Kalvoda, Tomáš; Šťovíček, Pavel A family of explicitly diagonalizable weighted Hankel matrices generalizing the Hilbert matrix. (English) Zbl 1359.47027 Linear Multilinear Algebra 64, No. 5, 870-884 (2016). The main object under consideration in the present paper is a three-parameter family \(B=B(a,b,c)=\|B_{kj}\|_{k,j\geq0}\) of weighted Hankel-type semi-infinite matrices with entries \[ B_{kj}=\frac{\Gamma(k+j+a)}{\Gamma(k+j+b+c)}\, \sqrt{\frac{\Gamma(k+b)\Gamma(k+c)\Gamma(j+b)\Gamma(j+c)}{\Gamma(k+a)k!\Gamma(j+a)j!}}, \] where \(a,b,c\) are positive parameters subject to \[ a<b+c, \quad b<a+c, \quad c\leq a+b. \] This family contains the celebrated Hilbert matrix \(H(\theta)=B(\theta,\theta,1)\). The authors construct a unitary mapping, diagonalizing \(B\), give a precise spectral analysis of the family. They show that the spectrum is pure absolutely continuous filling the interval \([0,M(a,b,c)]\) with \[ M(a,b,c)=\frac1{\Gamma(b+c-a)}\,\Gamma^2\Bigl(\frac{b+c-a}2\Bigr). \] Under certain relaxed assumptions on \(a,b,c\), there is a finite point spectrum, which is computed explicitly. Reviewer: Leonid Golinskii (Kharkov) Cited in 6 Documents MSC: 47B37 Linear operators on special spaces (weighted shifts, operators on sequence spaces, etc.) 47B36 Jacobi (tridiagonal) operators (matrices) and generalizations 47A10 Spectrum, resolvent 33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.) Keywords:weighted Hankel-type matrices; diagonalization; absolutely continuous spectrum; point spectrum; Hilbert matrix PDFBibTeX XMLCite \textit{T. Kalvoda} and \textit{P. Šťovíček}, Linear Multilinear Algebra 64, No. 5, 870--884 (2016; Zbl 1359.47027) Full Text: DOI arXiv References: [1] Rosenblum M, II. Proc. Amer. Math. Soc 9 pp 581– (1958) [2] Yafaev DR, Funktional. Anal. Prilozhen 44 pp 65– (2010) [3] DOI: 10.1002/cpa.3160110407 · Zbl 0084.10901 · doi:10.1002/cpa.3160110407 [4] DOI: 10.1007/978-3-642-86712-5 · doi:10.1007/978-3-642-86712-5 [5] DOI: 10.1090/S0002-9904-1962-10726-5 · Zbl 0105.04201 · doi:10.1090/S0002-9904-1962-10726-5 [6] DOI: 10.2140/pjm.2005.219.323 · Zbl 1100.15009 · doi:10.2140/pjm.2005.219.323 [7] DOI: 10.1016/0024-3795(82)90247-6 · Zbl 0484.15009 · doi:10.1016/0024-3795(82)90247-6 [8] DOI: 10.1016/0024-3795(81)90138-5 · Zbl 0477.15005 · doi:10.1016/0024-3795(81)90138-5 [9] DOI: 10.1007/978-0-387-21681-2 · doi:10.1007/978-0-387-21681-2 [10] Hardy GH, Inequalities (1934) [11] DOI: 10.1016/0024-3795(87)90139-X · Zbl 0636.15013 · doi:10.1016/0024-3795(87)90139-X [12] DOI: 10.4153/CMB-1990-010-7 · Zbl 0661.15023 · doi:10.4153/CMB-1990-010-7 [13] DOI: 10.1007/978-3-642-05014-5 · Zbl 1200.33012 · doi:10.1007/978-3-642-05014-5 [14] Rosenblum M, I. Proc. Amer. Math. Soc 9 pp 137– (1958) [15] Varadarajan VS, Geometry of quantum theory, 2. ed. (1985) [16] DOI: 10.1007/978-94-011-3154-4 · doi:10.1007/978-94-011-3154-4 [17] Abramowitz M, Handbook of mathematical functions (1972) [18] DOI: 10.1137/0511064 · Zbl 0454.33007 · doi:10.1137/0511064 [19] DOI: 10.1007/s00365-012-9157-z · Zbl 1268.47040 · doi:10.1007/s00365-012-9157-z 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.