Multiplication Operators on the Lipschitz Space of a Tree

Abstract

In this paper, we study the multiplication operators on the space of complex-valued functions f on the set of vertices of a rooted infinite tree T which are Lipschitz when regarded as maps between metric spaces. The metric structure on T is induced by the distance function that counts the number of edges of the unique path connecting pairs of vertices, while the metric on C is Euclidean. After observing that the space L of such functions can be endowed with a Banach space structure, we characterize the multiplication operators on L that are bounded, bounded below, and compact. In addition, we establish estimates on the operator norm and on the essential norm, and determine the spectrum. We then prove that the only isometric multiplication operators on L are the operators whose symbol is a constant of modulus one. We also study the multiplication operators on a separable subspace of L we call the little Lipschitz space. Mathematics Subject Classification (2000). Primary 47B38, 05C05.

Discrete analogue of generalized Hardy spaces and multiplication operators on homogenous trees

“…Colonna multiplication operators on Lipschitz space, weighted Lipschitz space, and iterated logarithmic Lipschitz space of a tree in [10], [4] and [3], respectively. In [2,11], the authors discussed about multiplication operators between Lipschitz type spaces and the space of bounded functions on a tree.…”

Some results in Möbius invariant spaces

“…Namely, for any f ∈ X , M g ( f ) = g f . See [12][13][14][15] for multipliers of analytic function spaces. Let T be a bounded linear operator on X .…”

The Forward and Backward Shift on the Lipschitz Space of a Tree

“…Lastly, for every v ∈ T , we denote by S v the sector determined by v, which consists of v and all its nchildren; i.e., S v := In [11], Colonna and Easley show that…”

“…In [11], Colonna and Easley introduced the Lipschitz space of a tree, L. This is the Banach space of complex-valued functions on a (countably infinite and locally finite) tree which are Lipschitz functions, when the tree is endowed with the edge-counting metric. This space may be considered as the discrete analogue of the classical Bloch space: the space of functions f : D → C which are Lipschitz when the unit disk D is given the hyperbolic or Bergman metric (see, e.g., [22]) and the set of complex numbers C is given the usual Euclidean metric.…”