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In this paper, the functions u ∈ B V φ [ 0 , 1 ] u\in B{V}_{\varphi }\left[0,1] which define compact and Fredholm multiplication operators M u {M}_{u} acting on the space of functions of bounded φ \varphi -variation are studied. All the functions u ∈ B V φ [ 0 , 1 ] u\hspace{-0.08em}\in \hspace{-0.08em}B{V}_{\varphi }\left[0,\hspace{-0.08em}1] which define multiplication operators M u : B V φ [ 0 , 1 ] → B V φ [ 0 , 1 ] {M}_{u}:B{V}_{\varphi }\left[0,1]\to B{V}_{\varphi }\left[0,1] with closed range are characterized.

For 1 < p < ∞, the Besov space Bp consists of those functions f which are analytic in the unit disc 𝔻 = {z ∈ 𝔺 : |z| < 1} and satisfy ∫𝔻(1 − |z|2)p−2|f ′(z)|p dA(z) < ∞. The space B2 reduces to the classical Dirichlet space 𝒟. It is known that if f ∈ 𝒟then |f ′(reiθ)| = o[(1 − r)−1/2], for almost every ∈ [0, 2π]. Hallenbeck and Samotij proved that this result is sharp in a very strong sense. We obtain substitutes of the above results valid for the spaces Bp (1 < p < ∞) an we give also an application of our them to questions concerning multipliers between Besov spaces.

In this paper, we show that, for the rational inner function $\theta (z,w)=\frac{zw+\overline{b}w+\overline{c}z+\overline{d}}{1+bz+cw+dzw}$ , $S_{z}$ is reducible on the quotient module $\mathcal{K}_{\theta }=H^{2}\ominus \theta H^{2}$ over the bidisk if and only if $\theta $ is the product of two one-variable inner functions.

Fix $N\in\mathbb{N}$ , and assume that, for every $n\in\{1,\ldots,N\}$ , the functions $f_{n}\colon[0,1]\to[0,1]$ and $g_{n}\colon[0,1]\to\mathbb{R}$ are Lebesgue-measurable, $f_{n}$ is almost everywhere approximately differentiable with $|g_{n}(x)|\lt |f'_{n}(x)|$ for almost all $x\in[0,1]$ , there exists $K\in\mathbb{N}$ such that the set $\{x\in[0,1]:\operatorname{card}{f_{n}^{-1}(x)}\gt K\}$ is of Lebesgue measure zero, $f_{n}$ satisfy Luzin’s condition N, and the set $f_{n}^{-1}(A)$ is of Lebesgue measure zero for every set $A\subset\mathbb{R}$ of Lebesgue measure zero. We show that the formula $Ph=\sum_{n=1}^{N}g_{n}\cdot (h\circ f_{n})$ defines a linear and continuous operator $P\colon L^{1}([0,1])\to L^{1}([0,1])$ , and then we obtain results on the existence and uniqueness of solutions $\varphi\in L^{1}([0,1])$ of the equation $\varphi=P\varphi+g$ with a given $g\in L^{1}([0,1])$ .

We introduce the notion of $\vec{s}$ -numbers for operators in Banach couples. We investigate variants of the approximation, Gelfand, and Kolmogorov numbers. In particular, we derive upper estimates of these numbers for operators between spaces generated by interpolation functors on Banach couples satisfying interpolation variants of approximation properties. We also study two-sided interpolation of entropy numbers.

We investigate the convergence rate of the generalized Bochner–Riesz means $S_{R}^{\delta,\gamma}$ on $L^{p}$ -Sobolev spaces in the sharp range of $\delta$ and $p$ ( $p\geq2$ ). We give the relation between the smoothness imposed on functions and the rate of almost-everywhere convergence of $S_{R}^{\delta,\gamma}$ . As an application, the corresponding results can be extended to the $n$ -torus $\mathbb{T}^{n}$ by using some transference theorems. Also, we consider the following two generalized Bochner–Riesz multipliers, $(1-\vert \xi \vert ^{\gamma_{1}})_{+}^{\delta}$ and $(1-\vert \xi \vert ^{\gamma_{2}})_{+}^{\delta}$ , where $\gamma_{1}$ , $\gamma_{2}$ , $\delta$ are positive real numbers. We prove that, as the maximal operators of the multiplier operators with respect to the two functions, their $L^{2}(|x|^{-\beta})$ -boundedness is equivalent for any $\gamma_{1}$ , $\gamma_{2}$ and fixed $\delta$ .

We give a partial converse to [8, Theorem 1.3] (as a resolution of [2, Problem 8.4] for the quasiconformal $Q$-composition) for $Q_{0 \lt \alpha \lt 2^{-1}} (\mathbb{R}^{n \geq 2})$, and yet demonstrate that if $f : \mathbb{R}^2 \to \mathbb{R}^2$ is a homeomorphism then the boundedness of $u \mapsto u \circ f$ on $Q_{2^{-1} \lt \alpha \lt 1} (\mathbb{R}^2) \subset BMO (\mathbb{R}^2)$ yields the quasiconformality of $f$.

This paper concerns the analysis of the structure of bi-contractive projections on spaces of vector valued continuous functions and presents results that extend the characterization of bi-contractive projections given by the first author. It also includes a partial generalization of these results to affine and vector valued continuous functions from a Choquet simplex into a Hilbert space.

We characterize two-weight norm inequalities for potential type integral operators in terms of Sawyer-type testing conditions. Our result is stated in a space of homogeneous type with no additional geometric assumptions, such as group structure or non-empty annulus property, which appeared in earlier works on the subject. One of the new ingredients in the proof is the use of a finite collection of adjacent dyadic systems recently constructed by the author and T. Hyt\"onen. We further extend the previous Euclidean characterization of two-weight norm inequalities for fractional maximal functions into spaces of homogeneous type., Comment: 33 pages, v8 (some typos corrected; clarified the relationship between the different constants present in the several steps of the proof of the main result; Lemma 6.18 modified; examples of spaces and operators included; fixed some technical details; Definition 2.14 and Lemma 2.15 modified; Lemma 6.17 corrected; measures allowed with point masses; some imprecise arguments clarified)

We correct an error in Part (c) of Proposition 3.3 in our paper "Some remarks on substitution and composition operators" [Rend. Istit. Mat. Univ. Trieste 53 (2021), Art. No. 6, pp. 1-25].

In this paper we study the linear substitution operator Sϕ(f) := f ◦ ϕ generated by some function ϕ : [0, 1] → [0, 1], as well as the nonlinear composition operator Cg(f) := g ◦ f generated by some function g : R → R. We will show that these operators have a very different (and sometimes quite surprising) behavior in the space of con-tinuous functions, Lipschitz functions, functions of bounded variation, and Baire class one functions. A main emphasis is put on examples and counterexamples which illustrate this behavior.

Let $\mathbb B_n$ be the open unit ball in $\mathbb C^n$. We characterize the spectra of pointwise multipliers $M_u$ acting on Banach spaces of analytic functions on $\mathbb B_n$ satisfying some general conditions. These spaces include Bergman-Sobolev spaces $A^p_{\alpha,\beta}$, Bloch-type spaces $\mathcal B_\alpha$, weighted Hardy spaces $H^p_w$ with Muckenhoupt weights and Hardy-Sobolev Hilbert spaces $H^2_\beta$. Moreover, we describe the essential spectra of multipliers in most of the aforementioned spaces, in particular, in those spaces for which the set of multipliers is a subset of the ball algebra., Comment: 19 pages. To appear in Taiwanese J. Math

For each [math] , the [math] -Bloch space consists of all analytic functions [math] on the unit disk satisfying [math] . We consider the following complex integral operators, namely the [math] -Cesàro operator ¶ C β ( f ) ( z ) = ∫ 0 z f ( w ) w ( 1 − w ) β d w ¶ and its generalization, acting from the [math] -Bloch space to itself, where [math] and [math] . We investigate the boundedness and compactness of the [math] -Cesàro operators and their generalizations. Also we calculate the essential norm and spectrum of these operators.

We introduce a convolution form, in terms of integration over the unit disc D, for operators on functions f in H(D), which correspond to Taylor expansion multipliers. We demonstrate advantages of the introduced integral representation in the study of mapping properties of such operators. In particular, we prove the Young theorem for Bergman spaces in terms of integrability of the kernel of the convolution. This enables us to refer to the introduced convolutions as Hadamard-Bergman convolution. Another, more important, advantage is the study of mapping properties of a class of such operators in Holder type spaces of holomorphic functions, which in fact is hardly possible when the operator is defined just in terms of multipliers. Moreover, we show that for a class of fractional integral operators such a mapping between Holder spaces is onto. We pay a special attention to explicit integral representation of fractional integration and differentiation. Russian Foundation for Fundamental ResearchRussian Foundation for Basic Research (RFBR) [18-01-00094-a] Russian Foundation for Basic ResearchRussian Foundation for Basic Research (RFBR) [18-01-00094-a, 19-01-00223-a] info:eu-repo/semantics/publishedVersion

We introduce the generalized weighted Morrey spaces and the generalized weighted weak Morrey spaces on RD-spaces. As applications, the boundedness of the Hardy–Littlewood maximal operator and the Calderón–Zygmund operator is established.

Let $C^{(\mathbf{n})}=C^{(\mathbf{n})}(\mathbb{D}\times \mathbb{D})$ be a Banach space of complex valued functions $f(x,y)$ that are continuous on the closed bidisc $\overline{\mathbb{D}\times \mathbb{D}}$ , where $\mathbb{D}=\{z\in \mathbb{C}:|z|\lt 1\}$ is the unit disc in the complex plane $\mathbb{C}$ and has $n$ th partial derivatives in $\mathbb{D}\times \mathbb{D}$ which can be extended to functions continuous on $\overline{\mathbb{D}\times \mathbb{D}}$ . The Duhamel product is defined on $C^{(\mathbf{n})}$ by the formula \begin{equation*}(f\circledast g)(z,w)=\frac{\partial ^{2}}{\partial z\partial w}\int _{0}^{z}\int _{0}^{w}f(z-u,w-v)g(u,v)\,dv\,du.\end{equation*} In the present paper we prove that $C^{(\mathbf{n})}(\mathbb{D}\times\mathbb{D})$ is a Banach algebra with respect to the Duhamel product $\circledast$ . This result extends some known results. We also investigate the structure of the set of all extended eigenvalues and extended eigenvectors of some double integration operator $W_{zw}$ . In particular, the commutant of the double integration operator $W_{zw}$ is also described.

Classical parabolic Riesz and parabolic Bessel type potentials are interpreted as negative fractional powers of the differential operators (−△+∂∕∂t) and (I−△+∂∕∂t). Here, △ is the Laplacian and I is the identity operator. We introduce some generalizations of these potentials, namely, we define the family of operators Aβ,𝜃α=(𝜃I+(−△)β∕2+∂∕∂t)−α for 𝜃≥0 and α,β>0, and investigate its behavior in the framework of Lp(ℝn+1)-spaces.

We define some metrics on the set of all bounded normal composition operators in [math] , and show that these metrics are equivalent with the metric induced by the usual operator norm.

Conditional heteroskedastic financial time series are commonly modelled by ARCH and GARCH. ARCH(1) and GARCH processes were recently extended to the function spaces C[0,1] and L2[0,1], their probabilistic features were studied and their parameters were estimated. The projections of the operators on finite-dimensional subspace were estimated, as were the complete operators in GARCH(1,1). An explicit asymptotic upper bound for the estimation errors was stated in ARCH(1). This article provides sufficient conditions for the existence of strictly stationary solutions, weak dependence and finite moments of ARCH and GARCH processes in various Lp[0,1] spaces, C[0,1] and other spaces. In L2[0,1] we deduce explicit asymptotic upper bounds of the estimation errors for the shift term and the complete operators in ARCH and GARCH and for the projections of the operators on a finite-dimensional subspace in ARCH. The operator estimaton is based on Yule-Walker equations. The estimation of the GARCH operators also involves a result concerning the estimation of the operators in invertible, linear processes which is valid beyond the scope of ARCH and GARCH. Through minor modifications, all results in this article regarding functional ARCH and GARCH can be transferred to functional ARMA.

Let $T$ be an operator and suppose that there exists a positive constant $C$ such that $$\left(\int_I|Tf(x)|^q\, dx\right)^{1/q}\leq C\left(\int_I|f(x)|^q\, dx\right)^{1/q}$$ for every $q$ which is near enough to $1$ and for every interval $I$ in $\mathbb{R}$ and $f\in L^{\infty}(\mathbb{R})$. Then we show that $T$ maps $uL^{\infty}$ to ${\rm{BMO}}_u$.

In this paper, we studied the abstract versions of Korovkin type approximation theorems via statistical relative $\mathcal{A-}$Summation process in modular spaces for double sequences. Then, we discuss the results which are obtained by special choice of the scale function and the matrix sequences and we give an application that shows our results are stronger than studied before. Finally, we study an extension to non-positive linear operators.

In this paper, we give a boundedness criterion for the potential operator ℐ α { {\mathcal I} }^{\alpha } in the local generalized Morrey space L M p , φ { t 0 } ( Γ ) L{M}_{p,\varphi }^{\{{t}_{0}\}}(\text{Γ}) and the generalized Morrey space M p , φ ( Γ ) {M}_{p,\varphi }(\text{Γ}) defined on Carleson curves Γ \text{Γ} , respectively. For the operator ℐ α { {\mathcal I} }^{\alpha } , we establish necessary and sufficient conditions for the strong and weak Spanne-type boundedness on L M p , φ { t 0 } ( Γ ) L{M}_{p,\varphi }^{\{{t}_{0}\}}(\text{Γ}) and the strong and weak Adams-type boundedness on M p , φ ( Γ ) {M}_{p,\varphi }(\text{Γ}) .

We show that in a scale of weighted Dirichlet spaces $$D_{\alpha }$$, including the Bergman space, given any finite Blaschke product B there exists an equivalent norm in $$D_{\alpha }$$ such that B satisfies the wandering subspace property with respect to such norm. This extends, in some sense, previous results by Carswell et al. (Indiana Univ Math J 51(4):931–961, 2002). As a particular instance, when $$B(z)=z^k$$ and $$|\alpha | \le \frac{\log (2)}{\log (k+1)}$$, the chosen norm is the usual one in $$D_\alpha $$.

We obtain the Lipschitz boundedness for a class of fractional multilinear operators $I_{\Omega,\alpha}^{A,m}$ with rough kernels $\Omega\in L_{s}(\mathbb S^{n-1})$, $s>n/(n-\alpha)$ on the local generalized Morrey spaces $LM_{p,\varphi}^{\{x_0\}}$, generalized Morrey spaces $M_{p,\varphi}$ and vanishing generalized Morrey spaces $VM_{p,\varphi}$, where the functions $A$ belong to homogeneous Lipschitz space $\dot{\Lambda}_{\beta}$, $0

We investigate pointwise convergence of entangled ergodic averages of Dunford-Schwartz operators $T_0,T_1,\ldots, T_m$ on a Borel probability space. These averages take the form \[ \frac{1}{N^k}\sum_{1\leq n_1,\ldots, n_k\leq N} T_m^{n_{\alpha(m)}}A_{m-1}T^{n_{\alpha(m-1)}}_{m-1}\ldots A_2T_2^{n_{\alpha(2)}}A_1T_1^{n_{\alpha(1)}} f, \] where $f\in L^p(X,\mu)$ for some $1\leq p, Comment: 16 pages

We characterize disjointness of supercyclic operators which map a holomorphic function to a partial sum of the Taylor expansion. In particular, we show that disjoint hypercyclicity equals disjoint supercyclicity for families of Taylor-type operators. Moreover, we give a sufficient condition to yield the disjoint supercyclicity for families of Taylor-type operators.

We obtain estimates for the essential norm of the composition operators acting on the general spaces $H_{\omega,p}$ of Hardy spaces. Our characterization is given in terms of generalized Nevanlinna counting functions.

In this paper, we consider impulsive integro-differential equations in Banach space and we establish the bound on the difference between two approximate solutions. We also discuss nearness and convergence of solutions of the problem under consideration. The impulsive integral inequality of Grownwall type is used to obtain results.

We consider complex dynamical systems showing metastable behavior, but no local separation of fast and slow time scales. The article raises the question of whether such systems exhibit a low-dimensional manifold supporting its effective dynamics. For answering this question, we aim at finding nonlinear coordinates, called reaction coordinates, such that the projection of the dynamics onto these coordinates preserves the dominant time scales of the dynamics. We show that, based on a specific reducibility property, the existence of good low-dimensional reaction coordinates preserving the dominant time scales is guaranteed. Based on this theoretical framework, we develop and test a novel numerical approach for computing good reaction coordinates. The proposed algorithmic approach is fully local and thus not prone to the curse of dimension with respect to the state space of the dynamics. Hence, it is a promising method for data-based model reduction of complex dynamical systems such as molecular dynamics.

This paper deals with the structure of contractive and bi-contractive projections on spaces of continuous functions defined on a compact and Hausdorff topological space.

In this paper,we study the properties of the multiplication operator acting on the bounded variation space BV[0, 1]. In particular,we show the existence of non-null compact multiplication operators on BV[0, 1] and non-invertible Fredholm multiplication operators on BV[0, 1].

We establish necessary and sufficient conditions for the one-dimensional differential Hardy inequality to hold, including the overdetermined case. The solution is given in terms different from those of the known results. Moreover, the least constant for this inequality is estimated.

Let us consider the following two norms in the vector space \,$\mathcal{P}$ \,of all complex polynomials: $$ \|p\|_{D_{r}}:= \sup\{|p(z)|: |z

By using the idea of $q$-calculus, we generalize the conic domain and define various subclasses of analytic functions which map the open unit disk $$ \mathbb {U}= \{ z:z\in \mathbb {C} \text { and } \lvert z \rvert \lt 1 \} $$ onto this generalized conic domain. We study here some conspicuous results such as: coefficients estimates, sufficient condition and convolution properties for these newly defined classes. We consider various corollaries and consequences of our main results. Most of our results are connected with earlier works related to this field.

Let $I$ be a countably infinite index set, and let $X$ be a Banach sequence space over $I$ . In this article, we characterize the disjoint hypercyclic and supercyclic weighted pseudoshift operators on $X$ in terms of the weights, the OP-basis, and the shift mappings on $I$ . Also, the shifts on weighted $L^{p}$ spaces of a directed tree and the operator weighted shifts on $\ell ^{2}(\mathbb{Z},\mathcal{K})$ are investigated as special cases.

In this paper we study Hausdorff operators on the Bergman spaces $A^{p}(\mathbb{U})$ of the upper half plane., 14 pages

A complete characterization of nearly-invariant subspaces of finite defect for the backward shift operator acting on the Hardy space is provided in the spirit of Hitt and Sarason's theorem. As a corollary we describe the almost-invariant subspaces for the shift and its adjoint., 11 pages. Revised version. To appear in J. Operator Theory

A necessary and sufficient condition for the core function $\varphi \in L^{1}(0,1)$ is established in order to define a bounded cordial Volterra integral operator $(V_{\varphi }u)(t)=\int _{0}^{t}t^{-1}\varphi (t^{-1}s)u(s)\,ds$ in a space $L^{p}(0,T)$ for a $p\in [1,\infty ]$. This condition implies the boundedness of $V_{\varphi }$ also in the Sobolev spaces $W^{m,p}(0,T)$, $m\geq 1$. The spectra of $V_{\varphi }$ in the spaces $L^{p}(0,T)$ and $W^{m,p}(0,T)$ are determined and the spectral properties of $V_{\varphi }$ are examined in $L^{p}(0,T)$.

[EN] We study the dynamics of the multi-dimensional Cesar degrees integral operator on L-P (I-n), for I the unit interval, 1 < p < infinity, and n >= 2, that is defined as C(f)(x(1),...,x(n)) = 1/x(1)x(2)...x(n) integral(x1)(0) ... integral(x1)(0) f(u(1),...,u(n))du(1)...du(n) for f is an element of L-p(I-n). This operator is already known to be bounded. As a consequence of the Eigenvalue Criterion, we show that it is hypercyclic as well. Moreover, we also prove that it is Devaney chaotic and frequently hypercyclic., The first author was supported by MEC, grant MTM201675963-P. The third author was supported by grant MTM2015-65825-P.

Every composition operator $C_{\varphi}$ on the Lipschitz space $\operatorname{Lip}_0(X)$ attains its norm. This fact is essentially known and we give in this paper a sequential characterization of the extremal functions for the norm of $C_{\varphi}$ on $\operatorname{Lip}_0(X)$. We also characterize the norm-attaining composition operators $C_{\varphi}$ on the little Lipschitz space $\operatorname{lip}_0(X)$ which separates points uniformly and identify the extremal functions for the norm of $C_{\varphi}$ on $\operatorname{lip}_0(X)$. We deduce that compact composition operators on $\operatorname{lip}_0(X)$ are norm-attaining whenever the sphere unit of $\operatorname{lip}_0(X)$ separates points uniformly. In particular, this condition is satisfied by spaces of little Lipschitz functions on Holder compact metric spaces $(X,d^{\alpha})$ with $0 \lt \alpha \lt 1$.

In this paper, we give some estimates of the essential norm for generalized weighted composition operators from $H^\infty $ to the logarithmic Bloch space. Moreover, we give a new characterization for the boundedness, compactness and essential norm of the generalized weighted composition operator from $H^\infty $ to the logarithmic Bloch space.

We establish spherical variants of the Gleason-Kahane-Zelazko and Kowalski-Slodkowski theorems, and we apply them to prove that every weak-2-local isometry between two uniform algebras is a linear map. Among the consequences, we solve a couple of problems posed by O. Hatori, T. Miura, H. Oka, and H. Takagi in 2007., L. Li was partly supported by NSF of China project no. 11301285. A. M. Peralta was partially supported by the Spanish Ministry of Economy and Competitiveness and European Re- gional Development Fund project no. MTM2014-58984-P and Junta de Andalucía grant FQM375. L. Wang was partly supported by NSF of China Grants No. 11371222, 11871303, and 11671133. Y.-S. Wang was partly supported by Taiwan MOST 104-2115-M-005-001-MY2.

[EN] Consider two continuous linear operators T: X-1 (mu) -> Y-1 (nu) and S: X-2 (mu) -> Y-2 (nu) between Banach function spaces related to different sigma-finite measures mu and nu. By means of weighted norm inequalities we characterize when T can be strongly factored through S, that is, when there exist functions g and h such that T(f) = gS(hf) for all f is an element of X-1 (mu). For the case of spaces with Schauder basis, our characterization can be improved, as we show when S is, for instance, the Fourier or Cesar operator. Our aim is to study the case where the map T is besides injective. Then we say that it is a representing operator-in the sense that it allows us to represent each element of the Banach function space X (mu) by a sequence of generalized Fourier coefficients-providing a complete characterization of these maps in terms of weighted norm inequalities. We also provide some examples and applications involving recent results on the Hausdorff-Young and the Hardy-Littlewood inequalities for operators on weighted Banach function spaces., The first author gratefully acknowledge the support of the Ministerio de Economia y Competitividad (project #MTM2015-65888-C4-1-P) and the Junta de Andalucia (project FQM-7276), Spain. The second author was supported by National Science Centre, Poland, project no. 2015/17/B/ST1/00064. The third author acknowledges with thanks the support of the Ministerio de Economia y Competitividad (project MTM2016-77054-C2-1-P), Spain.

[EN] This paper deals with bilinear operators acting in pairs of Banach function spaces that factor through the pointwise product. We find similar situations in different contexts of the functional analysis, including abstract vector lattices¿orthosymmetric maps, C¿-algebras¿zero product preserving operators, and classical and harmonic analysis¿integral bilinear operators. Bringing together the ideas of these areas, we show new factorization theorems and characterizations by means of norm inequalities. The objective of the paper is to apply these tools to provide new descriptions of some classes of bilinear integral operators, and to obtain integral representations for abstract classes of bilinear maps satisfying certain domination properties., The first author was supported by TUBITAK-The Scientific and Technological Research Council of Turkey, Grant No. 2211/E. The second author was supported by Ministerio de Economia y Competitividad (Spain) and FEDER, Grant MTM2016-77054-C2-1-P.

Let $X,Y$ be compact Hausdorff spaces and $A,B$ be either closed subspaces of $C(X)$ and $C(Y)$, respectively, containing constants or positive cones of such subspaces. In this paper we study surjections $T:A \longrightarrow B$ satisfying the norm condition $\|T(f) T(g) -1 \|_Y=\|fg-1\|_X$ for all $f,g \in A$, where $\|\cdot\|_X$ and $\|\cdot\|_Y$ denote the supremum norms. We show that under a mild condition on the strong boundary points of $A$ and $B$ (and the assumption $T(i)=i T(1)$ in the subspace case), the map $T$ is a weighted composition operator on the set of strong boundary points of $B$. This result is an improvement of the known results for uniform algebra case to closed linear subspaces and their positive cones.

Recently we have introduced a product-type operator and studied it on some spaces of analytic functions on the unit disc. Here we start investigating the operator on the space of analytic functions on the upper half-plane. We characterize the boundedness and compactness of the operator between Hardy and α-Bloch spaces on the domain.

In this note complex symmetric composition operators $C_\varphi$ on the Bergman space $A^2(\mathbb{D})$ are studied. It is shown that if an operator $C_\varphi$ is complex symmetric on $A^2(\mathbb{D})$ then either $\varphi\colon \mathbb{D}\to\mathbb{D}$ has a Denjoy--Wolff point in $\mathbb{D}$ or is an elliptic automorphism of the disc. Moreover in the latter case $\varphi$ is either a rotation or has an order smaller than six.

The aim of this article is to obtain common fixed points of set mappings defined on a family of sets endowed with a graph satisfying $\Theta-$contraction. Our results generalize and extend various results in the existing literature. We also provide some non trivial examples to support the validity of our main results. Some applications to the construction of common fixed points of set mappings in $\varepsilon$-chainable metric space are also discussed.

Wiener-type variation spaces, also known as $\operatorname{BV}_{p}$ -spaces ( $1\leq p\lt \infty$ ), are complete normed linear spaces. A function $g$ is called a multiplier from $\operatorname{BV}_{p}$ to $\operatorname{BV}_{q}$ if the pointwise multiplication $fg$ belongs to $\operatorname{BV}_{q}$ for each $f\in\operatorname{BV}_{p}$ . In this article, we characterize the multipliers from $\operatorname{BV}_{p}$ to $\operatorname{BV}_{q}$ for the cases $1\leq q\lt p$ and $1\leq p\leq q$ .

In this article, we investigate the surjective linear isometries between the differentiable function spaces $C_{0}^{p}(X,E)$ and $C_{0}^{q}(Y,F)$ , where $X$ , $Y$ are open subsets of $\mathbb{R}$ and $E$ , $F$ are strictly convex Banach spaces with dimension greater than $1$ . We show that such isometries can be written as weighted composition operators.