×

Weighted integral Hankel operators with continuous spectrum. (English) Zbl 1517.47046

Summary: Using the Kato-Rosenblum theorem, we describe the absolutely continuous spectrum of a class of weighted integral Hankel operators in \(L^2(\mathbb{R}_+)\). These self-adjoint operators generalise the explicitly diagonalisable operator with the integral kernel \(s^\alpha t^\alpha (s + t)^{-1-2 \alpha}\), where \(\alpha > -1/2\). Our analysis can be considered as an extension of J. S. Howland’s 1992 paper [Indiana Univ. Math. J. 41, No. 2, 427–434 (1992; Zbl 0773.47013)] which dealt with the unweighted case, corresponding to \(\alpha = 0\).

MSC:

47B35 Toeplitz operators, Hankel operators, Wiener-Hopf operators
47A10 Spectrum, resolvent

Citations:

Zbl 0773.47013
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] A. B. ABUSAKSAKA, J. R. PARTINGTON, Diffusive systems and weighted Hankel operators, Oper. Matrices 11 no. 1 (2017), 125-132.; · Zbl 1367.47032
[2] A. B. ALEKSANDROV, V. V. PELLER, Distorted Hankel integral operators, Indiana Univ. Math. J., 53 no. 4 (2004), 925-940.; · Zbl 1071.47029
[3] J. S. HOWLAND, Spectral theory of operators of Hankel type, II, Indiana Univ. Math. J. 41 no. 2 (1992), 427-434.; · Zbl 0773.47013
[4] S. JANSON, J. PEETRE, Paracommutators - boundedness and Schatten-von Neumann properties, Trans. Amer. Math. Soc. 305 (1988), 467-504.; · Zbl 0644.47046
[5] T. KALVODA, P. STOVICEK, A family of explicitly diagonalizable weighted Hankel matrices generalizing the Hilbert matrix, Linear Multilinear Algebra 64 no. 5 (2016), 870-884.; · Zbl 1359.47027
[6] S. POWER, Hankel operators on Hilbert space, Research Notes in Math. 64, Pitman, Boston, 1982.; · Zbl 0489.47011
[7] A. PUSHNITSKI, D. YAFAEV, Spectral and scattering theory of self-adjoint Hankel operators with piecewise continuous symbols, J. Operator Theory 74 no. 2 (2015), 417-455.; · Zbl 1389.47040
[8] A. PUSHNITSKI, D. YAFAEV, Sharp estimates for singular values of Hankel operators, Integr. Equ. Oper. Theory 83 no. 3 (2015), 393-411.; · Zbl 1329.47031
[9] M. REED, B. SIMON, Methods of Modern Mathematical Physics. III: Scattering Theory. Academic Press, New York 1979.; · Zbl 0405.47007
[10] R. ROCHBERG, Trace ideal criteria for Hankel operators and commutators, Indiana Univ. Math. J. 31 no. 6 (1982), 913-925.; · Zbl 0514.47020
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.