On the Uniqueness of Wave Operators Associated with Non-Trace Class Perturbations

2004 ◽  
Vol 47 (1) ◽  
pp. 144-151
Author(s):  
Jingbo Xia
Keyword(s):  
Wave Operator ◽  
Trace Class ◽  
Uniqueness Result ◽  
Orthogonal Projections ◽  
Wave Operators ◽  
Adjoint Operators ◽  
Norm Ideal

AbstractVoiculescu has previously established the uniqueness of the wave operator for the problem of -perturbation of commuting tuples of self-adjoint operators in the case where the norm ideal has the property , where {Pn} is any sequence of orthogonal projections with rank(Pn) = n. We prove that the same uniqueness result holds true so long as is not the trace class. (It is well known that there is no such uniqueness in the case of trace-class perturbation.)

scholarly journals Analytic Scattering Theory for Jacobi Operators and Bernstein–Szegö Asymptotics of Orthogonal Polynomials

2018 ◽  
Vol 30 (08) ◽  
pp. 1840019 ◽  
Author(s):  
D. R. Yafaev
Keyword(s):  
Jacobi Matrix ◽  
Trace Class ◽  
Spectral Shift ◽  
Shift Function ◽  
Discrete Schrödinger Operator ◽  
Wave Operators ◽  
Jacobi Operators ◽  
Matrix Formula ◽  
Asymptotics Of Orthogonal Polynomials ◽  
Szego Asymptotics

We study semi-infinite Jacobi matrices [Formula: see text] corresponding to trace class perturbations [Formula: see text] of the “free” discrete Schrödinger operator [Formula: see text]. Our goal is to construct various spectral quantities of the operator [Formula: see text], such as the weight function, eigenfunctions of its continuous spectrum, the wave operators for the pair [Formula: see text], [Formula: see text], the scattering matrix, the spectral shift function, etc. This allows us to find the asymptotic behavior of the orthonormal polynomials [Formula: see text] associated to the Jacobi matrix [Formula: see text] as [Formula: see text]. In particular, we consider the case of [Formula: see text] inside the spectrum [Formula: see text] of [Formula: see text] when this asymptotic has an oscillating character of the Bernstein–Szegö type and the case of [Formula: see text] at the end points [Formula: see text].

The length metric on the set of orthogonal projections and new estimates in the subspace perturbation problem

2015 ◽  
Vol 2015 (708) ◽  
pp. 1-15 ◽  
Author(s):  
Konstantin A. Makarov ◽  
Albrecht Seelmann
Keyword(s):  
Hilbert Space ◽  
Orthogonal Projections ◽  
Bounded Linear ◽  
Length Space ◽  
Metric Properties ◽  
Perturbation Problem ◽  
Subspace Perturbation Problem ◽  
Adjoint Operators ◽  
Spectral Subspaces ◽  
Operator Angle

AbstractWe consider the problem of variation of spectral subspaces for bounded linear self-adjoint operators in a Hilbert space. Using metric properties of the set of orthogonal projections as a length space, we obtain a new estimate on the norm of the operator angle associated with two spectral subspaces for isolated parts of the spectrum of the perturbed and unperturbed operators, respectively. In particular, recent results by Kostrykin, Makarov and Motovilov from [Proc. Amer. Math. Soc. 131, 3469–3476] and [Trans. Amer. Math. Soc. 359, 77–89] are strengthened.

On the almost sure approximation of self-adjoint operators in L2 (0, 1)

1996 ◽  
Vol 119 (3) ◽  
pp. 537-543
Author(s):  
L. J. Ciach ◽  
R. Jajte ◽  
A. Paszkiewicz
Keyword(s):  
Adjoint Operator ◽  
Almost Sure Convergence ◽  
The Other ◽  
Orthogonal Projections ◽  
Weak Operator Topology ◽  
Weak Operator ◽  
Image Position ◽  
Adjoint Operators ◽  
Orthogonal Systems ◽  
Martingale Convergence

There are several important theorems concerning the almost sure convergence of (monotone) sequences of orthogonal projections in L2-spaces. Let us mention here the martingale convergence theorems or the results on the developments of functions with respect to orthogonal systems. On the other hand every self-adjoint operator with the spectrum on the interval [0, 1] is a limit of some sequence of orthogonal projections in the weak operator topology (see [1]). This paper is devoted to a problem of approximation of a self-adjoint operator A acting in L2 (0, 1) by a sequence Pn of orthogonal projections in the sense that

scholarly journals An elementary proof of the symplectic spectral theorem

2019 ◽  
Vol 12 (03) ◽  
pp. 1950033
Author(s):  
Camilo Sanabria Malagón
Keyword(s):  
Elementary Proof ◽  
Vector Spaces ◽  
Inner Product ◽  
Spectral Theorem ◽  
Linear Combinations ◽  
Orthogonal Projections ◽  
Finite Dimensional ◽  
Lagrangian Subspace ◽  
The Matrix ◽  
Adjoint Operators

The classical spectral theorem completely describes self-adjoint operators on finite-dimensional inner product vector spaces as linear combinations of orthogonal projections onto pairwise orthogonal subspaces. We prove a similar theorem for self-adjoint operators on finite-dimensional symplectic vector spaces over perfect fields. We show that these operators decompose according to a polarization, i.e. as the product of an operator on a Lagrangian subspace and its dual on a complementary Lagrangian. Moreover, if all the eigenvalues of the operator are in the base field, then there exists a Darboux basis such that the matrix representation of the operator is [Formula: see text] blocks block-diagonal, where the first block is in Jordan normal form and the second block is the transpose of the first one.

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