Towards a Model Theory for 2-Hyponormal Operators
by
Raúl E. Curto
The University of Iowa

An operator T is subnormal if there exist operators A and B such that

^ T   : = æ ç è T A 0 B ö ÷ ø

is normal, i.e.,

ì ï í ï î [ T*, T ] : = T*T - TT* = AA* A*T = BA* [ B, B* ] = A*A

(sub)

On the other hand, T is k-hyponormal if M_k(T) : = ( T^*jTⁱ )_{i, j=0}^k >= 0. The Bram-Halmos Criterion states that T is subnormal if and only if T is k-hyponormal for every k >= 1. It is then natural to ask whether k-hyponormal operators admit an extension [^T] with one or more of the properties listed in (sub).

Definition. T is said to be weakly subnormal if there exist A and B such that the first two conditions in (1) hold: [ T^*, T ] = AA^* and A^*T = BA^*. [^T] is said to be a partially normal extension of T.

In joint work with Woo Young Lee, we have obtained the following result.

Theorem. Let \alpha be a strictly increasing weight sequence, and let W_\alpha be the associated unilateral weighted shift. If W_\alpha is 2-hyponormal then W_\alpha is weakly subnormal. More precisely, there exists a weight sequence \beta such that

^ T   : = æ ç è W\alpha [ W\alpha*, W\alpha ]1/2 0 W\beta ö ÷ ø

is partially normal, and \sigma([^T]) = \sigma(W_\alpha).

As a consequence, we obtain an operator matricial description of Stampfli's construction of the minimal normal extension of a subnormal weighted shift. The proof of the above theorem is based on the following intriguing result.

Proposition. Let \alpha be strictly increasing, let u_n : = \alpha_n² - \alpha_n-1², and let \Theta_n : = \fracu_n+1u_n+2 for n >= 0. If W_\alpha is 2-hyponormal then

1 <= \Thetan <= \Theta0 ( \frac|| W\alpha ||2\alpha0\alpha1 )2

for sufficiently large n. In particular, u_n is eventually decreasing.

http://www.math.uiowa.edu/~curto

Date received: May 5, 1999

Copyright © 1999 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cacw-64.