48 research outputs found
When is hyponormality for 2-variable weighted shifts invariant under powers?
For 2-variable weighted shifts W_{(\alpha,\beta)}(T_1, T_2) we study the
invariance of (joint) k- hyponormality under the action (h,\ell) ->
W_{(\alpha,\beta)}^{(h,\ell)}(T_1, T_2):=(T_1^k,T_2^{\ell}) (h,\ell >=1). We
show that for every k >= 1 there exists W_{(\alpha,\beta)}(T_1, T_2) such that
W_{(\alpha,\beta)}^{(h,\ell)}(T_1, T_2) is k-hyponormal (all h>=2,\ell>=1) but
W_{(\alpha,\beta)}(T_1, T_2) is not k-hyponormal. On the positive side, for a
class of 2-variable weighted shifts with tensor core we find a computable
necessary condition for invariance. Next, we exhibit a large nontrivial class
for which hyponormality is indeed invariant under all powers; moreover, for
this class 2-hyponormality automatically implies subnormality. Our results
partially depend on new formulas for the determinant of generalized Hilbert
matrices and on criteria for their positive semi-definiteness
A new approach to the 2-variable subnormal completion problem
We study the Subnormal Completion Problem (SCP) for 2-variable weighted
shifts. We use tools and techniques from the theory of truncated moment
problems to give a general strategy to solve SCP. We then show that when all
quadratic moments are known (equivalently, when the initial segment of weights
consists of five independent data points), the natural necessary conditions for
the existence of a subnormal completion are also sufficient. To calculate
explicitly the associated Berger measure, we compute the algebraic variety of
the associated truncated moment problem; it turns out that this algebraic
variety is precisely the support of the Berger measure of the subnormal
completion
k-hyponormality of multivariable weighted shifts
We characterize joint k-hyponormality for 2-variable weighted shifts. Using
this characterization we construct a family of examples which establishes and
illustrates the gap between k-hyponormality and (k+1)-hyponormality for each
k>=1. As a consequence, we obtain an abstract solution to the Lifting Problem
for Commuting Subnormals.Comment: 13 pages; to appear in J. Funct. Ana
Hyponormality and subnormality for powers of commuting pairs of subnormal operators
Let H_0 (resp. H_\infty denote the class of commuting pairs of subnormal
operators on Hilbert space (resp. subnormal pairs), and for an integer k>=1 let
H_k denote the class of k-hyponormal pairs in H_0. We study the hyponormality
and subnormality of powers of pairs in H_k. We first show that if (T_1,T_2) is
in H_1, then the pair (T_1^2,T_2) may fail to be in H_1. Conversely, we find a
pair (T_1,T_2) in H_0 such that (T_1^2,T_2) is in H_1 but (T_1,T_2) is not.
Next, we show that there exists a pair (T_1,T_2) in H_1 such that T_1^mT_2^n is
subnormal (all m,n >= 1), but (T_1,T_2) is not in H_\infty; this further
stretches the gap between the classes H_1 and H_\infty. Finally, we prove that
there exists a large class of 2-variable weighted shifts (T_1,T_2) (namely
those pairs in H_0 whose cores are of tensor form) for which the subnormality
of (T_1^2,T_2) and (T_1,T_2^2) does imply the subnormality of (T_1,T_2)
When is hyponormality for 2-variable weighted shifts invariant under powers?
Abstract. For 2-variable weighted shifts W(α,β) ≡ (T1, T2) we study the invariance of (joint) khyponormality under the action (h, ℓ) 7→ W (h,ℓ) (α,β) := (T h 1 , T ℓ 2 ) (h, ℓ ≥ 1). We show that for every k ≥ 1 there exists W(α,β) such that W (h,ℓ) (α,β) is k-hyponormal (all h ≥ 2, ℓ ≥ 1) but W(α,β) is not k-hyponormal. On the positive side, for a class of 2-variable weighted shifts with tensor core we find a computable necessary condition for invariance. Next, we exhibit a large nontrivial class for which hyponormality is indeed invariant under all powers; moreover, for this class 2-hyponormality automatically implies subnormality. Our results partially depend on new formulas for the determinant of generalized Hilbert matrices and on criteria for their positive semi-definiteness