Towards a model theory for 2-hyponormal operators |
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Authors: | Raúl E Curto and Woo Young Lee |
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Institution: | (1) Department of Mathematics, University of Iowa, 52242 Iowa City, IA;(2) Department of Mathematics, Sungkyunkwan University, 440-746 Suwon, Korea |
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Abstract: | We introduce the notion ofweak subnormality, which generalizes subnormality in the sense that for the extension
ofT
we only require that
hold forf
; in this case we call
a partially normal extension ofT. After establishing some basic results about weak subnormality (including those dealing with the notion of minimal partially normal extension), we proceed to characterize weak subnormality for weighted shifts and to prove that 2-hyponormal weighted shifts are weakly subnormal. Let {
n
}
n=0
be a weight sequence and letW
denote the associated unilateral weighted shift on
. IfW
is 2-hyponormal thenW
is weakly subnormal. Moreover, there exists a partially normal extension
on
such that (i)
is hyponormal; (ii)
; and (iii)
. In particular, if is strictly increasing then
can be obtained as whereW
is a weighted shift whose weight sequence {
n
·
n=0
is given by In this case,
is a minimal partially normal extension ofW
. In addition, ifW
is 3-hyponormal then
can be chosen to be weakly subnormal. This allows us to shed new light on Stampfli's geometric construction of the minimal normal extension of a subnormal weighted shift. Our methods also yield two additional results: (i) the square of a weakly subnormal operator whose minimal partially normal extension is always hyponormal, and (ii) a 2-hyponormal operator with rank-one self-commutator is necessarily subnormal. Finally, we investigate the connections of weak subnormality and 2-hyponormality with Agler's model theory.Supported by NSF research grant DMS-9800931.Supported by the Brain Korea 21 Project from the Korean Ministry of Education. |
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Keywords: | 2000 Mathematics Subject Classification" target="_blank">2000 Mathematics Subject Classification Primary 47B20 47B35 47B37 Secondary 47-04 47A20 47A57 |
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