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

Abstract

Abstract. In this article, we define discrete analogue of generalized Hardy spaces and its separable subspace on a homogenous rooted tree and study some of its properties such as completeness, inclusion relations with other spaces, separability, growth estimate for functions in these spaces and their consequences. Equivalent conditions for multiplication operators to be bounded and compact are also obtained. Furthermore, we discuss about point spectrum, approximate point spectrum and spectrum of multiplication operators and discuss when a multiplication operator is an isometry.

Composition operators on Hardy spaces of the homogenous rooted trees

“…The following results proved by the present authors in [15] are needed elsewhere. In [15], authors raised the question whether T 2 is a Hilbert space or not. The answer is indeed No!.…”

“…Recently, some classes of operators including Toeplitz operators with symbol from the Lipschitz space of a tree were considered in [11]. In [15], the present authors defined discrete analogue (T p ) of generalized Hardy spaces on homogeneous rooted tree and studied multiplication operators on them. Study of composition operators on T p spaces were initiated by the present authors in [16].…”

Composition Operators on the Discrete Analogue of Generalized Hardy Space on Homogenous Trees

“…p in T p spaces. The following results proved by the present authors [11] are needed for our present investigation.…”

“…Also, there a number of articles dealing with composition operators on different transform spaces, see for example [1,2,7]. In this article, Ω will be a homogeneous rooted tree and X the discrete analogue of generalized Hardy space introduced in [11].…”

“…In [6], Colonna et al studied composition operators on Lipschitz functions on a tree with edge counting metric to the complex plane with Euclidean metric, which is a discrete analogue of Bloch space, because Bloch space is also consisting of only Lipschitz functions on the unit disk under Hyperbolic metric to the complex plane with Euclidean metric. In [11], the present authors defined discrete analogue (T p ) of generalized Hardy spaces on homogeneous rooted tree. In the same article multiplication operators on T p spaces are studied.…”