Let $T$ be a bounded, linear operator on a complex, separable, infinite dimensional Hilbert space $H$. We assume that $T$ is an essentially isometric (resp. normal) operator, that is, $I_{H}-T^{\ast }T$ (resp. $TT^{\ast }-T^{\ast }T)$ is compact. For the compactness of $S$ from the commutant of $T,$ some necessary and sufficient conditions are found on $S.$ Some related problems are also discussed.

Let $C_{A}^{\left( n\right) }:=C_{A}^{\left( n\right) }\left( \mathbb{D}\times\mathbb{D}\right) $ denote the subspace of functions in the Banach space $C^{\left( n\right) }\left( \overline{\mathbb{D}% \times\mathbb{D}}\right) $ which are analytic in the bi-disc $\mathbb{D}% \times\mathbb{D}$. We consider the subspace $B_{zw}$ consisting from the functions $f\in C_{A}^{\left( n\right) }$ which can be represented in the form $f\left( z,w\right) =g\left( zw\right) ,$ where $g$ is a single variable function from the disc algebra $C_{A}\left( \mathbb{D}\right) $. We prove that $B_{zw}$ is a Banach algebra under the Duhamel multiplication \[ \left( f\circledast g\right) \left( zw\right) =\frac{\partial^{2}% }{\partial z\partial w}\underset{0}{\overset{z}{\int}}\underset{0}{\overset {w}{\int}}f\left( \left( z-u\right) \left( w-v\right) \right) g\left( uv\right) dvdu \] and describe its maximal ideal space. We also consider the Hardy type operator $f\rightarrow xy\underset{0}{\overset{x}{\int}}\underset{0}{\overset{y}{\int}% }f\left( t\tau\right) d\tau dt$ and discuss its some properties.

To a generalized tight continuous frame in a Hilbert space $\mathcal{H}$, indexed by a locally compact space $\Sigma$ endowed with a Radon measure, one associates a coorbit theory converting spaces of functions on $\Sigma$ in spaces of vectors comparable with $\mathcal{H}$. If the continuous frame is provided by the action of a suitable family of bounded operators on a fixed window, a symbolic calculus emerges, assigning operators in $\mathcal{H}$ to functions on $\Sigma$. We give some criteria of relative compactness for sets and for families of compact operators, involving tightness properties in terms of objects canonically associated to the frame. Particular attention is dedicated to a magnetic version of the pseudodifferential calculus.

Under suitable assumptions on the delay operator $F$, we give necessary and sufficient conditions for the inhomogeneous abstract degenerate delay equations: $ (Mu)'(t)=Au(t)+Fu_{t}+f(t), \ (t\in \T)$ to have $L^p$-maximal regularity.

We consider pseudoquotient extensions of positive linear functionals on a commutative Banach algebra $\mathcal{A}$ and give conditions under which the constructed space of pseudoquotients can be identified with all Radon measures on the structure space $\hat{\mathcal{A}}$.

For a compact $K$, a necessary condition for $C(K)$ to have the Controlled Separable Complementation Property is that $K$ be monolithic. In this paper, we prove that when $K$ contains no copy of $[0,\omega^\omega]$ and the set of points which admit a countable neighborhood base is a cofinite subset of $K$, then monolithicity of $K$ is sufficient for $C(K)$ to enjoy the Controlled Separable Complementation Property. We also show that, for this type of compacta $K$, the space $C(K)$ is separably extensible.

The main purpose of this paper is to characterize norm-additive in modulus, not necessarily linear, maps defined between function algebras (not necessarily unital or uniformly closed). In fact, for function algebras $A$ and $B$ on locally compact Hausdorff spaces $X$ and $Y$, respectively, we study surjections $T, S:A\longrightarrow B$ satisfying $\||Tf|+|Sg|\|_Y= \||f|+|g|\|_X$ for all $f,g\in A$.

We present vector-valued versions of two theorems due to A. Jimenez-Vargas, by showing that, if $B(X,E)$ and $B(Y,F)$ are certain vector-valued Banach algebras of continuous functions and $T:B(X,E)\to B(Y,F)$ is a separating linear operator, then $\widehat{T}:\widehat{B(X,E)}\to \widehat{B(Y,F)}$, defined by $\widehat{T}\hat{f}=\widehat{Tf}$, is a weighted composition operator, where $\widehat{Tf}$ is the Gelfand transform of $Tf$. Furthermore, it is shown that, under some conditions, every bijective separating map $T:B(X,E)\to B(Y,F)$ is biseparating and induces a homeomorphism between the character spaces $M(B(X,E))$ and $M(B(Y,F))$. In particular, a complete description of all biseparating, or disjointness preserving linear operators between certain vector-valued Lipschitz algebras is provided. In fact, under certain conditions, if the bijections $T:Lip^{\alpha}(X,E)\to Lip^{\alpha}(Y,F)$ and $T^{-1}$ are both disjointness preserving, then $T$ is a weighted composition operator in the form $Tf(y)=h(y)(f(\phi(y))),$ where $\phi$ is a homeomorphism from $Y$ onto $X$ and $h$ is a map from $Y$ into the set of all linear bijections from $E$ onto $F$. Moreover, if $T$ is multiplicative then $M(E)$ and $M(F)$ are homeomorphic.

We investigate generalized amenability, contractibility, biprojectivity and biflatness properties of various classes of abstract Segal algebras with respect to the Banach algebra $\mathcal{A}$. Moreover, we verify some of the previous available results about Segal algebras, for abstract Segal algebras.

We give a characterization of convex functions in terms of difference among values of a function. As an application, we propose an estimation of operator monotone functions: If $A \geq B \ge 0$, $A-B$ is invertible and $f$ is operator monotone on $(0, \infty)$, then $ f(A) - f(B) \ge f(\|B\|+ \epsilon) - f(\|B\|) > 0$, where $\epsilon = \|(A-B)^{-1}\|^{-1}$. Moreover it gives a simple proof to Furuta's theorem: If $\log A$ is strictly greater than $\log B$ for invertibel operators $A, \ B \geq 0$ and $f$ is operator monotone on $(0, \infty)$, then there exists a positive number $\beta$ such that $ f(A^\alpha)$ is strictly greater than $f(B^\alpha)$ for all positive numbers $ \alpha \le \beta$.

We will find the $p\alpha-$dual for $X_{T}$, where $X$ is one of the spaces $c$, $c_{0}$, $\ell_{\infty}$ and $T$ is a triangle matrix. This will be achieved in two ways: firstly, under some conditions for the inverse matrix $S$ of $T$ and secondly, for arbitrary triangles $T$.

We prove that a dictionary for a Banach space $X$ has the strong coefficient quantization property if it has the same property when it restricted on the unit ball of $X$. We also obtain the same result for the strong net quantization property.

The Lupaş $q$-transform emerges in the study of the limit $q$-Lupaş operator. This transform is closely connected to the theory of positive linear operators of approximation theory, the $q$-boson operator calculus, the methods of summation of divergent series, and other areas. Given $q\in (0,1), \;f\in C[0,1],$ the Lupaş $q$-transform of $f$ is defined by: $$(\Lambda_qf)(z):=\frac{1}{(-z;q)_\infty}\cdot\sum_{k=0}^\infty\frac{f(1-q^k)q^{k(k-1)/2}} {(q;q)_k} z^k,$$ where $$(a;q)_k:=\prod_{j=0}^{k-1}\left(1-aq^j\right),\; (a;q)_\infty:=\prod_{j=0}^{\infty}\left(1-aq^j\right),\;\;k\in {\Bbb N}_0,\;a\in {\Bbb C}.$$ The analytical and approximation properties of $\Lambda_q$ have already been examined. In this paper, some properties of the Lupaş $q$-transform related to continuous linear operators in normed linear spaces are investigated.

The generalization of the Bernstein polynomials based on Polya distribution is considered in the present article. Here, we introduce a mixed summation-integral type operators having Polya and Bernstein basis functions in summation and integration respectively. We establish some direct results which include an asymptotic formula, local and global approximation results for these operators in terms of modulus of continuity.

The paper outlines a covariant theory of operators related to groups and homogeneous spaces. A methodical use of groups and their representations allows to obtain results of algebraic and analytical nature. The consideration is systematically illustrated by a representative collection of examples.

For any $\lambda\geq1$, $\mathbb{R}^2_\lambda$ is Banaś-Frączek space, the exact value of the Jordan-von Neumann constant $C_{NJ}(\mathbb{R}^2_\lambda)$ is investigated. By careful calculations, $C_{NJ}(\mathbb{R}^2_\lambda)=2-\frac{1}{\lambda^{2}}$ is given.

Let $\mathbb{D}$ be the open unit disk with its boundary $\partial\mathbb{D}$ in the complex plane $\mathbb{C}$ and $dA(z)=\frac{1}{\pi}dx\, dy,$ the normalized area measure on $\mathbb{D}.$ Let $L_{a}^{2}(\mathbb{D}, dA)$ be the Bergman space consisting of analytic functions on $\mathbb{D}$ that are also in $L^2(\mathbb{D}, dA).$ In this paper we obtain certain distance estimates for bounded linear operators defined on the Bergman space.

Let $\{a_n\}_{n=1}^\infty$ be a sequence of group invertible elements of a unital $C^*$ algebra $\mathcal{A}$ that converges to $a$. We present some equivalent conditions for the group invertibility of $a$ and for the convergence of $\{a_n^\#\}_{n=1}^\infty$ to $a^\#$.

A new geometric notion on a nonempty and convex pair of subsets of a convex metric space $X$, called semi-normal structure, is introduced and used to investigate the existence of best proximity pairs for a new class of mappings, called strongly noncyclic relatively C-nonexpansive. We also study the structure of minimal sets of strongly noncyclic relatively C-nonexpansive mappings in the setting of convex metric spaces.

We investigate the class $\mathcal{B}^{loc}(\mathbb{R}^{n})$ of exponents $p(\cdot)$ for which the local Hardy-Littlewood maximal operator is bounded in variable exponent Lebesgue spaces $L^{p(\cdot)}(\mathbb{R}^{n})$. Littlewood-Paley square function characterization of $L^{p(\cdot)}(\mathbb{R}^{n})$ spaces with the above class of exponent are also obtained.