Abstract: Abstract In this paper, we begin with the study of elements in \(C^*\)-algebras which are mapped to Schatten class ideals through the faithful left regular representation. We further give some functorial properties of Schatten classes on the category of representations of a \(C^*\)-algebra and category of unitary representations of a group. PubDate: 2020-03-04

Abstract: Abstract We obtain some necessary or sufficient conditions for an operator on a complex separable Hilbert space to be expressible as a product of two normal operators. For example, it is shown that if T is such a product, then \(\dim \ker T \ge \dim \ (\ker T^*\ \cap \ \mathrm {ran}\, T^*)\). On the other hand, any operator T satisfying \(\dim \ker T^* \ge \dim \ker T\) and \(\dim \ (\ker T\ \cap \ \overline{\mathrm {ran}\, T})\ge \dim \ (\ker T^*\ \cap \ \mathrm {ran}\, T^*\)) is a product of two normal operators. Such results complement our previous ones on the products of finitely many normal operators. We also obtain characterizations for products of two essentially normal operators. PubDate: 2020-03-02

Abstract: Abstract In this note, we get solutions focusing on integral inequalities of Hermite–Hadamard type and unusual attached to it, with \(\eta\)-G-pre invex functions by the way of fractional integral operator selected by Raina R. K. PubDate: 2020-02-19

Abstract: Abstract We study probability inequalities leading to tail estimates in a general semigroup \({\mathscr {G}}\) with a translation-invariant metric \(d_{\mathscr {G}}\). (An important and central example of this in the functional analysis literature is that of \({\mathscr {G}}\) a Banach space.) Using our prior work Khare and Rajaratnam (Ann Prob 45(6A):4101–4111, 2017) that extends the Hoffmann–Jørgensen inequality to all metric semigroups, we obtain tail estimates and approximate bounds for sums of independent semigroup-valued random variables, their moments, and decreasing rearrangements. In particular, we obtain the “correct” universal constants in several cases, extending results in the Banach space literature by Johnson et al. (Ann Prob 13(1):234-253, 1985), Hitczenko (Ann Prob 22(1):453–468, 1994), and Hitczenko and Montgomery-Smith (Ann Prob 29(1):447-466, 2001). Our results also hold more generally, in a very primitive mathematical framework required to state them: metric semigroups \({\mathscr {G}}\). This includes all compact, discrete, or (connected) abelian Lie groups. PubDate: 2020-02-18

Abstract: Abstract We apply several numerical radius inequalities to the Frobenius companion matrices of monic matrix polynomials to derive new bounds for the eigenvalues of these polynomials. One of our bounds improves upon an earlier bound due to Higham and Tisseur. PubDate: 2020-02-18

Abstract: Abstract In this paper, we investigate the existence of weak solution for a fractional type problems driven by a nonlocal operator of elliptic type in a fractional Orlicz–Sobolev space, with homogeneous Dirichlet boundary conditions. We first extend the fractional Sobolev spaces \(W^{s,p}\) to include the general case \(W^sL_A\), where A is an N-function and \(s\in (0,1)\). We are concerned with some qualitative properties of the space \(W^sL_A\) (completeness, reflexivity and separability). Moreover, we prove a continuous and compact embedding theorem of these spaces into Lebesgue spaces. PubDate: 2020-02-17

Abstract: Abstract In this article, we introduce binomial weighted sequence spaces \(b_p^{r,s}(w)\)\((1\le p<\infty )\), where \(w=(w_n)\) is a non-negative decreasing sequence of real numbers, and investigate some topological and inclusion properties of the new spaces. We give an upper estimation of \(\left\ A\right\ _{\ell _p(w),b_p^{r,s}(w)}\), where A is the Hausdorff matrix operator or Nörlund matrix operator. A Hardy type formula is established in the case of Hausdorff matrix operator. In the final section, we give an upper estimation for the transpose of Nörlund matrix as an operator from \(\ell _p\) to \(b_p^{r,s}\). PubDate: 2020-02-17

Abstract: Abstract In this paper, by virtue of an expression of matrix geometric means for positive semidefinite matrices via the Moore-Penrose inverse, we show matrix versions of the Hölder-McCarthy inequality, the Hölder inequality and quasi-arithmetic power means via matrix geometric means, and their reverses for positive definite matrices via the generalized Kantorovich constant. PubDate: 2020-02-13

Abstract: Abstract For a Banach algebra A and a semisimple commutative Banach algebra B, we prove that every 3-homomorphism \(\varphi :A\longrightarrow B\) is automatically continuous, and we generalize this result for almost 3-homomorphism. We also investigate the automatic continuity of almost 3-Jordan homomorphisms and show that if A is a commutative Banach algebra, then every almost 3-Jordan homomorphism \(\varphi :A\longrightarrow \mathbb {C}\) is automatically continuous. PubDate: 2020-02-05

Abstract: Abstract By means of a fixed point method we discuss the deformation of two-variable and multivariate operator means of positive definite matrices/operators. It is shown that the deformation of any operator mean in the Kubo–Ando sense becomes again an operator mean in the same sense. The operator means deformed by the weighted power means with two parameters are particularly examined. PubDate: 2020-02-05

Abstract: Abstract This paper is concerned with asymmetric truncated Toeplitz operators, which are compressions of multiplication operators acting between two model spaces. We give necessary and sufficient conditions so that the product of two asymmetric truncated Toeplitz operators is also an asymmetric truncated Toeplitz operator. Then we define analogue classes to the Sedlock classes, which consist on the algebras of truncated Toeplitz operators, and deal with the case of two inner functions such that one divides the other. Some examples of the product of rank-one asymmetric truncated Toeplitz operators involving inner functions are also given. PubDate: 2020-02-01

Abstract: Abstract The weighted Moore–Penrose inverse of a tensor via the Einstein product is introduced in the literature, very recently. The objective of this paper is to study the reverse-order law for the weighted Moore–Penrose inverse. In this context, a few properties of the weighted Moore–Penrose inverse of an arbitrary order tensor via the Einstein product are discussed. Some new sufficient conditions for the reverse-order law of even-order square tensors are obtained. Several characterizations of the reverse-order law for the weighted Moore–Penrose inverse of tensors via the same product are also presented. PubDate: 2020-02-01

Abstract: Abstract In this paper, we establish interior \(C^{1,\alpha }\) estimates for solutions of the linearized Monge–Ampère equation $$\begin{aligned} {\mathcal {L}}_{\phi }u:=\text {tr}[\varPhi D^2 u]=f, \end{aligned}$$where the density of the Monge–Ampère measure \(g:=\text {det}D^2\phi \) satisfies a VMO-type condition and \(\varPhi :=(\text {det}D^2\phi )(D^2\phi )^{-1}\) is the cofactor matrix of \(D^2\phi \). PubDate: 2020-02-01

Abstract: Abstract In this article, we study the relationship between p- (V) subsets and p-\( (V^{*})\) subsets of dual spaces. We investigate the Banach space X with the property that the adjoint of every p-convergent operator \( T:X\rightarrow Y \) is weakly q-compact, for every Banach space Y. Moreover, we define the notion of q-reciprocal Dunford–Pettis\( ^{*} \) property of order p on Banach spaces and obtain a characterization of Banach spaces with this property. Also, the stability of reciprocal Dunford–Pettis property of order p for projective tensor product is given. PubDate: 2020-02-01

Abstract: Abstract Let \({\mathbb {D}}\) denote the open unit disk in \({\mathbb {C}}\). For an integer \(n\ge 0\), let \(V_n\) be the space defined recursively by $$\begin{aligned} V_0=\big \{f:{\mathbb {D}}\rightarrow {\mathbb {C}}: f \text { analytic}, f(z) =O\big ((1- z )^{-1}\big )\big \}, \end{aligned}$$and for \(n\ge 1\), \(f\in V_n\) if and only if \(f'\in V_{n-1}\). In this work, we characterize the bounded and the compact weighted composition operators between \(V_n\) and the Bloch space and the space BMOA of analytic functions of bounded mean oscillation, respectively. We also show that the bounded (respectively, compact) weighted composition operators mapping the space BMOA into \(V_n\) are precisely the same as the bounded (respectively, compact) weighted composition operators mapping the Bloch space into \(V_n\). PubDate: 2020-02-01

Abstract: Abstract The continuous wavelet transform with rotations in higher dimensions is used to prove the regularity of distributions u under \(Qu = v\), where v is a distribution with compact support of class \(C^{\infty }(\mathbb {R}^n)\) in a neighborhood of some given point \(x = b_0\) in \(\mathbb {R}^n\) and where \(Q = \sum _{ \alpha = p} c_{\alpha }\partial ^{\alpha }\) is a linear partial differential operator of order \(p > 0\) with constant coefficients \(c_{\alpha }\), under the assumption that the continuous wavelet transform with rotations converges in a neighborhood of \(x = b_0\). PubDate: 2020-02-01

Abstract: Abstract Let (X, d) be a compact metric sapce and \(\mathrm{Lip}(X,d)\) denote the Banach space of all scalar-valued Lipschitz functions f on (X, d) endowed with the norm \(\Vert f\Vert _{X,L_{(X,d)}}=\max \{ \Vert f\Vert _X,L_{(X,d)}(f)\}\), where \(\Vert f\Vert _X=\sup \{ f(x) : x \in X \}\) and \(L_{(X,d)}(f)\) is the Lipschitz constant of f on (X, d). Applying the extreme point techniques, linear Lipschitz isometries between \(\mathrm{Lip}(X,d)\)-spaces have been characterized in Jiménez-Vargas and Villegas-Vallecillos (J Math 34:1165–1184, 2008). In this paper we introduce norm-attaining unit functions in Lipschitz spaces and apply this consept for characterizing into and onto linear Lipschitz isometries T from \(\mathrm{Lip}(X,d)\) to \(\mathrm{Lip}(Y,\rho )\). In particular, we generalize the result given by Jiménez-Vargas and Villegas-Vallecillos in the surjectivity case of T by omitting the nonvanishing condition for \( T1_{X} \) on Y. PubDate: 2020-02-01

Abstract: Abstract An extension of Chernoff’s product formula for one-parameter functions taking values in the space of bounded linear operators on a Banach space is given. Essentially, the n-th one-parameter function in the product formula is mapped by the n-th iterate of a contraction acting on the space of the one-parameter functions. The motivation to study this specific product formula lies in the growing field of dynamical control of quantum systems, involving the procedure of dynamical decoupling and also the Quantum Zeno effect. PubDate: 2020-02-01

Abstract: Abstract In this paper we state and prove a new compact embedding theorem for fractional Sobolev spaces with variable exponents. As a consequence, we obtain version of the Rellich–Kondrachov theorem in this setting. PubDate: 2020-02-01

Abstract: Abstract We focus in this paper on the study of the decomposition of partial isometries with finite ascent satisfying suitable conditions on the kernel sequence. We show that a propagation phenomena occurs for such partial isometries and we give a structure theorem for partial isometries with finite ascent. We also exhibit several examples illustrating different situations. PubDate: 2020-02-01