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 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.

Generalized Kantorovich modifications of positive linear operators

<p style='text-indent:20px;'>Starting with a positive linear operator we apply the Kantorovich modification and a related modification. The resulting operators are investigated. We are interested in the eigenstructure, Voronovskaya formula, the induced generalized convexity, invariant measures and iterates. Some known results from the literature are extended.</p>

Idempotent Fourier multipliers acting contractively on $$H^p$$ spaces

We describe the idempotent Fourier multipliers that act contractively on $$H^p$$ H p spaces of the d-dimensional torus $$\mathbb {T}^d$$ T d for $$d\ge 1$$ d ≥ 1 and $$1\le p \le \infty $$ 1 ≤ p ≤ ∞ . When p is not an even integer, such multipliers are just restrictions of contractive idempotent multipliers on $$L^p$$ L p spaces, which in turn can be described by suitably combining results of Rudin and Andô. When $$p=2(n+1)$$ p = 2 ( n + 1 ) , with n a positive integer, contractivity depends in an interesting geometric way on n, d, and the dimension of the set of frequencies associated with the multiplier. Our results allow us to construct a linear operator that is densely defined on $$H^p(\mathbb {T}^\infty )$$ H p ( T ∞ ) for every $$1 \le p \le \infty $$ 1 ≤ p ≤ ∞ and that extends to a bounded operator if and only if $$p=2,4,\ldots ,2(n+1)$$ p = 2 , 4 , … , 2 ( n + 1 ) .

The fragmentation equation with size diffusion: Well posedness and long-term behaviour

The dynamics of the fragmentation equation with size diffusion is investigated when the size ranges in $(0,\infty)$ . The associated linear operator involves three terms and can be seen as a nonlocal perturbation of a Schrödinger operator. A Miyadera perturbation argument is used to prove that it is the generator of a positive, analytic semigroup on a weighted $L_1$ -space. Moreover, if the overall fragmentation rate does not vanish at infinity, then there is a unique stationary solution with given mass. Assuming further that the overall fragmentation rate diverges to infinity for large sizes implies the immediate compactness of the semigroup and that it eventually stabilizes at an exponential rate to a one-dimensional projection carrying the information of the mass of the initial value.

Projecting onto rectangular matrices with prescribed row and column sums

In 1990, Romero presented a beautiful formula for the projection onto the set of rectangular matrices with prescribed row and column sums. Variants of Romero’s formula were rediscovered by Khoury and by Glunt, Hayden, and Reams for bistochastic (square) matrices in 1998. These results have found various generalizations and applications.In this paper, we provide a formula for the more general problem of finding the projection onto the set of rectangular matrices with prescribed scaled row and column sums. Our approach is based on computing the Moore–Penrose inverse of a certain linear operator associated with the problem. In fact, our analysis holds even for Hilbert–Schmidt operators, and we do not have to assume consistency. We also perform numerical experiments featuring the new projection operator.

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.

Permutable Symmetric Hadamard Matrices in Quaternion Algebra and Engineering Applications

In this paper, aiming to develop the group and out-of-group formalization of the symmetry concept, the preservation of a matrix symmetry after row permutation is considered by the example of the maximally permutable \emph{normalized} Hadamard matrices which row and column elements are either plus or minus one. These matrices are used to extend the additive decomposition of a linear operator into symmetric and skew-symmetric parts using several commuting operations of the Hermitian conjugation type, for the quaternionic generalization of a vector cross product, as well as for creating educational puzzles and other applications.

Identification of Matrix Diffusion Coefficient in a Parabolic PDE

An inverse problem of identifying the diffusion coefficient in matrix form in a parabolic PDE is considered. Following the idea of natural linearization, considered by Cao and Pereverzev (2006), the nonlinear inverse problem is transformed into a problem of solving an operator equation where the operator involved is linear. Solving the linear operator equation turns out to be an ill-posed problem. The method of Tikhonov regularization is employed for obtaining stable approximations and its finite-dimensional analysis is done based on the Galerkin method, for which an orthogonal projection on the space of matrices with entries from L 2 ⁢ ( Ω ) L^{2}(\Omega) is defined. Since the error estimates in Tikhonov regularization method rely heavily on the adjoint operator, an explicit representation of adjoint of the linear operator involved is obtained. For choosing the regularizing parameter, the adaptive technique is employed in order to obtain order optimal rate of convergence. For the relaxed noisy data, we describe a procedure for obtaining a smoothed version so as to obtain the error estimates. Numerical experiments are carried out for a few illustrative examples.

An evaluation of the data space dimension in phase retrieval: results in Fresnel zone

<div>In this paper, we address the problem of computing the dimension of data space in phase retrieval problem. Starting from the quadratic formulation of the phase retrieval, the analysis is performed in two steps. First, we exploit the lifting technique to obtain a linear representation of the data. Later, we evaluate the dimension of data space by computing analytically the number of relevant singular values of the linear operator that represents the data. The study is done with reference to a 2D scalar geometry consisting of an electric current strip whose square amplitude of the electric radiated field is observed on a two-dimensional extended domain in Fresnel zone.</div>