We obtain the estimate of the Lebesgue measure of the spectrum for the direct integral of matrix-valued functions. These estimates are applicable for a wide class of discrete periodic operators. For example these results give new and sharp spectral bounds for 1D periodic Jacobi matrices and 2D discrete periodic Schrodinger operators.

In this paper we introduce operator preinvex functions and establish a Hermite--Hadamard type inequality for such functions. We give an estimate of the right hand side of a Hermite--Hadamard type inequality in which some operator preinvex functions of selfadjoint operators in Hilbert spaces are involved. Also some Hermite--Hadamard type inequalities for the product of two operator preinvex functions are given.

The complete characterization of the Hardy-type $L^p - L^q$ inequalities on the weighted cones of quasi-concave functions for all $p,q \in (0,\infty)$ is given.

In this paper we investigate weighted composition operators between weak and strong vector-valued Bergman spaces and Hardy spaces, and give some estimates of their norms.

We shall investigate the boundedness of the intrinsic square functions and their commutators on generalized weighted Orlicz--Morrey spaces $M^{\Phi,\varphi}_{w}({\mathbb{R}}^n)$. In all the cases, the conditions for the boundedness are given in terms of Zygmund-type integral inequalities on weights $\varphi$ without assuming any monotonicity property of $\varphi(x,\cdot)$ with $x$ fixed.

In this paper we study some new converses of the operator version for the Jensen inequality and the scalar Lah--Ribarič inequality for continuous convex functions and we will give applications of those results to quasi-arithmetic means and power means for selfadjoint operators. Furthermore, we will also give improvements of the obtained results.

We give a characterization of hypercyclic abelian semigroup $\mathcal{G}$ of affine maps on $\mathbb{C}^{n}$. If $\mathcal{G}$ is finitely generated, this characterization is explicit. We prove in particular that no abelian group generated by $n$ affine maps on $\mathbb{C}^{n}$ has a dense orbit.

In this paper we define non-archimedean measures and integral operators taking values in a locally convex space. We show the relation between these two concept. We define what we called integral function respect to an integral operator. We give necessary and sufficient condition in order to know when a function is integrable with respect to an integral operator. In the second part, we define a kind of absolutely continuous relation between measures in this context. After that, we formulate a type of Radon--Nikodym Theorem between vector measures and a scalar measures which are absolutely continuous.

We introduce $q$-frequently hypercyclic operators and derive a sufficient criterion for a continuous operator to be $q$-frequently hypercyclic on a locally convex space. Applications are given to obtain $q$-frequently hypercyclic operators with respect to the norm-, $F$-norm- and weak*- topologies. Finally, the frequent hypercyclicity of the non-convolution operator $T_\mu$ defined by $T_\mu(f)(z)=f'(\mu z)$, $|\mu|\geq 1$ on the space $H(\mathbb{C})$ of entire functions equipped with the compact-open topology is shown.

We present some spectral radius inequalities for nonnegative matrices. Using the ideas of Audenaert, we then prove the inequality which may be regarded as a Cauchy--Schwarz inequality for spectral radius of nonnegative matrices $$ \rho(A \circ B) \leq [\rho(A \circ A)]^{\frac{1}{2}}[\rho(B\circ B)]^{\frac{1}{2}}. $$ In addition, new proofs of some related results due to Horn and Zhang, Huang are also given. Finally, we interpolate Huang's inequality by proving $$ \rho(A_{1}\circ A_{2} \circ \cdots \circ A_{k}) \leq [\rho(A_{1}A_2\cdots A_{k})]^{1-\frac{2}{k}}[\rho((A_{1}\circ A_{1})\cdots (A_{k}\circ A_{k})]^{\frac{1}{k}} \leq \rho(A_{1}A_2 \cdots A_{k}).$$

In a normed space we introduce an exact and approximate orthogonality relation. We consider classes of linear mappings approximately preserving this kind of orthogonality. We show that, in particular, the property that a linear mapping approximately preserves the $B$-orthogonality is equivalent to that it approximately preserves the $\rho,\rho_+$-orthogonality (although these orthogonalities need not be equivalent). Moreover, we show that every approximately orthogonality preserving linear mapping is necessarily a scalar multiple of an almost isometry.

In several works, the theory of strongly continuous groups is used to build a framework for solving random homogenization problems. Following this idea, we present a detailed and comprehensive framework enabling one to solve homogenization problems in algebras with mean value, regardless of whether they are ergodic or not. We also state and prove a compactness result for Young measures in these algebras. As an important achievement we study the homogenization problem associated with a stochastic Ladyzhenskaya model for incompressible viscous flow, and we present and solve a few examples of homogenization problems related to nonergodic algebras.

Let $\mathcal A$ and $\mathcal B$ be Banach algebras and let $T:{\mathcal B}\rightarrow{\mathcal A}$ be a continuous homomorphism. Recently, we introduced a product ${\mathcal M}:={\mathcal A}\times_T{\mathcal B}$, which is a strongly splitting Banach algebra extension of $\mathcal B$ by $\mathcal A$. In the present paper, we characterize biprojectivity, approximate biprojectivity and biflatness of ${\mathcal A}\times_T{\mathcal B}$ in terms of ${\mathcal A}$ and ${\mathcal B}$. We also study some notions of amenability such as approximate amenability and pseudo-amenability of ${\mathcal A}\times_T{\mathcal B}$.

We present simple proofs for Burnside's asymptotic formula and for its extension to positive real numbers.

The goal of this paper is to present characterizations for absolute continuity of representable positive functionals on general $^*$-algebras. From the results we give a new and very different proof to our recently published Lebesgue decomposition theorem for representable positive functionals. On unital $C^*$-algebras and measure algebras of compact groups further characterizations are included in the paper. As an application of our results, we answer Gudder's problem on the uniqueness of the Lebesgue decomposition in the case of commutative $^*$-algebras and measure algebras of compact groups. Another application to faithful positive functionals defined on the latter $^*$-algebras is also included.

The paper is concerned with weak approximation properties which are weaker than the classical approximation property. For $\lambda \geq 1$, we prove that a Banach space $X$ has the $\lambda$-bounded weak approximation property ($\lambda$-BWAP) if and only if every locally $1$-complemented subspace of $X$ has the $\lambda$-BWAP, and that if $X$ has the $\lambda$-BWAP and $Z$ is a locally $\mu$-complemented subspace of $X$, then $Z$ has the $(2\mu+4)\mu\lambda$-BWAP. It also follows that $X$ has the weak approximation property (WAP) if and only if every locally complemented subspace of $X$ has the WAP.

Casazza and Christensen [Canad. Math. Bull., 51 (2008), 348--358] introduced and studied the reconstruction property in Banach spaces. In this paper, we discuss different types of convergence of series related to the reconstruction property in Banach space. First we discuss the uniform convergence of series associated with the reconstruction property in Banach spaces. Necessary and sufficient conditions for the uniform convergence of certain series related to the reconstruction property in Banach spaces are given. A sufficient condition for a Banach space to be finite dimensional in terms of the uniform convergence of a series related to the reconstruction property in Banach spaces is obtained. Motivated by a series of papers by Casazza, we discuss unconditional convergence of series associated with the reconstruction property in Banach spaces. A necessary condition in this direction is given. An absolute type reconstruction property in Banach spaces is also discussed which depends on the absolute convergence of series related to the reconstruction property in Banach spaces.

Conditions such that a locally $k$-convex inductive limit of a sequence of $k_n$-normed algebras is a locally $m$-($k$-convex) algebra, are given. It is shown that every locally pseudoconvex inductive limit $E$ of a sequence of commutative locally $m$-pseudoconvex algebras is a commutative locally\break $m$-pseudoconvex algebra if the multiplication in $E$ is jointly continuous.

A discrete group $\mathrm{G}$ is called rigidly symmetric if for every $C^*$-algebra $\mathcal{A}$ the projective tensor product $\ell^1(\mathrm{G})\widehat\otimes A$ is a symmetric Banach $^*$-algebra. For such a group we show that the twisted crossed product $\ell^1_{\alpha,\omega}(\mathrm{G};\mathcal{A})$ is also a symmetric Banach $^*$-algebra, for every twisted action $(\alpha,\omega)$ of $\mathrm{G}$ in a $C^*$-algebra $\mathcal{A}$. We extend this property to other types of decay, replacing the $\ell^1$-condition. We also make the connection with certain classes of twisted kernels, used in a theory of integral operators involving group $2$-cocycles. The algebra of these kernels is studied, both in intrinsic and in represented version.

Consider a pseudo-$H$-space $E$ endowed with a separately continuous biadditive associative multiplication which induces a grading on $E$ with respect to an abelian group $G$. We call such a space a graded pseudo-$H$-ring and we show that it has the form $E = cl(U + \sum_j I_j)$ with $U$ a closed subspace of $E_1$ (the summand associated to the unit element in $G$), and any $I_j$ runs over a well described closed graded ideal of $E$, satisfying $I_jI_k = 0$ if $j \neq k$. We also give a context in which graded simplicity of $E$ is characterized. Moreover, the second Wedderburn-type theorem is given for certain graded pseudo-$H$-rings.