Jacobi matrices on trees

Agnieszka M. Kazun¹ ; Ryszard Szwarc²

¹ Institute of Mathematics University of Wrocław Pl. Grunwaldzki 2/4 50-384 Wrocław, Poland
² Institute of Mathematics University of Wrocław Pl. Grunwaldzki 2/4 50-384 Wrocław, Poland and Institute of Mathematics and Computer Science University of Opole Oleska 48 45-052 Opole, Poland

Colloquium Mathematicum, Tome 118 (2010) no. 2, pp. 465-497

Résumé

Symmetric Jacobi matrices on one sided homogeneous trees are studied. Essential selfadjointness of these matrices turns out to depend on the structure of the tree. If a tree has one end and infinitely many origin points the matrix is always essentially selfadjoint independently of the growth of its coefficients. In case a tree has one origin and infinitely many ends, the essential selfadjointness is equivalent to that of an ordinary Jacobi matrix obtained by restriction to the so called radial functions. For nonselfadjoint matrices the defect spaces are described in terms of the Poisson kernel associated with the boundary of the tree.

Détail

DOI : 10.4064/cm118-2-7

Keywords: symmetric jacobi matrices sided homogeneous trees studied essential selfadjointness these matrices turns out depend structure tree tree has end infinitely many origin points matrix always essentially selfadjoint independently growth its coefficients tree has origin infinitely many ends essential selfadjointness equivalent ordinary jacobi matrix obtained restriction called radial functions nonselfadjoint matrices defect spaces described terms poisson kernel associated boundary tree