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Disintegration-of-measure techniques for commuting multivariable weighted shifts. (English) Zbl 1092.47025
Let $${\mathcal H}$$ be a Hilbert space and $${\mathcal B}({\mathcal H})$$ be the $$C^*$$-algebra of all bounded operators on $${\mathcal H}$$. For $$S,T\in {\mathcal B}({\mathcal H})$$, let $$[S,T]:=ST-RS$$. One says that a pair $$(T_1,T_2)$$ of operators $$T_1,T_2\in{\mathcal B}({\mathcal H})$$ is (jointly) hyponormal if the operator matrix $$\left( \begin{smallmatrix} [T_1^*,T_1] & [T_2^*,T_1] \cr [T_1^*,T_2] & [T_2^*,T_2] \end{smallmatrix} \right)$$ is positive on $${\mathcal H}\oplus{\mathcal H}$$, a pair $$(T_1,T_2)$$ is normal if $$T_1T_2=T_2T_1$$ and $$T_1,T_2$$ are normal; and a pair $$(T_1,T_2)$$ is subnormal if $$(T_1,T_2)$$ is the restriction of a normal pair to a common invariant subspace. Given a commuting pair of subnormal operators $$T_1$$ and $$T_2$$ on a Hilbert space $${\mathcal H}$$, one asks whether the pair $$(T_1,T_2)$$ is the restriction of a commuting normal pair acting on a larger Hilbert space $${\mathcal K}$$.
The authors study this problem for commuting pairs of weighted shifts on $$\ell^2({\mathbb Z}_+)$$. In that case the existence of the above lifting to a larger space, as proved by N. P. Jewell and A. R. Lubin [J. Oper. Theory 1, 207–223 (1979; Zbl 0431.47016)], is equivalent to the existence of a positive regular Borel probability measure on $${\mathbb R}_+^2$$ (the Berger measure) interpolating the moments generated by the weight sequences. The authors find a new necessary condition for the existence of a Berger measure for a subnormal $$2$$-variable weighted shift in terms of Berger measures for $$1$$-variable weighted shifts associated to horizontal rows and vertical columns of the $$2$$-variable shift. This condition is not sufficient.
The authors produce an example of a commuting, hyponormal, $$2$$-variable weighted shift $$(T_1,T_2)$$ such that $$T_1$$ and $$T_2$$ are subnormal weighted shifts, with mutually absolutely continuous Berger measures for almost all horizontal and vertical slices, and such that $$(T_1,T_2)$$ does not admit a lifting.

##### MSC:
 47B20 Subnormal operators, hyponormal operators, etc. 47B37 Linear operators on special spaces (weighted shifts, operators on sequence spaces, etc.) 47A13 Several-variable operator theory (spectral, Fredholm, etc.) 28A50 Integration and disintegration of measures 44A60 Moment problems 47A20 Dilations, extensions, compressions of linear operators
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