In this paper, we study the Hardy-Rellich type inequalities and uncertainty principle on the geodesic sphere. Firstly, we derive $L^p$-Hardy inequalities via divergence theorem, which are in turn used to establish the $L^p$-Rellich inequalities. We also establish Heisenberg uncertainty principle on the sphere via the Hardy-Rellich type inequalities. The best constants appearing in the inequalities are shown to be sharp.

As generalizations of the arithmetic and the geometric means, for positive real numbers $a$ and $b$, the power difference mean $J_{q}(a,b)=\frac{q}{q+1}\frac{a^{q+1}-b^{q+1}}{a^{q}-b^{q}}$, the Lehmer mean $L_{q}(a,b)=\frac{a^{q+1}+b^{q+1}}{a^{q}+b^{q}}$ and the Heron mean $K_{q}(a,b)=(1-q)\sqrt{ab}+q\frac{a+b}{2}$ are well known.

In this paper, concerning our recent results on estimations of the power difference mean, we obtain the greatest value $\alpha=\alpha(q)$ and the least value $\beta=\beta(q)$ such that the double inequality for the Lehmer mean $$K_{\alpha}(a,b)< L_{q}(a,b)< K_{\beta}(a,b)$$ holds for any $q \in \mathbb{R}$. We also obtain an operator version of this estimation. Moreover, we discuss generalizations of the results on estimations of the power difference and the Lehmer means.This argument involves refined Heinz operator inequalities by Liang and Shi.

‎Suppose that $\hat{b}_m\downarrow 0,\ \{\hat{b}_m\}_{m=1}^\infty\notin l^2,$‎ ‎and $b_n=2^{-\frac{m}{2}}\hat{b}_m$ for all $ n\in(2^m,2^{m+1}].$‎ ‎In this paper‎, ‎it is proved that any measurable and almost everywhere finite‎ ‎function $f(x)$ on $[0,1]$ can be corrected on a set of arbitrarily small measure‎ ‎to a bounded measurable function $\widetilde{f}(x)$; so that the nonzero Fourier-Haar‎ ‎coefficients of the corrected function present some subsequence of $\{b_n\}$‎, ‎and its‎ ‎Fourier-Haar series converges uniformly on $[0,1]$‎.

‎Let $C[0,T]$ denote the space of continuous real-valued functions on $[0,T]$‎. ‎In this paper we introduce two Banach algebras‎: ‎one of them is defined on $C[0,T]$ and the other is a space of equivalence classes of measures over paths of bounded variation on $[0,T]$‎. ‎We establish an isometric isomorphism between them and evaluate analytic Feynman integrals of the functions in the Banach algebras‎, ‎which play significant roles in the Feynman integration theories and quantum mechanics‎.

‎In this paper‎, ‎we obtain results for factorizability of quantum channels‎. ‎Firstly‎, ‎we prove that if a tensor $T\otimes S_k$ of a quantum channel $T$ on $M_n(\mathbb{C})$ with the completely depolarizing channel $S_k$ is written as a convex combination of automorphisms on the matrix algebra $M_n(\mathbb{C})\otimes M_k(\mathbb{C})$ with rational coefficients‎, ‎then the quantum channel $T$ has an exact factorization through some matrix algebra with the normalized trace‎. ‎Next‎, ‎we prove that if a quantum channel has an exact factorization through a finite dimensional von Neumann algebra with a convex combination of normal faithful tracial states with rational coefficients‎, ‎then it also has an exact factorization through some matrix algebra with the normalized trace‎.

‎We consider a coupled system describing the interaction between acoustic and elastic regions‎, ‎where the coupling occurs not via material properties but through an interaction on an interface separating the two regimes‎. ‎Evolutionary well-posedness in the sense of Hadamard well-posedness supplemented by causal dependence is shown for a natural choice of generalized interface conditions‎. ‎The results are obtained in a real Hilbert space setting incurring no regularity constraints on the boundary and almost none on the interface of the underlying regions‎.

We construct a topology on the standard Hilbert module $H_{\mathcal{A}}$ over a unital $C^*$-algebra and topology on $H_ \mathcal {A} ^{＃}$ (the extension of the module $H_{\mathcal{A}}$ by the algebra $\mathcal{A}^{**}$) such that any “compact” operator (i.e. any operator in the norm closure of the linear span of the operators of the form $z\mapsto x \langle y,z \rangle$, $x,y\in H_{\mathcal{A}}$ (or $x,y \in H_ \mathcal {A} ^{＃}$)) maps bounded sets into totally bounded sets.

‎‎‎Main properties of the regular (or extended) spectrum of elements in topological algebras (introduced by L‎. ‎Waelbroeck and G‎. ‎R‎. ‎Allan for unital locally convex algebras) are presented‎. ‎Descriptions of the relationship between the usual spectrum and the regular spectrum of elements in topological algebras with jointly continuous multiplication are given‎. ‎It is shown that the usual spectrum and the regular spectrum of elements coincide for Hausdorff locally convex Waelbroeck algebras‎. ‎Main properties of the disolvent map of elements in topological algebras are studied‎.

‎‎‎Let $T$ be an adjointable operator between two Hilbert $C^*$-modules‎, ‎and let $T^*$ be the adjoint operator of $T$‎. ‎The polar decomposition of $T$ is characterized as $T=U(T^*T)^\frac{1}{2}$ and $\mathcal{R}(U^*)=\overline{\mathcal{R}(T^*)}$‎, ‎where $U$ is a partial isometry‎, ‎$\mathcal{R}(U^*)$ and $\overline{\mathcal{R}(T^*)}$ denote the range of $U^*$ and the norm closure of the range of $T^*$‎, ‎respectively‎. ‎Based on this new characterization of the polar decomposition‎, ‎an application to the study of centered operators is carried out‎.

‎‎In this paper‎, ‎we relate the existence of certain projections‎, ‎commuting with a bounded linear operator $T\in L(X)$ acting on Banach space $X$‎, ‎with the generalized Kato decomposition of $T$‎. ‎We also relate the existence of these projections with some properties of the quasi-nilpotent part $H_0(T)$ and the analytic core $K(T)$‎. ‎Further results are given for the isolated points of some parts of the spectrum‎.