Two-moment characterization of spectral measures on the real line

Abstract

Kiukas, Lahti and Ylinen asked the following general question. When is a positive operator measure projection valued? A version of this question formulated in terms of operator moments was posed in a recent paper of the present authors. Let $T$ be a selfadjoint operator and $F$ be a Borel semispectral measure on the real line with compact support. For which positive integers $p<q$ do the equalities $T^k={\int }_{\mathbb{R}}{x}^{k}F(dx)$, $k=p,q$, imply that $F$ is a spectral measure? In the present paper, we completely solve the second problem. The answer is affirmative if $p$ is odd and $q$ is even, and negative otherwise. The case $(p,q)=(1,2)$ closely related to intrinsic noise operator was solved by several authors including Kruszyński and de Muynck as well as Kiukas, Lahti and Ylinen. The counterpart of the second problem concerning the multiplicativity of unital positive linear maps on $C^{\ast }$-algebras is also solved.

References in corpus (4)

• Positivity of $2\times 2$ block matrices of operators
• How sharp are PV measures?
• Quasinormality of powers of commuting pairs of bounded operators
• Subnormal $n$th roots of quasinormal operators are quasinormal

Cited by in corpus (1)

• The Local Operator Moment Problem on $\mathbb{R}$