Weakly n-hyponormal Weighted Shifts and Their Examples

Abstract

The gap between hyponormal and subnormal Hilbert space operators can be studied using the intermediate classes of weakly n-hyponormal and (strongly) n-hyponormal operators. The main examples for these various classes, particularly to distinguish them, have been the weighted shifts. In this paper we first obtain a characterization for a weakly n-hyponormal weighted shift Wα with weight sequence α, from which we extend some known results for quadratically hyponormal (i.e., weakly 2-hyponormal) weighted shifts to weakly n-hyponormal weighted shifts. In addition, we discuss some new examples for weakly n-hyponormal weighted shifts; one illustrates the differences among the classes of 2-hyponormal, quadratically hyponormal, and positively quadratically hyponormal operators. Mathematics Subject Classification (2000). 47B37, 47B20.

Some Quadratically Hyponormal Weighted Shifts

“…The proof is rather computational, so we first give an indication of the overall strategy. It is known that QH(W α(x) ) contains the interval [.63, .742] from results for positive quadratic hyponormality in [11], as well as containing the point .625 from [13]. To establish quadratic hyponormality we will consider the d n (x, t) := det D n (t) with a variable x; the key computational fact is that each is no worse than a quartic in the variable x.…”

“…Again non-negativity is easy for t ≥ 1 since c (11,12) ); we conjecture that it is, at least, in QH(W α(x) ). To prove this, it would be enough to show that for any strictly larger x, the "iteration" argument above works with sufficiently many iterations.…”

Weak and Quadratic Hyponormality of 2-Variable Weighted Shifts and Their Examples

“…. , n ( [8]). Because the structure of ntuple operators can be extended from the study of 2-tuple operators, many operator theorists have concentrated their studies to the structure of n-tuple operators (see [3], [4], [5], [6], [7], [9], [10], [11], [12], [13], etc.).…”

On Semi-cubically Hyponormal Weighted Shifts with First Two Equal Weights

“…Stampfli's result has been used to attempt the construction of nonsubnormal polynomially hyponormal weighted shifts (cf. [1], [3], [4], [7], [12], [15], [17]). In [3], Choi proved that if a weighted shift W α is polynomially hyponormal with the first two weights equal, then W α has the flatness property.…”