On the Estimate of the Distance to Non-Invertibility

Israel Journal of Mathematics, 1981

Proyecciones, 2022

In this paper we are concerned with the spectral analysis for the classes of finte rank perturbations of diagonal operators in the form A = D +F where D is a diagonal operator and F = u 0 1 ⊗v 1 +...+u 0 m ⊗ v m is an operator of finite rank in the non-archimedean Banach space of countable type. We compute the spectrum of A using the theory of Fredholm operators in non archimedean setting and the concept of essential spectrum for linear operators.

Proceedings - Mathematical Sciences, 2014

Let A(x) be a holomorphic family of bounded self-adjoint operators on a separable Hilbert space H and let A(x) n be the orthogonal compressions of A(x) to the span of first n elements of an orthonormal basis of H. The problem considered here is to approximate the spectrum of A(x) using the sequence of eigenvalues of A(x) n. We show that the bounds of the essential spectrum and the discrete spectral values outside the bounds of essential spectrum of A(x) can be approximated uniformly on all compact subsets by the sequence of eigenvalue functions of A(x) n. The known results for a bounded selfadjoint operator, are translated into the case of a holomorphic family of operators. Also an attempt is made to predict the existence of spectral gaps that may occur between the bounds of essential spectrum of A(0) = A and study the effect of holomorphic perturbation of operators in the prediction of spectral gaps. As an example, gap issues of some block Toeplitz-Laurent operators are discussed. The pure linear algebraic approach is the main advantage of the results here.

Numerical Functional Analysis and Optimization, 2019

The aim of this article is to prove several new numerical radius inequalities for n  n operator matrices on a Hilbert space. Let H 1 ; H 2 ;. .. ; H n be complex Hilbert spaces, and let T ¼ ½T ij be an n  n operator matrix with T ij 2 BðH j ; H i Þ. Among other inequalities, it shown that ARTICLE HISTORY

Filomat, 2014

Let X and Y two complex Banach spaces and (A, B) a pair of bounded linear operators acting on X with value on Y. This paper is concerned with spectral analysis of the pair (A, B). We establish some properties concerning the spectrum of the linear operator pencils A − λB when B is not necessarily invertible and λ ∈ C. Also, we use the functional calculus for the pair (A, B) to prove the corresponding spectral mapping theorem for (A, B). In addition, we define the generalized Kato essential spectrum and the closed range spectra of the pair (A, B) and we give some relationships between this spectrums. As application, we describe a spectral analysis of quotient operators.

Advances in operator theory, 2019

We obtain norm estimates for the resolvent of A and a bound for the spectral variation of A. In addition, the representation for the resolvents of the considered operators is established via multiplicative operator integrals. That representation can be considered as a generalization of the representation for the resolvent of a normal operator in a Hilbert space. It is also shown that the considered operators are Kreiss-bounded. Applications to integral operators in L p are also discussed. In particular, bounds for the upper and lower spectral radius of integral operators are derived.

Linear Algebra and its Applications, 2010

Journal of Mathematical Analysis and Applications, 2012

The classical Eckart-Young formula for square matrices identifies the distance to singularity of a matrix. The main purpose of this paper is to get generalizations of this formula. We characterize the distance to non-surjectivity of a linear operator W ∈ L(X, Y) in finitedimensional normed spaces X, Y , under the assumption that the operator W is surjective (i.e. W X = Y) and subjected to structured perturbations of the form W + MΔN. As an application of these results, we shall derive formulas of the controllability radius for a descriptor controllable system [E, A, B]: Eẋ = Ax + Bu, t 0, under the assumption that systems matrices E, A, B are subjected to structured perturbations and to multiperturbations.

Rocky Mountain Journal of Mathematics, 2003

A nonsingularity criterion for block matrices is derived. It improves the well-known results in the case of matrices which are "close" to block triangular ones. Moreover, an estimate for the norm of the inverse matrices is derived.