[Paper Review] On the spectral Theory of Operator Measures

M. M. Malamud, Семен Маркович Маламуд|ArXiv.org|Jun 21, 2002

Spectral Theory in Mathematical Physics4 references3 citations

TL;DR

This paper develops a spectral theory for nonorthogonal operator measures on Hilbert spaces, providing an inner characterization of the $ L_2(\Sigma, H) $ space via direct integral decomposition and establishing Hellinger spectral types through subspaces and vector chains. A key contribution is the construction of Hellinger chains and an analog of the Jordan decomposition for operator measures of weakly bounded variation, extending classical results to nonorthogonal settings.

ABSTRACT

In the first section we provide a solution to the M. G. Krein problem about an inner description of the space $L_2(Σ,H).$ In the second section we introduce the multiplicity function for an operator measure. Making use of the description of the space $L_2(Σ,H)$ we establish the correctness of the definition and give a criterion for a spectral measure to be a dilation of a given operator measure. In the third section we prove that the set of principal vectors of an operator measure is an everywhere dense $G_δ.$ This implies, in particular, that there are a lot of principal vectors in any cyclic subspace of a selfadjoint operator. In the 4th section we introduce Hellinger spectral types for an arbitrary operator measure and prove the existence of subspaces, realizing them. In the 5-th section we give an answer to an old question of Paulsen and provide an analytic description of completely bounded operator measures. The results of this section belong to the second author.

Motivation & Objective

• To provide an inner description of the Hilbert space $ L_2(\Sigma, H) $, addressing a problem posed by M. G. Kreín.
• To develop a theory of Hellinger spectral types for nonorthogonal operator measures, generalizing concepts from orthogonal measures.
• To establish the existence of subspaces realizing specific Hellinger spectral types, including vectors of maximal type.
• To prove an analog of the Jordan Theorem for operator measures, characterizing those of weakly bounded variation as differences of positive measures.
• To extend the Beresetskii-Gelfand-Kostyuchenko (BGK) theorem to nonorthogonal measures and provide a simplified proof for finite-dimensional cases.

Proposed method

• Use the BGK theorem to show that for $ T ∈ \mathfrak{S}_2(H) $ with trivial kernel, the operator measure $ \Sigma_T(\Delta) = T^*\Sigma(\Delta)T $ admits a density $ \Psi(t) = d\Sigma_T/d\rho \in \mathfrak{S}_1(H) $ $ \rho $-a.e.
• Define the Hilbert space $ \widetilde{\mathfrak{H}}_t $ as the completion of $ \mathcal{D}(T^{-1}) $ under the seminorm $ \|h\|_{\widetilde{\mathfrak{H}}_t} = \|\Psi(t)^{1/2}T^{-1}h\| $, and let $ \mathfrak{H}_t $ be the quotient space.
• Prove that $ L_2(\Sigma, H) $ is isometrically isomorphic to the direct integral $ \int_{\mathbb{R}}^\oplus \mathfrak{H}_t \, d\rho(t) $, with norm identity $ \|f\|^2_{L_2} = \int_\mathbb{R} \|\Psi(t)^{1/2}T^{-1}f(t)\|^2 d\rho(t) $.
• Define the multiplicity function $ N_\Sigma(t) = \sup_n \text{rank}(\Psi_n(t)) $, where $ \Psi_n(t) $ is the $ n \times n $ matrix of Radon-Nikodym derivatives.
• Introduce Hellinger subspaces $ H_k $ as nested subspaces where the $ k $-th spectral type is realized, and characterize them via the non-vanishing of $ \det \Psi_k(t) $.
• Establish a Jordan-type decomposition for operator measures of weakly bounded variation by showing $ \Sigma = \Sigma_1 - \Sigma_2 $ iff $ \sup_\pi \sum_i \|T^*\Sigma(\Delta_i)T\|_1 < \infty $ for all $ T \in \mathfrak{S}_2(H) $.

Experimental results

Research questions

• RQ1 How can the space $ L_2(\Sigma, H) $ be characterized intrinsically, independent of vector-valued function representations?
• RQ2 What is the structure of Hellinger spectral types for nonorthogonal operator measures, and how can they be realized via subspaces?
• RQ3 Under what conditions can an operator measure of weakly bounded variation be decomposed as a difference of two positive operator measures?
• RQ4 How do the spectral types of a selfadjoint operator relate to the structure of cyclic subspaces and the associated operator measure?
• RQ5 What is the role of the BGK theorem in constructing a direct integral representation of $ L_2(\Sigma, H) $, and how does it generalize to nonorthogonal measures?

Key findings

• The space $ L_2(\Sigma, H) $ is isometrically isomorphic to the direct integral $ \int_{\mathbb{R}}^\oplus \mathfrak{H}_t \, d\rho(t) $, with the norm identity $ \|f\|^2_{L_2} = \int_\mathbb{R} \|\Psi(t)^{1/2}T^{-1}f(t)\|^2 d\rho(t) $, providing an inner characterization of the space.
• The set of vectors of maximal type in a cyclic subspace $ L $ is a $ G_\delta $-dense set of second category, implying genericity of maximal type vectors.
• For any $ h \in \Omega_\Sigma $, there exists a Hellinger chain $ \{\lambda h\} = H_1 \subset H_2 \subset \cdots \subset H_m $ with $ \dim H_k = k $, such that $ \Gamma_k(P_L\Sigma|_L) = \Gamma_k(\Sigma) $.
• A chain $ H_1 \subset \cdots \subset H_m $ is a Hellinger chain if and only if $ \Gamma_k(\Sigma) = \{ t : \det \Psi_k(t) \neq 0 \} \mod \Sigma $, linking spectral types to the non-vanishing of minors of the density matrix.
• The multiplicity function $ N_\Sigma(t) $ is independent of the choice of orthonormal basis in $ H $, and $ m(\Sigma) = \esssup N_\Sigma(t) \mod \rho $.
• An operator measure-charge $ \Sigma $ of weakly bounded variation admits a Jordan decomposition $ \Sigma = \Sigma_1 - \Sigma_2 $ with $ \Sigma_i \geq 0 $ if and only if $ \sup_\pi \sum_i \|T^*\Sigma(\Delta_i)T\|_1 < \infty $ for all $ T \in \mathfrak{S}_2(H) $, generalizing the classical Jordan decomposition to operator-valued measures.