we introduce the concepts of left and right decomposably regular operators, left and right decomposably Fredholm operators in the setting of $B(X, Y)$, and the corresponding holomorphic versions in the setting of $B(X)$. By using Harte's techniques, we obtain various characterizations of these classes of operators. As the applications of these characterizations, we can compute the topological interiors and closures of them.

We give necessary and sufficient conditions for the boundedness and compactness of weighted composition operators between spaces of vector-valued Lipschitz functions. We then show that a bounded separating linear operator between these spaces is indeed a weighted composition operator.

Let $\mathcal{H}$ be an infinite dimensional Hilbert space, $\mathcal{B}_{p}\left(\mathcal{H}\right)$ the $p$-Schatten class of $\mathcal{H}$ and $U_{p}\left(\mathcal{H}\right)$ be the Banach-Lie group of unitary operators which are $p$-Schatten perturbations of the identity. Let $A$ be a bounded selfadjoint operator in $\mathcal{H}$. We show that $$\mathcal{O}_A:=\left\{UA : U \in U_{p}\left(\mathcal{H}\right) \right\}$$ is a smooth submanifold of the affine space $A + \mathcal{B}_{p}\left(\mathcal{H}\right)$ if only if $A$ has closed range. Furthermore, it is a homogeneous reductive space of $U_{p}\left(\mathcal{H}\right)$. We introduce two metrics: one via the ambient Finsler metric induced as a submanifold of $A + \mathcal{B}_{p}\left(\mathcal{H}\right)$, the other, by means of the quotient Finsler metric provided by the homogeneous space structure. We show that $\mathcal{O}_A$ is a complete metric space with the rectifiable distance of these metrics.

In this paper we define the sequence space $E_{n}^{q}(M,p,s)$ by using an Orlicz function and we study various properties and obtain some inclusion relations involving this space. We give some relations between $N_{k}-$lacunary statistical convergence and strongly $N_{k}-$lacunary convergence.

We derive an equivalent definition for the gap between two complemented submodules of a Hilbert $C^*$-module which is same as the one for closed subspaces of a Banach space. This gives an alternative way of defining gap between two regular operators. We give an alternative proof of the latter result. We also derive the McIntosh formula for computing the gap between two regular operators between Hilbert $C^*$-modules which is analogous to that of unbounded operators between Hilbert spaces.

In the paper, we prove the following assertions: (1) If $T$ is an algebraically quasi-$*$-$A$ operator, then $T$ is polaroid. (2) If $T$ or $T^{*}$ is an algebraically quasi-$*$-$A$ operator, then Weyl's theorem holds for $f(T)$ for every $f \in H(\sigma(T))$. (3) If $T^{*}$ is an algebraically quasi-$*$-$A$ operator, then a-Weyl's theorem holds for $f(T)$ for every $f \in H(\sigma(T))$.

In this paper we establish the best lower bound for the weighted Jensen's discrete inequality with ordered variables applied to a convex function $f$, in the case when the bound depends on $f$, weights and three fixed variables. Some applications for particular cases of interest are provided.

In this paper we show that, using combinatorial inequalities and Matrix-Averages, we can generate Musielak-Orlicz spaces, i.e., we prove that $\underset{\pi}{\mbox{Ave}} \max\limits_{1 \leq i \leq n} |x_i y_{i\pi(i)}| \sim \|x\|_{\Sigma M_i}$, where the Orlicz functions $M_1,\ldots,M_n$ depend on the matrix $(y_{ij})_{i,j=1}^n$. We also provide an approximation result for Musielak--Orlicz norms which already in the case of Orlicz spaces turned out to be very useful.

In this paper, we establish sharp maximal function inequalities for the Toeplitz type operator related to some general fractional integral operators. As an application, we obtain the boundedness of the operator on Lebesgue, Morrey and Triebel-Lizorkin spaces. The operators include Littlewood-Paley operator, Marcinkiewicz operator and Bochner-Riesz operator.

Let $\varphi,\psi$ be the analytic self-maps of the unit ball ${\mathbb B}$, we characterize the Hilbert-Schmidt differences of two composition operator $C_\varphi$ and $C_\psi$ on weighted Bergman space $A_\alpha^2$, and give some conclusions about the topological structure of $\mathcal{C}(A_\alpha^2)$, the space of all bounded composition operators on $A_\alpha^2$ endowed with operator norm.

In this paper we establish a Fubini theorem for functionals on a function space. We then establish some relationships as applications of our Fubini theorem. Finally, we present some historical remarks.

We consider permanence of spectral properties of Banach space operators asymptotically intertwined by the identity operator.

The aim of this note is to give a type of characterization of Banach spaces in terms of the stability of functional equations. More precisely, we prove that a normed space $X$ is complete if there exists a functional equation of the type $$\sum_{i=1}^{n}a_if(\varphi_i(x_1,\ldots,x_k))=0 \qquad(x_1,\ldots,x_k\in D)$$ with given real numbers $a_1,\ldots,a_n$, given mappings $\varphi_1\ldots,\varphi_n\colon D^k\to D$ and unknown function $f\colon D\to X$, which has a Hyers--Ulam stability property on an infinite subset $D$ of the integers.