Weighted version of two-point integral quadrature formula is obtained using $w-$harmonic sequences of functions. Improved version of Guessab and Schmeisser's result is given with new integral inequalities under various regular conditions. As special cases, the generalizations of quadrature formulae of Gauss type are established.

A general formulation of the Jensen-Mercer operator inequality for operator convex functions, continuous fields of operators and unital fields of positive linear mappings is given. As consequences, a global upper bound for Jensen's operator functional and some properties of the quasi-arithmetic operator means and quasi-arithmetic operator means of Mercer's type are obtained.

An operator $T \in B(H)$ is said to be absolute-$(p, r)$-paranormal if $\| |T|^{p} |T^{*}|^{r} x \|^{r} \|x\| \geq \| |T^{*}|^{r} x\|^{p + r}$ for all $x \in H$ and for positive real number $p > 0$ and $r > 0$, where $T=U |T|$ is the polar decomposition of $T$. In this paper, we discuss some properties of absolute-$(p, r)$-paranormal operators and show that Weyl's theorem holds for algebraically absolute-$(p, r)$-paranormal operators.

The purpose of this paper is to introduce the concept of a cone symmetric space and to investigate relationship between (cone) metric spaces and (cone) symmetric spaces. Among other things, we shall also extend some fixed point results from metric spaces to cone metric spaces (Theorem 3.3), and to symmetric spaces (Theorems 3.2 and 3.5) under some new contraction conditions.

Let $\mathcal{H}$ be a separable Hilbert space and let $\mathcal{A} \in \mathcal{B}(\mathcal{H})$. In this note the notion of a total decomposition is introduced, and it is shown that sometimes the block numerical ranges corresponding to a total decomposition approximate $\sigma(A),$ sometimes not.

We define a new metric on the set of all closed linear operators between Hilbert spaces and investigate its properties. In particular, we show that the set of all closed operators with a closed range is an open subset of the set of all closed operators and the map $T\mapsto T^\dagger$ is an isometry in this metric. We also investigate the relationships between the topology induced by this metric and the gap metric.

In this work we prove improved converse theorems of trigonometric approximation in variable exponent Lebesgue spaces with some Muckenhoupt weights.

New characterizations of inner product spaces among normed spaces involving the notion of strong convexity are given. In particular, it is shown that the following conditions are equivalent: (1) $(X,\|\cdot\|)$ is an inner product space; (2) $f:X\to \R$ is strongly convex with modulus $c>0$ if and only if $f-c\|\cdot\|^2$ is convex; (3) $\|\cdot\|^2$ is strongly convex with modulus $1$.

Let $A$ be any n-by-n normal matrix and let $k>0$ be an integer. By using the concept of the joint numerical range $W(A, A^2, \cdots, A^k),$ an analytic description of $V^k(A)$ for normal matrices will be presented. Additionally, new proof for Theorem 2.2 of Davis, Li and Salemi [Linear Algebra Appl., 428 (2008), pp. 137-153] is given.

Due to an example indicated to us in September 2009 we have to add one more restriction to the suppositions on the imprimitivity bimodules treated in Proposition 4.1, Theorem 5.1, Theorem 6.2 and Proposition 6.3. In the situation when the Banach-Saks property holds for the imprimitivity bimodule we can describe all possible additional examples violating the newly invented supposition. So the classification of Hilbert $C^*$-modules with the Banach-Saks property is complete. Beyond that, there is still an open problem for a certain class of imprimitivity bimodules with the weak or uniform weak Banach-Saks property which might violate the additional condition.

We discuss two new concepts of convergence in $L^{p}$-spaces, the so-called weak $\Sigma $-convergence and strong $\Sigma $-convergence, which are intermediate between classical weak convergence and strong convergence. We also introduce the concept of $\Sigma $-convergence for Radon measures. Our basic tool is the classical Gelfand representation theory. Apart from being a natural generalization of well-known two-scale convergence theory, the present study lays the foundation of the mathematical framework that is needed to undertake a systematic study of deterministic homogenization problems beyond the usual periodic setting. A few homogenization problems are worked out by way of illustration.

n this paper, we introduce the notion of $\phi$-weakly compatible mapping for a pair of mappings. A fixed point theorem for two pairs of $\phi$-weakly compatible mappings satisfying a rational type contraction in a normed space is also established. Subsequently we use our result to find existence of solutions of variational inequalities.

The notion of amalgamation of product systems has been introduced in [7] which generalizes the concept of Skeide product, introduced by Skeide, of two product systems via a pair of normalized units. In this paper we show that amalgamation leads to a setup where a product system is generated by two subsystems and conversely whenever a product system is generated by two subsystems, it could be realized as an amalgamated product. We parameterize all contractive morphism from a Type I product system to another Type I product system and compute index of amalgamated product through contractive morphisms.

We show that $L^p$-spaces over atomless measure spaces can be characterized in terms of a $p$-concavity type geometric property that is related with the Daugavet property.

Let $A$ be a separable unital $C^*$-algebra and let $\Theta_A$ be the canonical contraction from the Haagerup tensor product of $A$ with itself to the space of completely bounded maps on $A$. In our previous paper we showed that if $A$ satisfies (a) the lengths of elementary operators on $A$ are uniformly bounded, or (b) the image of $\Theta_A$ equals the set of all elementary operators on $A$, then $A$ is necessarily SFT (subhomogeneous of finite type). In this paper we extend this result; we show that if $A$ satisfies (a) or (b) then the codimensions of $2$-primal ideals of $A$ are also finite and uniformly bounded. Using this, we provide an example of a unital separable SFT algebra which does not satisfy (a) nor (b). However, if the primitive spectrum of a unital SFT algebra $A$ is Hausdorff, we show that such an $A$ satisfies (a) and (b).