In this paper we study sequence spaces equations (SSE) with operators, which are determined by an identity whose each term is a sum or a sum of products of sets of the form $\chi _{a}\left( T\right)$ and $\chi _{f\left( x\right) }\left( T\right)$ where $f$ maps $U^{+}$ to itself, $\chi $ is either of the symbols $s$, $s^{0}$, or $s^{\left( c\right) }$. Then we solve five (SSE) of the form $\chi _{a}+\chi _{x}^{\prime }=\chi _{b}^{\prime }$, where $\chi $, $\chi ^{\prime }$ are either $s^{{{}0}}$, $s^{\left( c\right) }$, or $s$. We apply the previous results to the solvability of the systems $s_{a}^{0}+s_{x}\left( \Delta \right) =s_{b}$, $s_{x}\supset s_{b}$ and $s_{a}+s_{x}^{\left( c\right) }\left( \Delta \right) =s_{b}^{\left( c\right) }$, $s_{x}^{\left( c\right) }\supset s_{b}^{\left( c\right) }$. Finally we solve the (SSE) with operators defined by $\chi _{a}\left( C\left( \lambda \right) D_{\tau }\right) +s_{x}^{\left( c\right) }\left( C\left( \mu \right) D_{\tau }\right) =s_{b}^{\left( c\right) }$ where $\chi $ is either $s^{0}$, or $s$.

In this paper we derive some improvements of means inequalities for Hilbert space operators. More precisely, we obtain refinements and reverses of the arithmetic-geometric operator mean inequality. As an application, we also deduce an improved variant for the refined arithmetic--Heinz mean inequality. We also present some eigenvalue inequalities for differences of certain operator means.

Let $T$ be a $(p,k)$-quasiposinormal operator on a complex Hilbert space $\mathcal{H}$, i.e $T^{*k}(c^{2}(T^{*} T)^{p}-(T T^{*})^{p})T^{k} \geq 0$ for a positive integer $p \in (0,1]$, some $c > 0$ and a positive integer $k$. In this paper, we prove that the spectral mapping theorem for Weyl spectrum holds for $(p, k)$ - quasiposinormal operators. We show that the Weyl type theorems holds for $(p,k)$- quasiposinormal. We prove that if $T^{*}$ is $(p,k)$-quasiposinormal, then generalized $a$-Weyl's theorem holds for $T$. Also we prove that $\sigma_{jp}(T)-\{0\} = \sigma_{ap}(T)-\{0\}$ holds for $(p,k)$-quasiposinormal operator.

Using a generalization of the semigroup theory of linear operators, we prove existence and uniqueness of mild solutions for the semilinear fractional order differential equation $${D}^{\alpha+1}_t u(t) + \mu {D}_t^{\beta} u(t) - Au(t) = f(t,u(t)), t\in (0,\infty), \alpha \in (0,\infty), \alpha \leq \beta \leq 1, \, \mu \geq 0, $$ with the property that the solution can be written as $u=f+h$ where $f$ belongs to the space of periodic (resp. almost periodic, compact almost automorphic, almost automorphic) functions and $h$ belongs to the space $ P_0(\mathbb{R}_{+},X):= \{ \phi\in BC(\mathbb{R}_{+},X) \, :\,\, \lim_{T \to \infty}\frac{1}{T} \int_{0}^{T}||\phi(s)||ds=0 \}$. Moreover, this decomposition is unique.

In this paper we study certain properties of Dobrushin's ergodicity coefficient for stochastic operators defined on noncommutative $L^1$-spaces associated with semi-finite von Neumann algebras. Such results extends the well-known classical ones to a noncommutative setting. This allows us to investigate the weak ergodicity of nonhomogeneous discrete Markov processes (NDMP) by means of the ergodicity coefficient. We provide a sufficient conditions for such processes to satisfy the weak ergodicity. Moreover, a necessary and sufficient condition is given for the satisfaction of the $L^1$-weak ergodicity of NDMP. It is also provided an example showing that $L^1$-weak ergodicity is weaker that weak ergodicity. We applied the main results to several concrete examples of noncommutative NDMP.

We introduce some special operator classes and study in terms of Berezin symbols their properties. In particular, we give some characterizations of compact operators and Schatten-von Neumann class operators in terms of Berezin symbols. We also consider some classes of compact operators on a Hilbert space $H,$ which are generalizations of the well known Schatten-von Neumann classes of compact operators. Namely, for any number $p \in (0,\infty)$ and the sequence $w:=(w_{n})_{n\geq0}$ of complex numbers $w_{n},$ $n\geq 0,$ we define the following classes of compact operators on $H $: $$S_{p}^{w}(H)=\left\{ K\in S_{\infty}(H):\sum_{n=0}^{\infty}(s_{n} (K))^{p}w_{n}^{p}\hbox{ is convergent series }\right\}, $$ where $s_{n}(K)$ denotes the $n$th singular number of the operator $K$. The characterizations of these classes are given in terms of Berezin symbols.

Let $\alpha \in (0, \infty)$ and $\beta \in (1, \infty)$. In the present work, the necessary and sufficient conditions for the boundedness and compactness of the integral operator of the form \begin{equation*} L_{\alpha, \beta} f(x):=v(x)\int_0^x \frac{\ln^{\beta-1}(\frac{x}{y})f(y)u(y)dy}{(x-y)^{1-\alpha}} ,\,\,\,\, x>0, \end{equation*} from $L^p\to L^q$, with locally integrable non-negative weight functions $u$ and $v$, in the case $p,q \in (0, \infty), p>\max(1/{\alpha},1),$ provided $u$ is non-increasing on $\mathbb{R}^+:=[0,\infty)$ are found.

If $\Sigma=(X,\sigma)$ is a topological dynamical system, where $X$ is a compact Hausdorff space and $\sigma$ is a homeomorphism of $X$, then a crossed product involutive Banach algebra $\ell^1$ is naturally associated with these data. If $X$ consists of one point, then $\ell^1$ is the group algebra of the integers, hence the general$\ell^1$could be regarded as a noncommutative $\ell^1$-algebra. In this paper, we study spectral synthesis for the closed ideals of $\ell^1$ in two versions, one modeled after $C(X)$and one modeled after $\ell^1(\mathbb{Z})$. We identify the closed ideals which are equal to (what is the analogue of) the kernel of their hull, and determine when this holds for all closed ideals, i.e., when spectral synthesis holds. In both models, this is the case precisely when $\Sigma$ is free.

In this paper we show that a bounded linear operator on a Banach space $X$ is polaroid if and only if $p(T)$ is polaroid for some polynomial $p$. Consequently, algebraically paranormal operators defined on Banach spaces are hereditarily polaroid. Weyl type theorems are also established for perturbations $f(T+K)$, where $T$ is algebraically paranormal, $K$ is algebraic and commutes with $T$, and $f$ is an analytic function, defined on an open neighborhood of the spectrum of $T+K$, such that $f$ is nonconstant on each of the components of its domain. These results subsume recent results in this area.

Multipliers have recently been introduced as operators for Bessel sequences and frames in Hilbert spaces. In this paper, we define the concept of $(X_{d}, X_{d}^{*})$ and $(l^{\infty}, X_{d}, X_{d}^{*})$-Bessel multipliers in Banach spaces and investigate the compactness of these multipliers. Also, we study the possibility of invertibility of $(l^{\infty}, X_{d}, X_{d}^{*})$-Bessel multiplier depending on the properties of its corresponding sequences and its symbol. Furthermore, we prove that every $(X_{d}, X_{d}^{*})$-Bessel multiplier is a $\lambda$-nuclear operator.

Matrix inequalities of Hölder type are obtained. Among other inequalities, it is shown that if $p,q \in (2,\infty) $ and $r>1$ with $1/p+1/q=1-1/r$, then for any $A_{i},B_{i}\in M_{n}\left(\mathbb{C} \right) $ and $\alpha _{i}\in \left[ 0,1\right] $ $\left( i=1,2,\cdots ,m\right) $ with $\sum\limits_{i=1}^{m}\alpha _{i}=1$, we have% \begin{equation*} \left\vert \sum\limits_{i=1}^{m}\alpha _{i}^{1/r}B_{i}A_{i}\right\vert \leq \left( \sum\limits_{i=1}^{m}\left\vert A_{i}\right\vert ^{p}\right) ^{1/p} \end{equation*}% whenever $\sum\limits_{i=1}^{m}\left\vert B_{i}^{\ast }\right\vert ^{q}\leq I $. Related unitarily invariant norm inequalities are also presented.

This paper is devoted to an investigation of frames and Riesz bases for general Banach sequence spaces. We establish various relationships between Bessel (respectively, frames) and Riesz sequences (respectively, Riesz bases), and then some of their applications are presented. Some recent results for Banach frames and atomic decompositions are sharpened with simple proofs. Banach spaces consisting of Bessel or Riesz sequences are introduced and it is shown that they are isometrically isomorphic to some Banach spaces of bounded linear operators, and that some subspaces of those Banach spaces are isometrically isomorphic to some Banach spaces of compact operators.

In this paper, we describe all maximal dissipative, maximal accretive and selfadjoint extensions of the minimal symmetric direct sum differential operators. Further using the equivalence of the Lax-Phillips scattering function and the Sz.-Nagy-Foiaş characteristic function we show that all eigen and associated functions of the maximal dissipative extension of the minimal symmetric direct sum operator are complete in $L_{w}^{2}(\Omega ),$ where $\Omega =\Omega _{1}\cup \Omega _{2},$ $\Omega _{1}=(0,c)$ and $\Omega _{2}=(c,\infty ).$

In this paper, we establish necessary and sufficient conditions for the solvability of the system of real quaternion matrix equations $$ \left\{ \begin{array} {l}{A}_{1}X=C_{1},~ \\ YB_{1}=D_{1}, \\ A_{2}Z=C_{2},ZB_{2}=D_{2},A_{3}ZB_{3}=C_{3}, \\ A_{4}X+YB_{4}+C_{4}ZD_{4}=E_{1}. \end{array}\right. $$ We also present an expression of the general solution to the system. The findings of this paper widely extend the known results in the literature.

In this note we describe some recent advances in the area of maximal function inequalities. We also study the behaviour of the centered Hardy--Littlewood maximal operator associated to certain families of doubling, radial decreasing measures, and acting on radial functions. In fact, we precisely determine when the weak type $(1,1)$ bounds are uniform in the dimension.