WEAKLY SUBNORMAL WEIGHTED SHIFTS NEED NOT BE 2-HYPONORMAL

Quartically Hyponormal Weighted Shifts Need Not Be 3-hyponormal

We give the first example of a quartically hyponormal unilateral weighted shift which is not 3-hyponormal.

Jointly Hyponormal Pairs of Commuting Subnormal Operators Need Not Be Jointly Subnormal

We construct three different families of commuting pairs of subnormal operators, jointly hyponormal but not admitting commuting normal extensions. Each such family can be used to answer in the negative a 1988 conjecture of R. Curto, P. Muhly and J. Xia. We also obtain a sufficient condition under which joint hyponormality does imply joint subnormality.

Propagation Phenomena for Hyponormal 2-variable Weighted Shifts

We study the class of hyponormal 2-variable weighted shifts with two consecutive equal weights in the weight sequence of one of the coordinate operators. We show that under natural assumptions on the coordinate operators, the presence of consecutive equal weights leads to horizontal or vertical flatness, in a way that resembles the situation for 1-variable weighted shifts. In 1variable, it is w...

One-step extensions of subnormal 2-variable weighted shifts

Existence of Non-subnormal Polynomially Hyponormal Operators

In 1950, P. R. Halmos, motivated in part by the successful development of the theory of normal operators, introduced the notions of subnormality and hyponormality for (bounded) Hilbert space operators. An operator T is subnormal if it is the restriction of a normal operator to an invariant subspace; T is hyponormal if T*T > TT*. It is a simple matrix calculation to verify that subnormality impl...