Abstract: Abstract This note is devoted to a weaker version of Bishop property \(\beta\) at a given complex number. We show in particular that this notion is a regularity and hence the induced spectrum satisfies all classical properties of the spectrum. PubDate: 2020-02-25

Abstract: Abstract For each analytic map \(\psi\) on the complex plane \(\mathbb {C}\), we study the Ritt’s resolvent growth condition for the composition operator \(C_{\psi} :f \rightarrow f\circ \psi\) on the Fock space \({\mathcal {F}}_2\). We show that \(C_{\psi}\) satisfies such a condition if and only if it is either compact or reduces to the identity operator. As a consequence, it is shown that the Ritt’s resolvent condition and the unconditional Ritt’s condition for \(C_{\psi}\) are equivalent. PubDate: 2020-02-17

Abstract: Abstract We introduce a concept of complete proximal normal structure and used to investigate the existence of a best proximity point for an arbitrary family of cyclic relatively nonexpansive mappings in the setting of strictly convex Banach spaces. We also prove that every bounded, closed and convex pair in uniformly convex Banach spaces as well as every compact and convex pairs in Banach spaces has complete proximal normal structure. Furthermore, we consider a class of cyclic relatively nonexpansive mappings in the sense of Suzuki and establish a new best proximity point theorem in the setting of Hilbert spaces. PubDate: 2020-02-17

Abstract: Abstract On the unit ball in the n dimensional complex Euclidean space, we introduce a new weighted Bergman space and Toeplitz operators. By Berezin transform and average functions, we estimate the essential norms of the Toeplitz operators. PubDate: 2020-02-17

Abstract: Abstract In this article, we investigate some classes of dominated orthogonally additive operators in lattice-normed spaces. We say that an orthogonally additive operator T from a lattice-normed space (V, E) to a lattice-normed space (W, F) is completely additive if, for every order summable family of pairwise disjoint elements \((v_{\alpha })_{\alpha \in \Lambda }\) of V, the family \((Tv_{\alpha })_{\alpha \in \Lambda }\) is order summable in W. The first part of the article is devoted to completely additive operators. We prove that a dominated orthogonally additive operator \(T:V\rightarrow W\) from a Banach–Kantorovich space V to a Banach–Kantorovich space W is completely additive if and only if so is its exact dominant \(\left\bracevert T\right\bracevert :E\rightarrow F\). One of the our main results asserts that an orthogonally additive map defined on a lateral ideal and dominated by a positive completely additive operator can be extended to a dominated completely additive operator defined on the whole space. In the second part of the article, we consider C-compact dominated orthogonally additive operators. We show that the C-compactness of a dominated orthogonally additive operator \(T:V\rightarrow W\) from a decomposable lattice-normed space (V, E) to an order continuous Banach lattice W implies the C-compactness of its exact dominant \(\left\bracevert T\right\bracevert :E\rightarrow W\). PubDate: 2020-02-14

Abstract: Abstract In this paper, we aim to introduce the notion of the spectral radius of bounded linear operators acting on a complex Hilbert space \(\mathcal {H}\), which are bounded with respect to the seminorm induced by a positive operator A on \(\mathcal {H}\). Mainly, we show that \(r_A(T)\le \omega _A(T)\) for every A-bounded operator T, where \(r_A(T)\) and \(\omega _A(T)\) denote respectively the A-spectral radius and the A-numerical radius of T. This allows to establish that \(r_A(T)=\omega _A(T)=\Vert T\Vert _A\) for every A-normaloid operator T, where \(\Vert T\Vert _A\) is denoted to be the A-operator seminorm of T. Moreover, some characterizations of A-normaloid and A-spectraloid operators are given. PubDate: 2020-02-14

Abstract: Abstract Let \({\mathcal {H}}\), \({\mathcal {K}}\) be Hilbert spaces over \({\mathbb {F}}\) with \(\dim {\mathcal {H}}\ge 3\), where \({\mathbb {F}}\) is the real or complex field. Assume that \(\varphi :{B}({\mathcal {H}})\rightarrow {B}({\mathcal {K}})\) is an additive surjective map and \(r\ge 3\) is a positive integer. It is shown that \(\varphi \) is r-nilpotent perturbation of scalars preserving in both directions if and only if either \(\varphi (A)=cTAT^{-1}+g(A)I\) holds for every \(A\in {B}({\mathcal {H}})\); or \(\varphi (A)=cTA^{*}T^{-1}+g(A)I\) holds for every \(A\in {B}({\mathcal {H}})\), where \(0\not =c\in {{\mathbb {F}}}\), \(T:{\mathcal {H}}\rightarrow {\mathcal {K}}\) is a \(\tau \)-linear bijective map with \(\tau :{\mathbb {F}}\rightarrow {\mathbb {F}}\) an automorphism and g is an additive map from \( B({\mathcal {H}})\) into \({{\mathbb {F}}}\). As applications, for any integer \(k\ge 5\), additive k-commutativity preserving maps and general completely k-commutativity preserving maps on \({B}({\mathcal {H}})\) are characterized, respectively. PubDate: 2020-02-13

Abstract: Abstract In this paper, on \({\mathbb {D}}\) we define Cowen–Douglas function introduced by the Cowen–Douglas operator \(M_\phi ^*\) on Hardy space \({\mathcal {H}}^2({\mathbb {D}})\), moreover, we give a necessary and sufficient condition to determine when \(\phi\) is a Cowen–Douglas function, where \(\phi \in {\mathcal {H}}^\infty ({\mathbb {D}})\) and \(M_{\phi }\) is the associated multiplication operator on \({\mathcal {H}}^{2}({\mathbb {D}})\). Then, we give some applications of Cowen–Douglas function on chaos, such as its application on the inverse problem of chaos for \(\phi (T)\), where \(\phi\) is a Cowen–Douglas function and T is the backward shift operator on the Hilbert space \({\mathcal {L}}^2({\mathbb {N}})\). PubDate: 2020-02-13

Abstract: Abstract In this work, we describe well-defined dissipative boundary conditions related with a singular third-order differential equation in lim-3 case at singular point. Using the characteristic function of the corresponding dissipative operator we introduce a completeness theorem. PubDate: 2020-02-13

Abstract: Abstract This paper is concerned with general \(n\times n\) upper-triangular operator matrices with given diagonal entries. The characterizations of perturbations of their left (right) essential spectrum and essential spectrum are given, based on the space decomposition technique. Moreover, some sufficient and necessary conditions are given under which the left (right) essential spectrum and the essential spectrum of such operator matrix, respectively, coincide with the union of the left (right) essential spectrum and the essential spectrum of its diagonal entries. PubDate: 2020-02-05

Abstract: Abstract The present paper deals with the A-statistical approximation processes of the general class of integral type linear positive operators including many well-known operators in the \(L_{p}\)-metric spaces. PubDate: 2020-01-31

Abstract: Abstract We derive some inequalities involving first four central moments of discrete and continuous distributions. Bounds for the eigenvalues and spread of a matrix are obtained when all its eigenvalues are real. Likewise, we discuss bounds for the roots and span of a polynomial equation. PubDate: 2020-01-31

Abstract: Abstract We study inductive limit in the category of ternary ring of operator (TRO). The existence of inductive limit in this category is proved and its behaviour with quotienting is discussed. For a TRO V, if A(V) is the linking \(C^{*}\)-algebra generated by V, then we investigate whether it commutes with inductive limits of TROs, in the sense that if \((V_n,f_n)\) is an Inductive system then \(\varinjlim A(V_n)=A(\varinjlim V_n)\). We show that some local properties such as simplicity, nuclearity and exactness behaves well with the inductive limit of TROs. We also discuss the commutativity of inductive limit of TROs with tensor products. PubDate: 2020-01-31

Abstract: Abstract Let \(B_s(H)\) denote the set of all bounded selfadjoint operators acting on a separable complex Hilbert space H of dimension \(\ge 2\). Also, let \({\mathcal {S}}{\mathcal {A}}_s(H)\) (esp. \({\mathcal {I}}{\mathcal {A}}_s(H)\)) denote the class of all singular (resp. invertible) algebraic operators in \(B_s(H)\). Assume \({\varPhi }:B_s(H)\rightarrow B_s(H)\) is a unital additive surjective map such that \({\varPhi }({\mathcal {S}}{\mathcal {A}}_s(H))={\mathcal {S}}{\mathcal {A}}_s(H)\) (resp. \({\varPhi }({\mathcal {I}}{\mathcal {A}}_s(H))={\mathcal {I}}{\mathcal {A}}_s(H)\)). Then \({\varPhi }(T)=\tau T\tau ^{-1}~\forall T\in B_s(H)\), where \(\tau\) is a unitary or an antiunitary operator. In particular, \({\varPhi }\) preserves the order \(\le\) on \(B_s(H)\) which was of interest to Molnar (J Math Phys 42(12):5904–5909, 2001). PubDate: 2020-01-16

Abstract: Abstract Let A be an infinite dimensional simple unital stably finite C*-algebra and B be a large subalgebra of A. In this paper, we show that B has local weak comparison if A has local weak comparison, and A has local weak comparison if \(M_{2}(B)\) has local weak comparison. As a consequence, we are able to prove that A has weak comparison if and only if B has weak comparison. PubDate: 2020-01-15

Abstract: Abstract In the present paper, we discuss the approximation properties of certain link integral modification of Ismail–May operators. We point out here that the operators of Ismail–May can also be derived from the Jain operators. We also establish some direct convergence estimates including error, difference estimates and an asymptotic formula in simultaneous approximation. In the end, we indicate through graphical representation the convergence of polynomial function with the link operators. PubDate: 2020-01-13

Abstract: Abstract We obtain two different expressions for the weighted core–EP inverse of the sum \(A+E\) for a given operator A and a perturbation operator E, under different conditions on the perturbing operator. As a consequence of these results, we get some relations between idempotents determined by the weighted core–EP inverses of a given operator and its perturbation and we present perturbation bounds for the weighted core–EP inverse. The application to the solution of a perturbed equation is given. We also apply these results to obtain perturbation results for the core–EP inverse. PubDate: 2020-01-01

Abstract: Abstract This paper is devoted to study the almost convergent sequence space \(\widehat{c}(\varPhi )\) derived by the Euler totient matrix. It is proved that the space \(\widehat{c}(\varPhi )\) and the space of all almost convergent sequences are linearly isomorphic. Further, the \(\beta \)-dual of the space \(\widehat{c}(\varPhi )\) is determined and Euler totient core of a complex-valued sequence has been defined. Finally, inclusion theorems related to this new type of core are obtained. PubDate: 2020-01-01

Abstract: Abstract We study Weyl-type perturbation theorems in the context of closed linear relations. General results on perturbations for dissipative relations are established. In the particular case of selfadjoint relations, we treat finite-rank perturbations and carry out a detailed analysis of the corresponding changes in the spectrum. PubDate: 2020-01-01

Abstract: Abstract The boundedness and the compactness of weighted composition operators from the minimal Möbius invariant space into n-th weighted-type spaces are studied in this paper. Moreover, some estimates for the essential norm of weighted composition operators are also investigated. PubDate: 2020-01-01