We are interested in the optimal growth in terms of Lp-averages of hypercyclic and $\mathcal{U}$-frequently hypercyclic functions for some weighted Taylor shift operators acting on the space of analytic functions on the unit disc. We unify the results obtained by considering intermediate notions of upper frequent hypercyclicity between $\mathcal{U}$-frequent hypercyclicity and hypercyclicity.

Let $\alpha : 1, 1, \sqrt{x} , \mathop{( \sqrt{u} , \sqrt{v} , \sqrt{w} )}\nolimits ^{\wedge } $ be a backward 3-step extension of a recursively generated weighted sequence of positive real numbers with $1\leq x\leq u\leq v\leq w$ and let ${W}_{\alpha } $ be the associated weighted shift with weight sequence $\alpha $. The set of positive real numbers $x$ such that ${W}_{\alpha } $ is quadratically hyponormal for some $u, v$ and $w$ is described, solving an open problem due to Curto and Jung [‘Quadratically hyponormal weighted shifts with two equal weights’, Integr. Equ. Oper. Theory 37 (2000), 208–231].

It is shown that most properties of (bounded) completely hyperexpansive operators remain valid for unbounded $2$-hyperexpansive operators. Powers of closed $2$-hyperexpansive operators are proved to be closed and $2$-hyperexpansive. Various parts of spectra of such operators are calculated. $2$-hyperexpansive weighted shifts are investigated. Examples of unbounded closed $2$-hyperexpansive ($2$-isometric) operators with invariant dense domains are established.

AMS 2000 Mathematics subject classification: Primary 47A05; 47B37. Secondary 47B20; 47B10

An operator $T$ on a Banach space $X$ is said to be weakly supercyclic (respectively $N$-supercyclic) if there exists a one-dimensional (respectively $N$-dimensional) subspace of $X$ whose orbit under $T$ is weakly dense (respectively norm dense) in $X$. We show that a weakly supercyclic hyponormal operator is necessarily a multiple of a unitary operator, and we give an example of a weakly supercyclic unitary operator. On the other hand, we show that hyponormal operators are never $N$-supercyclic. Finally, we characterize $N$-supercyclic weighted shifts.

In this note we examine the relationships between a subnormal shift, the measure its moment sequence generates, and those of a large family of weighted shifts associated with the original shift. We examine the effects on subnormality of adding a new weight or changing a weight. We also obtain formulas for evaluating point mass at the origin for the measure associated with the shift. In addition, we examine the relationship between the measure associated with a subnormal shift and those of a family of shifts substantially different from the original shift.

We study the stability of linear filters associated with certain types of linear difference equations with variable coefficients. We show that stability is determined by the locations of the poles of a rational transfer function relative to the spectrum of an associated weighted shift operator. The known theory for filters associated with constant-coefficient difference equations is a special case.