Weakly \(k\)-hyponormal and polynomially hyponormal commuting operator pairs (Q2018929)

The authors introduce the notion of weak \(k\)-hyponormality and polynomial hyponormality for commuting operator pairs on a Hilbert space and investigate their relationship with \(k\)-hyponormality and subnormality. They provide examples of 2-variable weighted shifts which are weakly 1-hyponormal but not hyponormal. By relating the weak \(k\)-hyponormality and \(k\)-hyponormality of a commuting operator pair to positivity of the restriction of some linear functionals to corresponding cones of functions, it is proved that there is an operator pair that is polynomially hyponormal but not 2-hyponormal, generalizing \textit{R. E. Curto} and \textit{M. Putinar}'s result [Bull. Am. Math. Soc., New Ser. 25, No. 2, 373--378 (1991; Zbl 0758.47027); J. Funct. Anal. 115, No. 2, 480--497 (1993; Zbl 0817.47026)] to the two-variable case.