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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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