We prove and discuss some power weighted Hardy-type inequalities on finite and infinite sets. In particular, it is proved that these inequalities are equivalent because they can all be reduced to an elementary inequality, which can be proved by Jensen inequality. Moreover, the corresponding limit (Pólya--Knopp type) inequalities and equivalence theorem are proved. All constants in these inequalities are sharp.

In the present paper, we investigate approximation to analytic function having property $B$ by complex linear operators sequence in parabolic domain. We apply Gergen, Dressel and Purcell's method to Chlodovsky type complex Szasz operators.

Some criteria for approximative compactness of every weakly$^{*}$ closed convex set of Orlicz function spaces equipped with the Luxemburg norm are given. Although, criteria for approximative compactness of Orlicz function spaces equipped with the Luxemburg norm were known, we can easily deduce them from our main results.

Let $H$ be an infinite dimensional complex Hilbert space and let $\phi$ be a surjective linear map on $B(H)$ with $\phi(I)-I\in{\mathcal{K}}(H)$, where $\mathcal{K}(H)$ denotes the closed ideal of all compact operators on $H$. If $\phi$ preserves the set of upper semi-Weyl operators and the set of all normal eigenvalues in both directions, then $\phi$ is an automorphism of the algebra $B(H)$. Also the relation between the linear maps preserving the set of upper semi-Weyl operators and the linear maps preserving the set of left invertible operators is considered.

An operator $T \in B( \mathcal{H}) $ is called $k$-quasi-$\ast$-class A if $T^{\ast k}(|T^{2}|-|T^{\ast}|^{2})T^{k} \geq 0$ for a positive integer $k$, which is a common generalization of $\ast$-class A and quasi-$\ast$-class A. In this paper, firstly we prove some inequalities of this class of operators; secondly we consider the tensor products for $k$-quasi-$\ast$-class A operators, giving a necessary and sufficient condition for $T\otimes S$ to be a $k$-quasi-$\ast$-class A operator when $T$ and $S$ are both non-zero operators; at last we prove that the spectrum is continuous on the class of all $k$-quasi-$\ast$-class A operators.

Let $\mathcal{H}$ be a complex Hilbert space, and $A_1,\ldots,A_N$ be bounded self-adjoint operators in $\mathcal{H}$ such that $A_i A_j$ is compact for any $i\neq j$. It is well-known that $\sigma_e(\sum_{i=1}^N A_i)\setminus\{0\}=(\cup_{i=1}^N\sigma_e(A_i))\setminus\{0\}$, where $\sigma_e(B)$ stands for the essential spectrum of a bounded self-adjoint operator $B$. In this paper we get necessary and sufficient conditions for $0\in\sigma_e(\sum_{i=1}^N A_i)$. This conditions are formulated in terms of the projection valued spectral measures of $A_i$, $i=1,\ldots,N$. Using this result, we obtain necessary and sufficient conditions for the sum of ranges of $A_i$, $i=1,\ldots,N$ to be closed.

We study composition operators acting between weighted Bergman spaces with admissible Békollé weights. The boundedness and compactness of composition operators are characterized in terms of the generalized Nevanlinna counting function associated with the inducing map of the composition operator and the associated weight function of Bergman space. For a special case, we also give the estimate of the essential norm.

In this paper, we consider orthonormal bases for finite dimensional normed linear spaces which are related to Birkhoff orthogonality. We also study the relationship between orthonormal bases and a structure of finite dimensional normed linear spaces.

We investigate additive results for $(p,q)$-outer generalized inverse of elements in Banach algebra, along with the representation of this inverse in a block matrix in the Banachiewicz--Schur form. Additionally, we investigate the $(p,q)$-pseudospectrum and $(p,q)$-condition spectrum of a block matrix $(a,0,0,b)$ in a Banach algebra.

In this paper, two infinite classes of inequalities for the $q$-gamma and the $q$-digamma functions are established by means of using the Taylor theorem. The bounds appeared in these inequalities are in terms of the Hurwitz and Riemann $q$-zeta functions.

We give a new characterization for the boundedness of composition operator followed by differentiation operator acting on Bloch-type spaces and calculate its essential norm in terms of the $n$-th power of the induced analytic self-map on the unit disk. From which some sufficient and necessary conditions of compactness of the operator follow immediately.

We yield operators $U$ and $V$ on Hilbert spaces that are parameterized by the orbits of certain interval maps that exhibit chaotic behavior and obey the (deformed) Baumslag--Solitar relation $$UV=e^{2\pi i \alpha} VU^n,\qquad \alpha\in \mathbb{R},\ n\in\mathbb{N}.$$ We then prove that the scalar $e^{2\pi i \alpha}$ can be removed whilst retaining the isomorphism class of the $C^*$-algebra generated by $U$ and $V$. Finally, we simultaneously unitarize $U$ and $V$ by gluing pairs of orbits of the underlying noninvertible dynamical system and investigate these unitary representations under distinct pairs of orbits.

Matrix rank and inertia optimization problems are a class of discontinuous optimization problems in which the decision variables are matrices running over certain matrix sets, while the ranks and inertias of the variable matrices are taken as integer-valued objective functions. In this paper, we establish a group of explicit formulas for calculating the maximal and minimal values of the rank and inertia objective functions of the Hermitian matrix-valued function $A_1 - B_1XB_1^{*}$ subject to the common Hermitian solution of a pair of consistent matrix equations $B_2XB^{*}_2 = A_2$ and $B_3XB_3^{*} = A_3$, and Hermitian solution of the consistent matrix equation $B_4X= A_4$, respectively. Many consequences are obtained, in particular, necessary and sufficient conditions are established for the triple matrix equations $B_1XB^{*}_1 =A_1$, $B_2XB^{*}_2 = A_2$ and $B_3XB^{*}_3 = A_3$ to have a common Hermitian solution, as well as necessary and sufficient conditions for the two matrix equations $B_1XB^{*}_1 =A_1$ and $B_4X = A_4$ to have a common Hermitian solution.

In this paper, we consider the problem of isometric extension in the unit sphere of the space $s_p(\alpha,H)$. We obtain that Lamperti isometric mapping of the unit sphere $S(s_p(\alpha,H))$ into itself can be extended to an isometry on the whole space $s_p(\alpha,H)$.

Let $X$ be a Banach space and let $Y$ be a separable Lindenstrauss space. We describe the Banach space $\mathcal{P}(Y,X)$ of absolutely summing operators as a general $\ell_1$-tree space. We also characterize the bounded approximation property and its weak version for $X$ in terms of the space of integral operators $\mathcal{I}(X,Z^*)$ and the space of nuclear operators $\mathcal{N}(X,Z^*)$, respectively, where $Z$ is a Lindenstrauss space, whose dual $Z^*$ fails to have the Radon-Nikodým property.

We present an isometric version of the complementably universal Banach space $\mathbb{P}$ with a Schauder decomposition. The space $\mathbb{P}$ is isomorphic to Pelczynski's space with a universal basis as well as to Kadec' complementably universal space with the bounded approximation property.

Let $\varphi: {\mathbb R^n}\times [0,\infty)\to[0,\infty)$ be such that $\varphi(x,\cdot)$ is nondecreasing, $\varphi(x,0)=0$, $\varphi(x,t)>0$ when $t>0$, $\lim_{t\to\infty}\varphi(x,t)=\infty$ and $\varphi(\cdot,t)$ is a Muckenhoupt $A_\infty({\mathbb R^n})$ weight uniformly in $t$. Let $\phi: [0,\infty)\to[0,\infty)$ be nondecreasing. In this article, the authors introduce the Musielak--Orlicz Morrey space $\mathcal M^{\varphi,\phi}(\mathbb R^n)$ and obtain the boundedness on $\mathcal M^{\varphi,\phi}(\mathbb R^n)$ of the intrinsic Lusin area function $S_{\alpha}$, the intrinsic $g$-function $g_{\alpha}$, the intrinsic $g_{\lambda}^*$-function $g^\ast_{\lambda, \alpha}$ and their commutators with ${\rm BMO}(\mathbb{R}^n)$ functions, where $\alpha\in(0,1]$, $\lambda\in(\min\{\max\{3,\,p_1\},3+2\alpha/n\},\infty)$ and $p_1$ denotes the uniformly upper type index of $\varphi$. Let $\Phi: [0,\infty)\to[0,\infty)$ be nondecreasing, $\Phi(0)=0$, $\Phi(t)>0$ when $t>0$, and $\lim_{t\to\infty}\Phi(t)=\infty$, $w\in A_\infty(\mathbb R^n)$ and $\phi: (0,\infty)\to(0,\infty)$ be nonincreasing. The authors also introduce the weighted Orlicz--Morrey space $M_w^{\Phi,\phi}(\mathbb R^n)$ and obtain the boundedness on $M_w^{\Phi,\phi}(\mathbb R^n)$ of the aforementioned intrinsic Littlewood--Paley functions and their commutators with ${\rm BMO}(\mathbb{R}^n)$ functions. Finally, for $q\in[1,\infty)$, the boundedness of the aforementioned intrinsic Littlewood--Paley functions on the Musielak-Orlicz Campanato space $\mathcal L^{\varphi,q}(\mathbb R^n)$ is also established.

For any trigonometric polynomial $\phi(\theta)$, we give a constructive algorithm by Sylvester elimination which produces matrices $C_1, C_2, C_3$ such that $\det(C_1+ \Re(\phi(\theta)) C_2+ \Im(\phi(\theta)) C_3)=0.$ For a typical trigonometric polynomial, we assert that $C_1$ is positive definite, and thus the typical polynomial curve admits

An operator $T$ on a complex Hilbert space $\mathcal{H}$ is said to be skew symmetric if there exists a conjugate-linear, isometric involution $C:\mathcal{H}\rightarrow\mathcal{H}$ such that $CTC=-T^*$. In this paper, using an interpolation theorem related to conjugations, we give a geometric characterization for a class of operators to be skew symmetric. As an application, we get a description of skew symmetric partial isometries.

We study the notions of acs, luacs and uacs Banach spaces which were introduced by Kadets et al. and form common generalisations of the usual rotundity and smoothness properties of Banach spaces. In particular, we are interested in (mainly infinite) absolute sums of such spaces. We also introduce some new classes of spaces that lie inbetween those of acs and uacs spaces and study their behaviour under the formation of absolute sums as well.