The Laguerre Type Operator in a Left Definite Hilbert Space

Krall, Allan M.; Everitt, W. N.; Littlejohn, Lance L.; Onyango-Otieno, Vitalis

doi:10.1006/jmaa.1995.1183

Cited by 6 publications

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“…The problem of describing the left-definite spaces has been discussed by a number of authors under the additional assumption that all functions in the maximal domain have "finite energy integrals." See, e.g., [1][2][3][4][5] and references cited therein. It is not known what happens if this additional assumption is dropped.…”

Section: Introductionmentioning

confidence: 99%

Left-definite spaces of singular Sturm–Liouville problems

Wei

2008

Journal of Mathematical Analysis and Applications

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Section: Introductionmentioning

confidence: 99%

Left-definite spaces of singular Sturm–Liouville problems

Wei

2008

Journal of Mathematical Analysis and Applications

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“…It is important for us to have an explicit expression for the LD spaces. The problem of describing LD spaces has been discussed by a number of authors under the additional assumption that all functions in the domain of the maximal operator for | w | −1 l have “finite energy integrals”; see, for example, 1, 2, 4–12, 15 and references cited therein. Recently, for singular Sturm‐Liouville problems, by the Friedrichs extension and LD self‐adjoint BC's, the first author and Fu 18 describe the left‐definite spaces in terms of a product space which consists of a weighted Sobolev space and a weighted vector space.…”

Section: Introductionmentioning

confidence: 99%

Left‐definite problems of regular self‐adjoint differential equations of even order

Wei

2010

Mathematische Nachrichten

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Key words Left-definiteness, left-definite space, boundary condition MSC (2010) Primary: 34B05, Secondary: 34L05, 47E05The left-definite boundary conditions of the problem consisting of a regular formally self-adjoint differential equation of even order and a self-adjoint boundary condition are characterized in terms of a certain fundamental set of solutions of the equation. Based on the left-definite boundary conditions, all the left-definite spaces of the problem are described explicitly.

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A General Left-Definite Theory for Certain Self-Adjoint Operators with Applications to Differential Equations

Littlejohn

Wellman

2002

Journal of Differential Equations

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DEDICATED TO W.N. EVERITT, AN INSPIRATION AND MENTOR TO BOTH AUTHORSWe show that any self-adjoint operator A (bounded or unbounded) in a Hilbert space H ¼ ðV ; ðÁ; ÁÞÞ that is bounded below generates a continuum of Hilbert spaces fH r g r>0 and a continuum of self-adjoint operators fA r g r>0 . For reasons originating in the theory of differential operators, we call each H r the rth left-definite space and each A r the rth left-definite operator associated with ðH ; AÞ. Each space H r can be seen as the closure of the domain DðA r Þ of the self-adjoint operator A r in the topology generated from the inner product ðA r x; yÞ ðx; y 2 DðA r ÞÞ. Furthermore, each A r is a unique self-adjoint restriction of A in H r . We show that the spectrum of each A r agrees with the spectrum of A and the domain of each A r is characterized in terms of another left-definite space. The Hilbert space spectral theorem plays a fundamental role in these constructions. We apply these results to two examples, including the classical Laguerre differential expression '½Á in which we explicitly find the leftdefinite spaces and left-definite operators associated with A, the self-adjoint operator generated by '½Á in L 2 ðð0; 1Þ; t a e Àt Þ having the Laguerre polynomials as eigenfunctions. # 2002 Elsevier Science (USA) Key Words: spectral theorem; self-adjoint operator; Hilbert space; Sobolev space; Dirichlet inner product; left-definite Hilbert space; left-definite self-adjoint operator; Laguerre polynomials; Stirling numbers of the second kind.Contents.

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The Laguerre Type Operator in a Left Definite Hilbert Space

Cited by 6 publications

References 0 publications

Left-definite spaces of singular Sturm–Liouville problems

Left-definite spaces of singular Sturm–Liouville problems

Left‐definite problems of regular self‐adjoint differential equations of even order

A General Left-Definite Theory for Certain Self-Adjoint Operators with Applications to Differential Equations

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