Criteria for positively quadratically hyponormal weighted shifts

Semi-cubic Hyponormality of Weighted Shifts with Stampfli Recursive Tail

“…The task at hand is to apply the machinery of Section 3 to the two (families of) matrices D [1] n (t) and D [2] n (t). This will involve replacing various objects ˇ in that section with their [1] and [2] versions. We will use expressions like G [j] , Q [j] , and η [j] with the obvious meanings.…”

“…We will use expressions like G [j] , Q [j] , and η [j] with the obvious meanings. It may be useful to note in advance that the limit lim n→∞ ηn = η < ∞ turns out to be the same for both " [1]" and " [2]" (see Lemma 4.2 above), as do certain quantities A and B. To ease the burden of notation slightly we omit the [j] superscripts on these quantities.…”

“…Then, if we apply Proposition 3.1 and Proposition 3.2 using (4.1) to D [1] n and D [2] n as in (2.2), Proposition 3.1 and Proposition 3.2 can be restated as follows.…”

“…There are several classes of operators in L(H) between subnormal and hyponormal which have been discussed for more than two decades (cf. [1], [6], [7], [8], [9], [17], [20]), including, for example, (strongly) n-hyponormal and weakly n-hyponormal operators. Some of these definitions were derived from the theory of multivariable operators.…”

“…∧ via the positive determinant coefficients property. Here we consider a recursive tail, which may be contrasted with [14] which considered related questions but with the tail arising from the Bergman shift (having weights 1 2 , 2 3 , 3 4 , . .…”