A functional model for nondissipative unbounded perturbations of an unbounded self-adjoint operator on a Hilbert space X is constructed. This model is analogous to the Nagy– Foias model of dissipative operators, but it is linearly similar and not unitarily equivalent to the operator. It is attached to a domain of parabolic type, instead of a half-plane. The transformation map from X to the mode...

We provide a comparative treatment of some aspects of spectral theory for self-adjoint and non-self-adjoint (but J-self-adjoint) Dirac-type operators connected with the defocusing and focusing nonlinear Schrödinger equation, of relevance to nonlinear optics. In addition to a study of Dirac and Hamiltonian systems, we also introduce the concept of Weyl–Titchmarsh half-line m-coefficients (and 2 ...

We discuss rank one perturbationsAα = A+α(φ, ·)φ,α ∈ R , A ≥ 0 self-adjoint. Let dμα(x) be the spectral measure defined by (φ, (Aα − z)−1φ) = ∫ dμα(x)/(x− z). We prove there is a measure dρ∞ which is the weak limit of (1 + α)dμα(x) as α → ∞. If φ is cyclic for A, then A∞, the strong resolvent limit of Aα, is unitarily equivalent to multiplication by x on L(R , dρ∞). This generalizes results kno...

Some new perturbation bounds of the positive (semi) definite polar factor and the (sub) unitary polar factor for the (generalized) polar decomposition under the general unitarily invariant norm and the spectral norm are presented. By applying our new bounds to the weighted cases, the known perturbation bounds for the weighted polar decomposition are improved. 2014 Elsevier Inc. All rights reser...

Given an r × r complex matrix T , if T = U |T | is the polar decomposition of T , then the Aluthge transform is defined by ∆ (T ) = |T |U |T |. Let ∆n(T ) denote the n-times iterated Aluthge transform of T , i.e. ∆0(T ) = T and ∆n(T ) = ∆(∆n−1(T )), n ∈ N. In this paper we make a brief survey on the known properties and applications of the Aluthge trasnsorm, particularly the recent proof of the...

Given two chains of subspaces in C, we study the set of unitary matrices that map the subspaces in the first chain onto the corresponding subspaces in the second chain, and minimize the value ‖U− In‖ for various unitarily invariant norms ‖ · ‖ on Cn×n. In particular, we give formula for the minimum value ‖U − In‖, and describe the set of all the unitary matrices in the set attaining the minimum...