On the Spectrum of the Bergman-Hilbert Matrix II

Davis, Chandler; Ghatage, Pratibha G.

doi:10.4153/cmb-1990-010-7

Cited by 4 publications

(3 citation statements)

References 4 publications

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“…As another application of the general results stated in Theorem 3 and Corollaries 4 and 5 let us consider the so called Bergman-Hilbert matrix A with the entries A j,k = (j + 1)(k + 1) (j + k + 1) 2 , j, k ∈ Z + , which has been introduced and studied as an operator on ℓ 2 (Z + ) in [4,3]. It is shown in [3,Prop. 2] that the essential spectrum of A equals the interval [0, 1].…”

Section: The Bergman-hilbert Matrixmentioning

confidence: 99%

A family of explicitly diagonalizable weighted Hankel matrices generalizing the Hilbert matrix

Kalvoda

Šťovı́ček

2015

Linear and Multilinear Algebra

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A three-parameter family B = B(a, b, c) of weighted Hankel matrices is introduced with the entriessupposing a, b, c are positive and a < b + c, b < a + c, c ≤ a + b. The famous Hilbert matrix is included as a particular case. The direct sum B(a, b, c) ⊕ B(a + 1, b + 1, c) is shown to commute with a discrete analog of the dilatation operator. It follows that there exists a three-parameter family of real symmetric Jacobi matrices, T (a, b, c), commuting with B(a, b, c). The orthogonal polynomials associated with T (a, b, c) turn out to be the continuous dual Hahn polynomials. Consequently, a unitary mapping U diagonalizing T (a, b, c) can be constructed explicitly. At the same time, U diagonalizes B(a, b, c) and the spectrum of this matrix operator is shown to be purely absolutely continuous and filling the interval [0, M (a, b, c)] where M (a, b, c) is known explicitly.If the assumption c ≤ a + b is relaxed while the remaining inequalities on a, b, c are all supposed to be valid, the spectrum contains also a finite discrete part lying above the threshold M (a, b, c). Again, all eigenvalues and eigenvectors are described explicitly.

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Section: The Bergman-hilbert Matrixmentioning

confidence: 99%

A family of explicitly diagonalizable weighted Hankel matrices generalizing the Hilbert matrix

Kalvoda

Šťovı́ček

2015

Linear and Multilinear Algebra

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“…According to Theorem 2.2, the essential spectrum and the a.c. spectrum of K(ϕ α ) is [0, ϕ α (0)], with multiplicity one. For α = 1 this is the Hilbert matrix, and for α = 2 this is a variant of the so-called Bergman-Hilbert matrix, considered in [6,4,11]. More generally, for all α ∈ N, the following matrix was considered in [11] (as part of a larger three-parameter family of infinite matrices):…”

Section: Examplesmentioning

confidence: 99%

The spectral density of Hardy kernel matrices

Pushnitski

2021

Preprint

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“…This form has been studied in connection with Hankel forms on Bergman spaces, by Ghatage [12] and Davis and Ghatage [10]. This explains the name of this form in the context of Hankel operators; for more details see [12].…”

Section: Cobos Kühn and Peetre Ieotmentioning

confidence: 99%

Multilinear Forms of Hilbert Type and Some Other Distinguished Forms

Cobos

Kühn

Peetre

2005

Integr. equ. oper. theory

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We give some new examples of bounded multilinear forms on the Hilbert spaces 2 and L2(0, ∞). We characterize those which are compact or Hilbert-Schmidt. In particular, we study m-linear forms (m ≥ 3) on 2 which can be regarded as the multilinear analogue of the famous Hilbert matrix. We also determine the norm of the permanent on K n , where K = R or C. Mathematics Subject Classification (2000). Primary 47A30, 47B10, 15A15.Keywords. Multilinear forms of the type of the Hilbert matrix; distinguished forms on L2(0, ∞); norm of the permanent.

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On the Spectrum of the Bergman-Hilbert Matrix II

Abstract: We study a class of matrices (introduced by T. Kato) with principal homogeneous part, and use Mellin transform of the homogeneous kernel to determine spectral density of the positive infinite matrices.

Cited by 4 publications

References 4 publications

A family of explicitly diagonalizable weighted Hankel matrices generalizing the Hilbert matrix

A family of explicitly diagonalizable weighted Hankel matrices generalizing the Hilbert matrix

The spectral density of Hardy kernel matrices

Multilinear Forms of Hilbert Type and Some Other Distinguished Forms

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