Abstract: We study the well-posedness of the third order degenerate differential equations with infinite delay$$(P_3): (Mu)'''(t) + (Lu)''(t) + (Bu)'(t)= Au(t) + \int _{-\infty }^t a(t-s)Au(s)ds + f(t){\text{ on }}[0, 2\pi ]$$in Lebesgue–Bochner spaces \(L^p(\mathbb{T};\; X)\) and periodic Besov spaces \(B_{p,\,q}^s(\mathbb{T};\; X)\), where A, B, L and M are closed linear operators on a Banach space X satisfying \(D(A)\subset D(B)\cap D(L)\cap D(M)\) and \( a\in L^1(\mathbb{R}_+)\). Using known operator-valued Fourier multiplier theorems, we give necessary and sufficient conditions for \((P_3)\) to be \(L^p\)-well-posed(or \(B_{p,q}^s\)-well-posed). Concrete examples are also given to support our main abstract results. PubDate: 2020-03-05

Abstract: Abstract The purpose of this paper is to provide a set of sufficient conditions so that the normalized form of the Fox–Wright functions have certain geometric properties like close-to-convexity, univalency, convexity and starlikeness inside the unit disc. In particular, we study some geometric properties for some class of functions related to the generalized hypergeometric functions. PubDate: 2020-02-18

Abstract: Abstract Let \(L_M\) be an Orlicz function space endowed with the Orlicz norm or the Luxemburg norm, and let X be a Banach space. In this paper we characterize the non-\(l_n^{(1)}\) point and the uniformly non-\(l_{n}^{(1)}\) point of Orlicz–Bochner function space \(L_M(\mu ,X)\). As the immediate consequences some criteria for non-square point and uniformly non-square point of \(L_M(\mu ,X)\) are obtained. PubDate: 2020-02-12

Abstract: Abstract In this manuscript, we transport the classical Operator Theory on complex Banach spaces to normed modules over absolutely valued rings. In some cases, we are able to extend classical results on complex Banach spaces to normed modules over normed rings. In order to make sure that bounded linear maps on normed modules coincide with the continuous linear maps, it is sufficient that the underlying ring be practical (a topological ring is practical if the invertibles approach zero). In order for other classical results to work on the scope of normed modules, it is sufficient that the underlying ring be normalizing. Normalizing rings is a new class of rings introduced in this manuscript. We provide a characterization of such rings as well as nontrivial examples. PubDate: 2020-02-12

Abstract: Abstract An important problem stemming from perturbation theory concerns description and understanding of operator \(\theta \)-Hölder functions. This article presents a survey of recent developments concerning operator \(\theta \)-Hölder functions with respect to symmetric quasi-norms. PubDate: 2020-02-12

Abstract: Abstract Let \({{\mathscr {M}}}\) be a \(II_1\) factor acting on the Hilbert space \({{\mathscr {H}}}\), and \({\mathscr {M}} _{\text {aff}}\) be the Murray–von Neumann algebra of closed densely-defined operators affiliated with \({{\mathscr {M}}}\). Let \(\tau \) denote the unique faithful normal tracial state on \(\mathscr {M}\). By virtue of Nelson’s theory of non-commutative integration, \({\mathscr {M}} _{\text {aff}}\) may be identified with the completion of \({{\mathscr {M}}}\) in the measure topology. In this article, we show that \(M_n({\mathscr {M}} _{\text {aff}}) \cong M_n({{\mathscr {M}}})_{\text {aff}}\) as unital ordered complex topological \(*\)-algebras with the isomorphism extending the identity mapping of \(M_n({{\mathscr {M}}}) \rightarrow M_n({{\mathscr {M}}})\). Consequently, the algebraic machinery of rank identities and determinant identities are applicable in this setting. As a step further in the Heisenberg–von Neumann puzzle discussed by Kadison–Liu (SIGMA Symmetry Integrability Geom. Methods Appl., 10:Paper 009, 40, 2014), it follows that if there exist operators P, Q in \({\mathscr {M}} _{\text {aff}}\) satisfying the commutation relation \(Q \; {{\hat{\cdot }}} \;P \; {\hat{-}} \;P \; {{\hat{\cdot }}} \;Q = {i\,}I\), then at least one of them does not belong to \(L^p({{\mathscr {M}}}, \tau )\) for any \(0 < p \le \infty \). Furthermore, the respective point spectrums of P and Q must be empty. Hence the puzzle may be recasted in the following equivalent manner - Are there invertible operators P, A in \({{\mathscr {M}}}_{\text {aff}}\) such that \(P^{-1} \; {{\hat{\cdot }}} \;A \; {{\hat{\cdot }}} \;P = I \; {\hat{+}} \;A\)' This suggests that any strategy towards its resolution must involve the study of conjugacy invariants of operators in \({{\mathscr {M}}}_{\text {aff}}\) in an essential way. PubDate: 2020-02-04

Abstract: Abstract We analyze f-frequently hypercyclic, q-frequently hypercyclic (\(q> 1\)), and frequently hypercyclic \(C_{0}\)-semigroups (\(q=1\)) defined on complex sectors,with values in separable infinite-dimensional Fréchet spaces. Some structural results are given for a general class of \({\mathcal F}\)-frequently hypercyclic \(C_{0}\)-semigroups, as well. We investigate generalized frequently hypercyclic translation semigroups and generalized frequently hypercyclic semigroups induced by semiflows on weighted function spaces. Several illustrative examples are presented. PubDate: 2020-02-04

Abstract: Abstract We establish weighted extrapolation theorems in classical and grand Lorentz spaces. As a consequence we have the weighted boundedness of operators of Harmonic Analysis in grand Lorentz spaces. We treat both cases: diagonal and off-diagonal ones. PubDate: 2020-01-31

Abstract: Abstract We study Birkhoff-James orthogonality and isosceles orthogonality of bounded linear operators between Hilbert spaces and Banach spaces. We explore Birkhoff-James orthogonality of bounded linear operators in light of a new notion introduced by us and also discuss some of the possible applications in this regard. We also study isosceles orthogonality of bounded (positive) linear operators on a Hilbert space and some of the related properties, including that of operators having disjoint support. We further explore the relations between Birkhoff-James orthogonality and isosceles orthogonality in a general Banach space. PubDate: 2020-01-24

Abstract: Abstract We consider subspaces of Morrey spaces defined in terms of various vanishing properties of functions. Such subspaces were recently used to describe the closure of \(C_0^\infty ({\mathbb {R}^n})\) in Morrey norm. We show that these subspaces are invariant with respect to some classical operators of harmonic analysis, such as the Hardy–Littlewood maximal operator, singular type operators and Hardy operators. We also show that the vanishing properties defining those subspaces are preserved under the action of Riesz potential operators and fractional maximal operators. PubDate: 2020-01-24

Abstract: Abstract A generalization of the products of composition, multiplication and differentiation operators is the Stević–Sharma operator \(T_{u_1,u_2,\varphi }\), defined by \(T_{u_1,u_2,\varphi }f=u_1\cdot f\circ \varphi +u_2\cdot f'\circ \varphi \), where \(u_1,u_2,\varphi \) are holomorphic functions on the unit disk \({\mathbb {D}}\) in the complex plane \({\mathbb {C}}\) and \(\varphi ({\mathbb {D}})\subset {\mathbb {D}}\). We are interested in the difference of Stević–Sharma operators which has never been considered so far. In this paper, we characterize its boundedness, compactness and order boundedness between Banach spaces of holomorphic functions. As an important special case, we obtain the above characterizations of the difference of weighted composition operators. Furthermore, we show the equivalence of order boundedness and Hilbert-Schmidtness for the difference of composition operators between Hardy or weighted Bergman spaces. PubDate: 2020-01-24

Abstract: Abstract On the basis of the characterizations of the boundedness and compactness of the Volterra type operator \(I_{g, \varphi }\) from mixed-norm spaces \(H(p,\, q,\, \phi )\) to Zygmund spaces \( \mathcal {Z}\), the authors provide a function-theoretic estimate for the essential norm of Volterra type operator \(I_{g, \varphi }\) by means of the definition of the essential norm of an operator and the properties of the analytic function. An estimate for the essential norm of the generalized integration operator $$\begin{aligned} I^{(n)}_{g, \varphi }f(z)=\int ^{z}_{0}f^{(n)}(\varphi (\xi ))g(\xi )d\xi , \ z\in \mathbb {D}, \end{aligned}$$from Bloch-type spaces to F(p, q, s) spaces is also obtained. PubDate: 2020-01-01

Abstract: Abstract In this paper we establish the subelliptic Picone type identities. As consequences, we obtain Hardy and Rellich type inequalities for anisotropic p-sub-Laplacians which are operators of the form $$\begin{aligned} {\mathcal {L}}_{p}f:= \sum _{i=1}^{N} X_i\left( X_i f ^{p_i-2} X_i f \right) ,\quad 1<p_i<\infty , \end{aligned}$$where \(X_i\), \(i=1,\ldots , N\), are the generators of the first stratum of a stratified (Lie) group. Moreover, analogues of Hardy type inequalities with multiple singularities and many-particle Hardy type inequalities are obtained on stratified groups. PubDate: 2020-01-01

Abstract: Abstract In this article, we introduce an explicit parallel algorithm for finding a common element of zeros of the sum of two accretive operators and the set of fixed point of a nonexpansive mapping in the framework of Banach spaces. We prove its strong convergence under some mild conditions. Finally, we provide some applications to the main result. The results presented in this paper extend and improve the corresponding results in the literature. PubDate: 2020-01-01

Abstract: Abstract In this paper, we study the existence of positive solutions for classes of nonlinear operator equations in Banach spaces ordered by cones. We establish new coincidence point theorems via a generalized monotone iterative method. As applications, we discuss the existence of positive solutions for systems of nonlinear matrix equations and a system of integral equations of Volterra type. PubDate: 2020-01-01

Abstract: Abstract We study the lattice of closed ideals in the algebra of continuous linear operators acting on pth Tandori and \(p'\)th Cesàro sequence spaces, \(1\leqslant p<\infty \), which we show are isomorphic to the classical sequence spaces \((\oplus _{n=1}^\infty \ell _\infty ^n)_p\) and \((\oplus _{n=1}^\infty \ell _1^n)_{p'}\), respectively. We also show that Tandori sequence spaces are complemented in certain Lorentz sequence spaces, and that the lattice of closed ideals for certain other Lorentz and Garling sequence spaces has infinite cardinality. PubDate: 2020-01-01

Abstract: Abstract We give a description of 2-local real isometries between \(C(X,\tau )\) and \(C(Y,\eta )\) where X and Y are compact Hausdorff spaces, X is also first countable and \(\tau \) and \(\eta \) are topological involutions on X and Y, respectively. In particular, we show that every 2-local real isometry T from \(C(X,\tau )\) to \( C(Y,\eta )\) is a surjective real linear isometry whenever X is also separable. PubDate: 2020-01-01

Abstract: Abstract We study, among others, upper, lower, upper modified and lower modified n-th von Neumann–Jordan constant and relationships between them. There are characterized uniformly non-\(l_{n}^{1}\) Banach spaces in terms of the upper modified n-th von Neumann–Jordan constant. Moreover, this constant is calculated explicitly for Lebesgue spaces \(L^{p}\) and \(l^{p}\)\((1\le p\le \infty ).\) Finally, it is shown that the sequence of n-th upper and modified upper von Neumann–Jordan constants for the space \(L^p\) as well as \(l^p\)\((2<p<\infty )\) converges to \(B_p^2\), where \(B_p\) is the best type (2, p) constant in the Khinthine inequality for the case \(2\le p<\infty \). PubDate: 2020-01-01

Abstract: Abstract Let \(\mathbb {Y}\) be either an Orlicz sequence space or a Marcinkiewicz sequence space. We take advantage of the recent advances in the theory of factorization of the identity carried on by Lechner (Stud Math 248(3):295–319, 2019) to provide conditions on \(\mathbb {Y}\) that ensure that, for any \(1\le p\le \infty \), the infinite direct sum of \(\mathbb {Y}\) in the sense of \(\ell _p\) is a primary Banach space. This way, we enlarge the list of Banach spaces that are known to be primary. PubDate: 2020-01-01

Abstract: Abstract The aim of this paper is to characterize the behavior of boundary limits of functions in Hardy space on angular domains. We investigate that the Cauchy integral of a function \(f\in L^{p}(\partial D_a)\) is in \(H^{p}(D_a).\) We also prove that the functions in \(H^{p}(D_a)\) are the Cauchy integral of their non-tangential boundary limits. In addition, we establish the orthogonality of non-tangential boundary limits of functions in \(H^{p}(D_a)\) and \(H^{q}(D_a)\). PubDate: 2020-01-01