Hyperexpansive composition operators

Abstract

The notion of a completely hyperexpansive operator has been introduced in [2] by Athavale. In this paper bounded and unbounded hyperexpansive composition operators are investigated. A unique semispectral measure is associated with a bounded completely hyperexpansive composition operator. Examples of bounded and unbounded densely defined 2-isometric (completely hyperexpansive, 2-hyperexpansive) composition operators with invariant domains are given.

On operators Cauchy dual to 2-hyperexpansive operators: the unbounded case

“…Proof. (1): It can be deduced from Proposition 2.2 and the discussion at the beginning of Example 4.4 of [11] that C φ D(C φ ) ⊂ D(C φ ). Set S ≡ C φ and notice that D ⊂ n≥0 D(S * k ), where D is as in P4.…”

“…C φ e i,j = e i+1,j+1 + √ a i e i,j+1 if i = j, e i,j+1 if i < j. Also, it can be deduced from the discussion at the beginning of Example 4.4 of [11] that C φ is a closed linear expansion. Thus the Cauchy dual operator C φ is an injective contraction (Lemma 2.3).…”

“…We illustrate the results of the present paper in the context of a class of composition operators defined on discrete measure spaces. The following example is borrowed from [11].…”

“…It was recorded in Example 4.4 of [11] that {e i,j : (i, j) ∈ X} is an orthonormal basis for L 2 (µ) and…”

“…The composition operator C φ enjoys the following properties, which can be easily deduced from [11,Example 4.4…”