Separating classes of composition operators via subnormal condition

Abstract

Abstract. Several classes have been considered to study the weak subnormalities of Hilbert space operators. One of them is n-hypnormality, which comes from the Bram-Halmos criterion for subnormal operators. In this note we consider E(n)-hyponormality, which is the parallel version corresponding to the Embry characterization for subnormal operators. We characterize E(n)-hyponormality of composition operators via k-th Radon-Nikodym derivatives and present some examples to distinguish the classes.

L(n)-HYPONORMALITY: A MISSING BRIDGE BETWEEN SUBNORMALITY AND PARANORMALITY

“…. , f n ∈ H. Based on Embry's characterization, McCullough and Paulsen introduced in [24] a new class of operators which, following [17], will be called E(n)-hyponormal: T is said to be E(n)-hyponormal (n ≥ 1) if inequality (1.2) holds for all f 0 , . .…”

“…As shown in [24], E(1)-hyponormality is essentially weaker than 1-hyponormality. Moreover, in view of [17], T is E(1)-hyponormal if and only if |T | 4 ≤ |T 2 | 2 , and so, by the Heinz inequality, such T must be an A-class operator, that is, |T | 2 ≤ |T 2 | (see [14, p. 166]). Hence, E(n)-hyponormality can be thought of as a bridge between subnormal operators and A-class operators.…”

Unbounded composition operators via inductive limits: Cosubnormal operators with matrix symbols, II

“…Motivated by these previous results and the criterion for subnormality of general unbounded operators due to Stochel and Szafraniec (see [27,Theorem 3]), we introduce in this paper classes S * n,r of unbounded operators closely related to cosubnormal operators (they resemble, in a sense, weak hyponormality classes studied in the case of bounded operators, cf. [22,20,15,19]) and investigate under what conditions composition operators with infinite matrix symbols belong to the classes. We use inductive limits to achieve our goal.…”

Weak Hyponomal Composition Operators Induced by a Tree

“…In particular, the p-hyponormality notion is contained in such studies. Jung-Lee-Park( [7]) constructed examples induced by some block matrix operators, and in [6], the classes of p-hyponormal operators are distinct with respect to positive real numbers p. In [1] and [2] Burnap-Jung-Lambert discussed some composition operator models on L 2 , and also they showed that the classes of p-hyponormal and p-paranormal operators are distinct for each p > 0. Nonetheless, new examples for the p-hyponormal operator classes are needed to study the relationship between the above two notions.…”