Spectral theory of operator measures in Hilbert space

Abstract

Abstract. In §2 the spaces L 2 (Σ, H) are described; this is a solution of a problem posed by M. G. Kreȋn.In §3 unitary dilations are used to illustrate the techniques of operator measures. In particular, a simple proof of the Naȋmark dilation theorem is presented, together with an explicit construction of a resolution of the identity. In §4, the multiplicity function N Σ is introduced for an arbitrary (nonorthogonal) operator measure in H. The description of L 2 (Σ, H) is employed to show that this notion is well defined. As a supplement to the Naȋmark dilation theorem, a criterion is found for an orthogonal measure E to be unitarily equivalent to the minimal (orthogonal) dilation of the measure Σ.In §5 it is proved that the set Ω Σ of all principal vectors of an arbitrary operator measure Σ in H is massive, i.e., it is a dense G δ -set in H. In particular, it is shown that the set of principal vectors of a selfadjoint operator is massive in any cyclic subspace.In §6, the Hellinger types are introduced for an arbitrary operator measure; it is proved that subspaces realizing these types exist and form a massive set.In §7, a model of a symmetric operator in the space L 2 (Σ, H) is studied.

Second‐order trace formulas

“…In particular, in finite-dimensional Hilbert space, we have explicit expressions of shift functions as done in the case of Krein trace formula for a pair of self-adjoint operators by Voiculescu [60] and Sinha and Mohapatra [37]. Moreover, our approach gives a better bound (32) for the shift function compared to (10).…”

“…\end{equation}$$It is important to note that, for the first time, semispectral measures in perturbation theory (in particular, in the context of double operator integrals) were used in [41]. In this connection, it is important to note that the existence of integrals with respect to a semispectral measure $$\mathcal {E}_T(\cdot)$$ appeared in the above formula (11) as well as direct functional calculus (without using the dilation reasoning) is discussed in detail in [32]. Moreover, a concept of multiplicity function for a semispectral measure is discussed there and it was established for the first time in [32] that the measures $$\mathcal {E}_T(\cdot)$$ and $$E_U(\cdot)$$ are spectrally equivalent.…”

“…In this connection, it is important to note that the existence of integrals with respect to a semispectral measure $$\mathcal {E}_T(\cdot)$$ appeared in the above formula (11) as well as direct functional calculus (without using the dilation reasoning) is discussed in detail in [32]. Moreover, a concept of multiplicity function for a semispectral measure is discussed there and it was established for the first time in [32] that the measures $$\mathcal {E}_T(\cdot)$$ and $$E_U(\cdot)$$ are spectrally equivalent. Note also that a simple proof of the classical Naimark's theorem [39] was obtained in [32].…”

“…Moreover, a concept of multiplicity function for a semispectral measure is discussed there and it was established for the first time in [32] that the measures $$\mathcal {E}_T(\cdot)$$ and $$E_U(\cdot)$$ are spectrally equivalent. Note also that a simple proof of the classical Naimark's theorem [39] was obtained in [32]. For more on semispectral measures and related stuff, we refer to [5].…”

An L2 model for selfadjoint elliptic differential operators with constant coefficients on bounded domains

“…Let Σ : R → L(L 2 (∂Ω)) be the nondecreasing operator function from the integral representation (2). The space L 2 Σ (L 2 (∂Ω)) is defined as in [2,7,12]. Very roughly speaking it consists of L 2 (∂Ω)-valued functions on R which are squareintegrable with respect to the measure dΣ.…”