Some Quadratically Hyponormal Weighted Shifts

Abstract

In the study of the gaps between subnormality and hyponormality both quadratic hyponormality and the related property positive quadratic hyponormality have been considered, especially for weighted shift operators. In particular, these have been studied for shifts with the first two weights equal and with Bergman tail or recursively generated tail. In this article, we characterize the allowed first two equal weights for quadratic hyponormality with Bergman tail, and the allowed first two equal weights for positive quadratic hyponormality with recursively generated tail. Mathematics Subject Classification (2000). 47B37, 47B20.

Quadratically hyponormal weighted shifts with recursive tail

“…If we want all the d n to be non-negative, surely there is an easy way this may occur: in [9], a weighted shift W α is defined to be positively quadratically hyponormal if c(n, n + 1) > 0 and c(n, i) ≥ 0, 0 ≤ i ≤ n, n ∈ N 0 . This class of operators has been studied in, for example, [1,6,17,20]. We specialize to a class of shifts due to Stampfli and arising in his consideration of the problem of completing an initial finite sequence of weights to yield the weight sequence for a subnormal shift [22].…”

Semi-cubic Hyponormality of Weighted Shifts with Stampfli Recursive Tail

“…Subsequently several papers have considered the problem of quadratic hyponormality of W α with first two weights equal (cf. [8], [9], [10], [14], [16], [22]). On the other hand, in [21], it was shown that if W α is cubically hyponormal with first two weights equal, then W α has flatness.…”

Backward 3-Step Extensions of Recursively Generated Weighted Shifts: A range of quadratic hyponormality

“…In particular, weak 2-hyponormality is referred to as quadratic hyponormality. Note that the structure of quadratically hyponormal weighted shifts has been studied at some length since 1990 (see [3,[6][7][8][9][10]).…”

“…Some of these and related studies have considered backward extensions, of length 1, 2 or 3, of the subnormal shift with weight sequence ( 12]; definitions are reviewed below). Many of these [3,6,9,11] have considered flatness (in which, under some condition, equality of two or three weights forces equality of all or almost all weights), and it is understood that to avoid trivialities there is no loss of generality in assuming that the weight sequence is strictly increasing except perhaps for the zeroth and first weights. An early question posed in [4] was to describe all quadratically hyponormal weighted shifts with the first two weights equal.…”