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Abstract. In this paper we prove the spectral theorem for quaternionic unbounded normal operators using the notion of S-spectrum. The proof technique consists of first establishing a spectral theorem for quaternionic bounded normal operators and then using a transformation which maps a quaternionic unbounded normal operator to a quaternionic bounded normal operator. With this paper we complete the foundation of spectral analysis of quaternionic operators. The S-spectrum has been introduced to define the quaternionic functional calculus but it turns out to be the correct object also for the spectral theorem for quaternionic normal operators. The fact that the correct notion of spectrum for quaternionic operators was not previously known has been one of the main obstructions to fully understanding the spectral theorem in this setting. A prime motivation for studying the spectral theorem for quaternionic unbounded normal operators is given by the subclass of unbounded anti-self adjoint quaternionic operators which play a crucial role in the quaternionic quantum mechanics.

Abstract. The quaternionic spectral theorem has already been considered in the literature, see e.g. [22], [31], [32], however, except for the finite dimensional case in which the notion of spectrum is associated to an eigenvalue problem, see [21], it is not specified which notion of spectrum underlies the theorem.In this paper we prove the quaternionic spectral theorem for unitary operators using the S-spectrum. In the case of quaternionic matrices, the S-spectrum coincides with the right-spectrum and so our result recovers the well known theorem for matrices. The notion of S-spectrum is relatively new, see [17], and has been used for quaternionic linear operators, as well as for n-tuples of not necessarily commuting operators, to define and study a noncommutative versions of the Riesz-Dunford functional calculus. The main tools to prove the spectral theorem for unitary operators are the quaternionic version of Herglotz's theorem, which relies on the new notion of q-positive measure, and quaternionic spectral measures, which are related to the quaternionic Riesz projectors defined by means of the S-resolvent operator and the S-spectrum. The results in this paper restore the analogy with the complex case in which the classical notion of spectrum appears in the Riesz-Dunford functional calculus as well as in the spectral theorem.

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The truncated matrix-valued K-moment problem on R d , C d , and T d will be considered. The truncated matrix-valued K-moment problem on R d requires necessary and sufficient conditions for a multisequence of Hermitian matrices {S γ } γ∈Γ (where Γ is a finite subset of N d 0) to be the corresponding moments of a positive Hermitian matrix-valued Borel measure σ, and also the support of σ must be contained in some given non-empty set K ⊆ R d , i.e., (0.1) S γ = R d ξ γ dσ(ξ), for all γ ∈ Γ, and (0.2) supp σ ⊆ K. Given a non-empty set K ⊆ R d and a finite multisequence, indexed by a certain family of finite subsets of N d 0 , of Hermitian matrices we obtain necessary and sufficient conditions for the existence of a minimal finitely atomic measure which satisfies (0.1) and (0.2). In particular, our result can handle the case when Γ = {γ ∈ N d 0 : 0 ≤ |γ| ≤ 2n + 1}. We will also discuss a similar result in the multivariable complex and polytorus setting.