caa a22 4500 475839129 CHVBK 20180406123856.0 cr unu---uuuuu 170329e20000501xx s 000 0 eng 10.1023/A:1006441812051 doi (NATIONALLICENCE)springer-10.1023/A:1006441812051 Laurent-Jacobi Matrices and the Strong Hamburger Moment Problem [Elektronische Daten] [Erik Hendriksen, Caspar Nijhuis] Let ℒ be the linear space of the Laurent polynomials and suppose that <⋅, ⋅<ℒ is a positive-definite Hermitian inner product in ℒ with the additional property that $$ \left\langle {f\left( z \right),g\left( z \right)} \right\rangle {/Mathcal L} = \left\langle {f\left( z \right)\overline {g\left( z \right)} ,1} \right\rangle {/Mathcal L} $$ . Starting from the five-term recurrence relation for orthogonal Laurent polynomials with respect to <⋅, ⋅<ℒ, we derive Laurent-Jacobi matrices $${/Mathcal J} $$ and $${/Mathcal K} $$ for the multiplication operator and its inverse in ℒ. These matrices are real and symmetric, and $${/Mathcal J} $$ generates a symmetric operator in the Hilbert space ℓ2 with natural basis { e n } n = 0 ∞. We show that this operator has deficiency indices (0, 0) or (1, 1) and that every self-adjoint extension A in ℓ2 has simple spectrum with generating vector e 0. Let E be the spectral measure of A. Then the measure μ e 0 given by μ e 0(Ω) =<E(Ω) e 0, e 0< for all Borel sets Ω in $${/Mathcal R} $$ , satisfies $$\int_{/Mathcal R} {f\overline g {d\mu }_{e_0 } } = \left\langle {f,g} \right\rangle {/Mathcal L} $$ forf,gℒ. In this way, we obtain a solution μ e 0 of the Strong Hamburger Moment Problem (SHMP) for which ℒ is dense in L 2(μ e 0). Some results concerning the relation between the deficiency indices andthe set of all solutions of the SHMP are established. Finally, we give an analogue of a theorem by M. H. Stone which tells us which self-adjoint operators are generatedby a Laurent-Jacobi matrix with deficiency indices (0, 0). Kluwer Academic Publishers, 2000 orthogonal Laurent polynomials nationallicence self-adjoint operator nationallicence moment problem nationallicence Hendriksen Erik Department of Mathematics, University of Amsterdam, Plantage Muidergracht 24, 1018 TV, Amsterdam, The Netherlands. e-mail aut Nijhuis Caspar Department of Mathematics, University of Amsterdam, Plantage Muidergracht 24, 1018 TV, Amsterdam, The Netherlands aut Acta Applicandae Mathematica Kluwer Academic Publishers 61/1-3(2000-05-01), 119-132 0167-8019 61:1-3<119 2000 61 10440 https://doi.org/10.1023/A:1006441812051 text/html Onlinezugriff via DOI 1 research-article jats NATIONALLICENCE 856 40 https://doi.org/10.1023/A:1006441812051 text/html Onlinezugriff via DOI NATIONALLICENCE 700 1- Hendriksen Erik Department of Mathematics, University of Amsterdam, Plantage Muidergracht 24, 1018 TV, Amsterdam, The Netherlands. e-mail aut NATIONALLICENCE 700 1- Nijhuis Caspar Department of Mathematics, University of Amsterdam, Plantage Muidergracht 24, 1018 TV, Amsterdam, The Netherlands aut NATIONALLICENCE 773 0- Acta Applicandae Mathematica Kluwer Academic Publishers 61/1-3(2000-05-01), 119-132 0167-8019 61:1-3<119 2000 61 10440 Metadata rights reserved Springer special CC-BY-NC licence nationallicence BK010053 XK010053 XK010000 NATIONALLICENCE NATIONALLICENCE NL-springer