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

Kód výsledku v IS VaVaI

<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F68407700%3A21340%2F16%3A00234545" target="_blank" >RIV/68407700:21340/16:00234545 - isvavai.cz</a>

Nalezeny alternativní kódy

RIV/68407700:21240/16:00234545

Výsledek na webu

<a href="http://dx.doi.org/10.1080/03081087.2015.1064348" target="_blank" >http://dx.doi.org/10.1080/03081087.2015.1064348</a>

DOI - Digital Object Identifier

<a href="http://dx.doi.org/10.1080/03081087.2015.1064348" target="_blank" >10.1080/03081087.2015.1064348</a>

Jazyk výsledku

angličtina

Název v původním jazyce

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

Popis výsledku v původním jazyce

A three-parameter family B = B(a, b, c) of weighted Hankel matrices is introduced where a, b, c are positive parameters fulfilling 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 analogue 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.

Název v anglickém jazyce

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

Popis výsledku anglicky

A three-parameter family B = B(a, b, c) of weighted Hankel matrices is introduced where a, b, c are positive parameters fulfilling 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 analogue 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.

Druh

J_x - Nezařazeno - Článek v odborném periodiku (Jimp, Jsc a Jost)

CEP obor

BA - Obecná matematika

OECD FORD obor

—

Projekt

<a href="/cs/project/GA13-11058S" target="_blank" >GA13-11058S: Spektrální analýza operátorů a její aplikace v kvantové mechanice</a><br>

Návaznosti

P - Projekt vyzkumu a vyvoje financovany z verejnych zdroju (s odkazem do CEP)

Rok uplatnění

2016

Kód důvěrnosti údajů

S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů

Název periodika

Linear and Multilinear Algebra

ISSN

0308-1087

e-ISSN

—

Svazek periodika

64

Číslo periodika v rámci svazku

5

Stát vydavatele periodika

US - Spojené státy americké

Počet stran výsledku

15

Strana od-do

870-884

Kód UT WoS článku

000373740700008

EID výsledku v databázi Scopus

2-s2.0-84957927039