The similarity problem for non-self-adjoint operators with absolutely continuous spectrum

2000 ◽  
Vol 34 (2) ◽  
pp. 143-145 ◽  
Author(s):  
A. V. Kiselev ◽  
M. M. Faddeev
Keyword(s):  
Continuous Spectrum ◽  
Absolutely Continuous Spectrum ◽  
Absolutely Continuous ◽  
Similarity Problem ◽  
Adjoint Operators

scholarly journals Weighted integral Hankel operators with continuous spectrum

2017 ◽  
Vol 4 (1) ◽  
pp. 121-129
Author(s):  
Emilio Fedele ◽  
Alexander Pushnitski
Keyword(s):  
Continuous Spectrum ◽  
Integral Kernel ◽  
Hankel Operators ◽  
Absolutely Continuous Spectrum ◽  
Absolutely Continuous ◽  
Weighted Integral ◽  
Adjoint Operators ◽  
Unweighted Case

Abstract Using the Kato-Rosenblum theorem, we describe the absolutely continuous spectrum of a class of weighted integral Hankel operators in L2(ℝ+). These self-adjoint operators generalise the explicitly diagonalisable operator with the integral kernel sαtα(s + t)-1-2α, where α > -1/2. Our analysis can be considered as an extension of J. Howland’s 1992 paper which dealt with the unweighted case, corresponding to α = 0.

Decay spectrum and decay subspace of normal operators

2001 ◽  
Vol 131 (6) ◽  
pp. 1245-1255 ◽  
Author(s):  
I. Antoniou ◽  
S. A. Shkarin
Keyword(s):  
Continuous Spectrum ◽  
Adjoint Operator ◽  
Unitary Operators ◽  
Absolutely Continuous Spectrum ◽  
Continuous Component ◽  
Absolutely Continuous ◽  
Direct Sum ◽  
Unique Decomposition ◽  
Normal Operators ◽  
Adjoint Operators

Let A be a self-adjoint operator on a Hilbert space. It is well known that A admits a unique decomposition into a direct sum of three self-adjoint operators Ap, Aac and Asc such that there exists an orthonormal basis of eigenvectors for the operator Ap, the operator Aac has purely absolutely continuous spectrum and the operator Asc has purely singular continuous spectrum. We show the existence of a natural further decomposition of the singular continuous component Asc into a direct sum of two self-adjoint operators and . The corresponding subspaces and spectra are called decaying and purely non-decaying singular subspaces and spectra. Similar decompositions are also shown for unitary operators and for general normal operators.

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