Jacobi matrices on trees
Volume 118 / 2010
Colloquium Mathematicum 118 (2010), 465-497 MSC: Primary 47B36; Secondary 42C05. DOI: 10.4064/cm118-2-7
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.