Properties of Chmielinski-orthogonality using Kadets-Klee property

The aim of this paper is to study new results of an approximate orthogonality of Birkhoff-James techniques in real Banach space , namely Chiemelinski orthogonality (even there is no ambiguity between the concepts symbolized by orthogonality) and provide some new geometric characterizations which is considered as the basis of our main definitions. Also, we explore relation between two different types of orthogonalities. First of them orthogonality in a real Banach space and the other orthogonality in the space of bounded linear operator . We obtain a complete characterizations of these two orthogonalities in some types of Banach spaces such as strictly convex space, smooth space and reflexive space. The study is designed to give different results about the concept symmetry of Chmielinski-orthogonality for a compact linear operator on a reflexive, strictly convex Banach space having Kadets-Klee property by exploring a new type of a generalized some results with Birkhoff James orthogonality in the space of bounded linear operators. We also exhibit a smooth compact linear operator with a spectral value that is defined on a reflexive, strictly convex Banach space having Kadets-Klee property either having zero nullity or not -right-symmetric.

On the Schrödinger Equation for Time-Dependent Hamiltonians with a Constant Form Domain

We study two seminal approaches, developed by B. Simon and J. Kisyński, to the well-posedness of the Schrödinger equation with a time-dependent Hamiltonian. In both cases, the Hamiltonian is assumed to be semibounded from below and to have a constant form domain, but a possibly non-constant operator domain. The problem is addressed in the abstract setting, without assuming any specific functional expression for the Hamiltonian. The connection between the two approaches is the relation between sesquilinear forms and the bounded linear operators representing them. We provide a characterisation of the continuity and differentiability properties of form-valued and operator-valued functions, which enables an extensive comparison between the two approaches and their technical assumptions.

On Numerical Radius Bounds Involving Generalized Aluthge Transform

In this paper, we establish some upper bounds of the numerical radius of a bounded linear operator S defined on a complex Hilbert space with polar decomposition S = U ∣ S ∣ , involving generalized Aluthge transform. These bounds generalize some bounds of the numerical radius existing in the literature. Moreover, we consider particular cases of generalized Aluthge transform and give some examples where some upper bounds of numerical radius are computed and analyzed for certain operators.

Some results about g-frames in Hilbert spaces

The concept of g-frame is a natural extension of the frame. This article mainly discusses the relationship between some special bounded linear operators and g-frames, and characterizes the properties of g-frames. In addition, according to the operator spectrum theory, the eigenvalues are introduced into the g-frame theory, and a new expression of the best frame boundary of the g-frame is given.

FURTHER INEQUALITIES FOR THE EUCLIDEAN OPERATOR RADIUS

By use of some non-negative Hermitian forms defined for n-tuple of bounded linear operators on the Hilbert space (H, h·, ·i) we establish new numerical radius and operator norm inequalities for sum of products of operators

On the justification of topological derivative for wave-based qualitative imaging of finite-sized defects in bounded media

PurposeThis work contributes to the general problem of justifying the validity of the heuristic that underpins medium imaging using topological derivatives (TDs), which involves the sign and the spatial decay away from the true anomaly of the TD functional. The author considers here the identification of finite-sized (i.e. not necessarily small) anomalies embedded in bounded media and affecting the leading-order term of the acoustic field equation.Design/methodology/approachTD-based imaging functionals are reformulated for analysis using a suitable factorization of the acoustic fields, which is facilitated by a volume integral formulation. The three kinds of TDs (single-measurement, full-measurement and eigenfunction-based) studied in this work are given expressions whose structure allows to establish results on their sign and decay properties. The latter are obtained using analytical methods involving classical identities on Bessel functions and Legendre polynomials, as well as asymptotic approximations predicated on spatial scaling assumptions.FindingsThe sign component of the TD imaging heuristic is found to be valid for multistatic experiments and if the sought anomaly satisfies a bound (on a certain operator norm) involving its geometry, its contrast and the operating frequency. Moreover, upon processing the excitation and data by applying suitably-defined bounded linear operatirs to them, the magnitude component of the TD imaging heuristic is proved under scaling assumptions where the anomaly is small relative to the probing region, the latter being itself small relative to the propagation domain. The author additionally validates both components of the TD imaging heuristic when the probing excitation is taken as an eigenfunction of the source-to-measurement operator, with a focusing effect analogous to that achieved in time-reversal based methods taking place. These findings extend those of earlier studies to the case of finite-sized anomalies embedded in bounded media.Originality/valueThe originality of the paper lies in the theoretical justifications of the TD-based imaging heuristic for finite-sized anomalies embedded in bounded media.

Analyticity of the Resolvent Function

Analytic dependence on a complex parameter appears at many places in the study of differential and integral equations. The display of analyticity in the solution of the Fredholm equation of the second kind is an early signal of the important role which analyticity was destined to play in spectral theory. The definition of the resolvent set is very explicit, this makes it seem plausible that the resolvent is a well behaved function. Let T be a closed linear operator in a complex Banach space X. In this paper we show that the resolvent set of T is an open subset of the complex plane and the resolvent function of T is analytic. Moreover, we show that if T is a bounded linear operator, the resolvent function of T is analytic at infinity, its value at infinity being 0 (where 0 is the bounded linear operator 0 in X). Consequently, we also show that if T is bounded in X then the spectrum of T is non-void.

Linear maps on ℬ(ℋ) preserving some operator properties

In this paper, for a complex Hilbert space ℋ with dim ℋ ≥ 2, we study the linear maps on ℬ(ℋ), the bounded linear operators on ℋ, that preserves projections and idempotents. As a result, we characterize the linear maps on ℬ(ℋ) that preserves involutions in both directions.

Perturbation Theory for Property (VE) and Tensor Product

Given a complex Banach space X, we investigate the stable character of the property (VE) for a bounded linear operator T:X→X, under commuting perturbations that are Riesz, compact, algebraic and hereditarily polaroid. We also analyze sufficient conditions that allow the transfer of property (VE) from the tensorial factors T and S to its tensor product.