Volume 30, Issue 6

The focus of this paper is to extend some stability results of the analysis of perturbed operators on the Banach space to the case of Fredholm perturbation theory in a Banach algebra relative to a bounded homomorphism. Our approach is to define some subsets in a Banach algebra to enrich the theory of perturbed left-right Browder and left-right Weyl elements in a Banach algebra relative to a bounded homomorphism.

Let { Λ i } i ∈ ℐ {\{\Lambda_{i}\}_{i\in\mathcal{I}}} be a g-frame for 𝒰 {\mathcal{U}} with respect to { 𝒱 i } i ∈ ℐ {\{\mathcal{V}_{i}\}_{i\in\mathcal{I}}} , where 𝒰 {\mathcal{U}} and { 𝒱 i } i ∈ ℐ {\{\mathcal{V}_{i}\}_{i\in\mathcal{I}}} are Hilbert spaces. The excess of { Λ i } i ∈ ℐ {\{\Lambda_{i}\}_{i\in\mathcal{I}}} is the largest cardinal number of the subsets 𝒥 {\mathcal{J}} of ℐ {\mathcal{I}} such that { Λ i } i ∈ ℐ ∖ 𝒥 {\{\Lambda_{i}\}_{i\in\mathcal{I\setminus J}}} is a g-frame for 𝒰 {\mathcal{U}} with respect to { 𝒱 i } i ∈ ℐ ∖ 𝒥 {\{\mathcal{V}_{i}\}_{i\in\mathcal{I\setminus J}}} . In this paper, we consider the excess of a g -frame and introduce the concept of m -uniform excess for g -frames. Also, we present constructions of m -uniform excess g -frames. Finally, we investigate the relationship between m -uniform excess g -frames and woven g -frames.

Our aim in this paper is to determine when a partially isometric matrix is normal. However, we do not restrict ourselves to the finite-dimensional case and we describe when a partial isometry in ℬ ⁢ ( ℋ ) {\mathcal{B}(\mathcal{H})} satisfies several strong and weak normal properties. In particular, we give elegant characterisations of normal partial isometries on infinite-dimensional Hilbert Spaces in terms of generalized inverses; we collect some spectral properties and explore when an operator in ℬ ⁢ ( ℋ ) {\mathcal{B}(\mathcal{H})} is similar to a normal partial isometry. We close the paper by treating when a partial isometry is hyponormal.

We propose to study the convergence of an iterative method used for solving non-linear least-square problems having differentiable as well as non-differentiable functions. We use the ω-condition on both first order divided difference of non-differentiable part and first order derivative of differentiable part to establish the condition for convergence of the method. We also present some numerical experiments as test beds for the proposed method. In all the numerical examples, we have compared our results with a well-known Gauss–Newton–Potra method and shown that our convergence analysis gives better error bounds.

Let 𝒜 {\mathcal{A}} and ℬ {\mathcal{B}} be two von Neumann algebras. In this paper, we show that a bijective unital map Φ : 𝒜 → ℬ {\Phi:\mathcal{A}\to\mathcal{B}} which preserves the bi-skew Jordan product is an additive ∗ {\ast} -isomorphism. Moreover, we show that Φ is a linear ∗ {\ast} -isomorphism or a conjugate linear ∗ {\ast} -isomorphism.

In this paper, quasi-statical boundary contact problems of couple-stress viscoelasticity for inhomogeneous anisotropic bodies with regard to friction are investigated. We prove the uniqueness theorem of weak solutions using the corresponding Green’s formulas and positive definiteness of the potential energy. To analyze the existence of solutions, we equivalently reduce the problem under consideration to a spatial variational inequality. We consider a special parameter-dependent regularization of this variational inequality which is equivalent to the relevant regularized variational equation depending on a real parameter, and study its solvability by the Galerkin approximate method. Some a priori estimates for solutions of the regularized variational equation are established and with the help of an appropriate limiting procedure, the existence theorem for the original contact problem with friction is proved.

This paper is a continuation of the so far performed studies on the concept of uniform statistical convergence. We first characterize two inequalities concerning the uniform statistical limit superior that lead to two core inclusion results for bounded real sequences. Using the inverse Fourier transformation, we also give a criterion on uniform statistical convergence, and study some factorization results of the space of uniformly statistically convergent sequences. These results are used to give a Korovkin-type approximation theorem.

We consider the optimal control problem of a non-linear system of differential inclusions with fast-oscillating parameters on semi-axes. Using the averaging method, we find an approximate solution for the optimal control of non-linear differential inclusions with fast-oscillating coefficients on a semi-axes. Thus, we prove the convergence of the optimal controls of the initial control problem to the optimal process of the averaged problem and the convergence of the corresponding cost functionals.

In this paper, we are concerned with positive solutions for the Dirichlet boundary value problem for equations and systems of Kirchhoff type. We obtain existence and localization results of positive solutions using Krasnosel’skiĭ’s fixed point theorem in cones and a weak Harnack-type inequality. The localization is given in terms of energy norm, being of interest from a physical point of view. In the case of systems, the results on the localization are established componentwise using the vector version of Krasnosel’skiĭ’s theorem, which allows some of the equations of the system to satisfy the compression condition and others the expansion one.

The reversible property of rings was introduced by Cohn and has important generalizations in noncommutative ring theory. In this paper, reversibility of rings is investigated in relation with quasinilpotents and idempotents, and our argument is spread out based on this. We call a ring R Qnil e -reversible if for any a , b ∈ R {a,b\in R} , being a ⁢ b = 0 {ab=0} implies b ⁢ a ⁢ e ∈ R qnil {bae\in R^{\rm qnil}} for a prescribed idempotent e ∈ R {e\in R} , where R qnil {R^{\rm qnil}} denotes the set of all quasinilpotent elements of R . In the first, we determine the set R qnil {R^{\rm qnil}} for some classes of rings to investigate the structure of Qnil e -reversible rings. In the second, we use R qnil {R^{\rm qnil}} to define Qnil e -reversibility of rings. The notion of Qnil e -reversible ring is a proper generalization of that of e -semicommutative ring, Qnil-semicommutative ring, e -reversible ring and right (left) quasi-duo ring. We obtain some relations between a ring and its quotient rings in terms of Qnil e -reversibility. Applications via some ring extensions and examples illustrating our results are provided.

The aim of this paper is to present the weighted generalized Moore–Penrose inverse of an operator between two Hilbert spaces as an extension of the Moore–Penrose inverse and the generalized Moore–Penrose inverse defined for an operator on a Hilbert space. Basic properties, characterizations and representations of the weighted generalized Moore–Penrose inverses are established. We extend some known results and give several new results for the generalized Moore–Penrose inverse. Applying the weighted generalized Moore–Penrose inverse, the solvability of some linear equations as well as general solution forms are obtained.

Let R be a commutative ring with A ⁢ ( R ) {A(R)} its set of annihilating-ideals. The extended annihilating-ideal graph of R , denoted by AG ¯ ⁢ ( R ) {\overline{\mathrm{AG}}(R)} , is an undirected graph with vertex set A ⁢ ( R ) * = A ⁢ ( R ) ∖ { 0 } {A(R)^{*}=A(R)\setminus\{0\}} and two vertices I 1 {I_{1}} and I 2 {I_{2}} are adjacent if and only if I 1 m ⁢ I 2 n = 0 {I_{1}^{m}I_{2}^{n}=0} with I 1 m ≠ 0 {I_{1}^{m}\neq 0} and I 2 n ≠ 0 {I_{2}^{n}\neq 0} , for some positive integers m and n . In this paper, we first study some basic properties of AG ¯ ⁢ ( R ) {\overline{\mathrm{AG}}(R)} and then we investigate the relationship between the extended annihilating-ideal graph AG ¯ ⁢ ( R ) {\overline{\mathrm{AG}}(R)} , the annihilator-ideal graph A I ⁢ ( R ) {A_{I}(R)} and the annihilating-ideal graph AG ⁢ ( R ) {\mathrm{AG}(R)} of a commutative ring R .

In this paper, we discuss umbral nature of 2-variable Hermite polynomials and obtain triple lacunary generating function for this polynomial by using the umbral method. Further, the lacunary generating function for 3-variable Hermite polynomials are also obtained.

In this paper, the orbits of observable physical quantities of position and momentum operators at the states of quantum Hilbert spaces are created. Also the Hilbert space of finite orbits and the Fréchet–Hilbert space of all orbits are created and the orbital operators corresponding to these observable operators in these spaces of orbits are defined and studied. We call this process the orbitization, and the obtained model the orbital quantum mechanics. The orbitization process is compared with the quantization process. The generalization of well-known canonical commutation relations for orbital operators corresponding to the position and momentum operators is established. Also, the Heisenberg uncertainty principle for the orbital operators is proved and the question of achieving equality in the Heisenberg inequality is considered.