Two-moment characterization of spectral measures on the real line

Abstract

In [30], 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 [43]. 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 = ∫ R 𝑥 𝑘 𝐹 (d 𝑥 ), 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 * -algebras is also provided.

Subnormal nth roots of quasinormal operators are quasinormal

“…The following lemma provides a necessary and sufficient condition for equality to hold in a Kadison-type inequality (see [61,Lemma 3.1] for a prototype of Lemma 3.1). Lemma 3.1 ([62, Lemma 3.2]).…”

“…Finally, the fourth approach (see (vii) and (viii)) concerns multiplicativity of UCP maps on C * -subalgebras generated by normal elements. This is inspired by the "moreover" part of Petz's theorem and its generalizations established first by Arveson in the finite-dimensional case and then by Brown in general (see[57],[9, Theorem 9.4] and[21, Corollary 2.8], respectively; see also[61, Remark 4.3]). For a closer look, we refer the reader to Corollary 5.1.…”