Suppose that λ − T is left-invertible in L(H) for all λ ∈ Ω, where Ω is an open subset of the complex plane. Then an operator-valued function L(λ) is a left resolvent of T in Ω if and only if T has an extension T̃ , the resolvent of which is a dilation of L(λ) of a particular form. Generalized resolvents exist on every open set U , with U included in the regular domain of T . This implies a form...

A new resolvent decomposition of a regular polynomial matrix is derived which is a refinement of the results given by Gohberg et al (1982) and Vardulakis (1991). Based on this, a complete solution of the regular Polynomial Matrix Descriptions (PMDs) is presented that reflects precisely the zero input response and the zero state response, and takes into account the impulsive properties of the sy...

We investigate the analytic perturbation of generalized inverses. Firstly we analyze the analytic perturbation of the Drazin generalized inverse (also known as reduced resolvent in operator theory). Our approach is based on spectral theory of linear operators as well as on a new notion of group reduced resolvent. It allows us to treat regular and singular perturbations in a unified framework. W...

The paper deals with infinite block matrices having compact off diagonal parts. Bounds for the spectrum are established and estimates for the norm of the resolvent are proposed. Applications to matrix integral operators are also discussed. The main tool is the π-triangular operators defined in the paper.

We extend our recent work with K. Zumbrun on long-time stability of multidimensional noncharacteristic viscous boundary layers of a class of symmetrizable hyperbolicparabolic systems. Our main improvements are (i) to establish the stability for a larger class of systems in dimensions d ≥ 2, yielding the result for certain magnetohydrodynamics (MHD) layers; (ii) to drop a technical assumption on...

We consider the problem of determining the unknown term in the right-hand side of a second-order differential equation with unbounded operator generating a cosine operator function from the overspecified boundary data. We obtain necessary and sufficient conditions of the unique solvability of this problem in terms of location of the spectrum of the unbounded operator and properties of its resol...

It has been established quite recently [1], that a contraction T , having finite defects (rank (I − T T ), rank (I − TT ) < ∞) and the spectrum σ(T ) not filling in the closed unit disk D, is similar to a normal operator if and only if C1(T ) = sup λ6∈σ(T ) ||(T − λ)|| · dist(λ, σ(T )) < ∞. It was shown later [9], that the “only if” part of the assertion is no longer true if T is not a contract...

In this paper, we consider the Lamé operator $$-\Delta ^*$$ and study resolvent estimate, uniform Sobolev Carleman estimate for . First, obtain sharp $$L^p$$ – $$L^q$$ estimates admissible p, q. This extends particular case $$q=\frac{p}{p-1}$$ due to Barceló et al. [4] Cossetti [8]. Secondly, show failure of For purpose directly analyze Fourier multiplier resolvent. allows us prove not only upp...

We investigate the asymptotic behaviour of large eigenvalues for a class of finite difference self-adjoint operators with compact resolvent in l2.