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We prove that for any self-adjoint operator A in a separable Hilbert space H and a given countable set Λ = {λi} i∈N of real numbers, there exist H−2-singular perturbationsà of A such that Λ ⊂ σp`ô. In particular, if Λ = {λ1, . . . , λn} is finite, then the operatorà solving the eigenvalues problem,Ãψ k = λ k ψ k , k = 1, . . . , n, is uniquely defined by a given set of orthonormal vectors {ψ k } n k=1 satisfying the condition span{ψ k } n k=1 ∩ dom`|A| 1/2´= {0}.

We give a matrix representation for the resolvent of the Friedrichs extension of some semibounded 2 x 2 operator matrices and study their essential spectrum. IntroductionRecently, V. Hardt, R. Mennlcken and S. Naboko [HMN] have investigated the spectral properties of some systems of singular differential operators of mixed order closely connected with one dimensional magnetohydrodynamic problems. In particular, they have studied the essential spectrum of a suitable self-adjoint extension L of the corresponding singular symmetric matrix differential operator L0. The question whether L is the Friedrichs extension of L0 remained open. In this note we give a positive answer to this question. For this purpose we consider abstract symmetric semibounded 2 x 2 matrix operators of the form acting in the product of Hilbert spaces 7-t = ~1 x ?-t2 and give an explicit formula for the resolvent of their Friedrichs extensions.In Section 1 we derive this representation of the resolvent of the Friedrichs extensions L of operators L0 of type (0.1) under some natural assumptions on the entries A, B, C and D. In the special case of the matrix differential operator studied in [HMN] we show that its Friedrichs extension coincides with the self-adjoint extension constructed in the [HMN]-paper. In Section 2 we consider the more special case when L0 is a relatively form-bounded perturbation of the diagonal operator 0(0 o)In Section 3 we prove an abstract result on the essential spectrum of matrix operators generalizing in the self-adjoint case results of [ALMS], [S] and [Ko]. In Section 4 we calculate the essential spectrum of a model mixed order differential operator.

For two types of stochastic particle systems in R d we show non-explosion in finite time by proving that their respective generators are L 1 ðmÞ-unique, where m is their respective invariant (in these cases even symmetrizing) measure. We also prove the much harder L 2 ðmÞ-uniqueness in both models. r 2004 Elsevier Inc. All rights reserved.

Let A be a self-adjoint operator in a separable Hilbert space H. Let {ψ k : k ≥ 1} be a given (finite or infinite) orthonormal system such that span{ψ k : k ≥ 1} ∩ dom(A) = {0} and let Λ := {λ k : k ≥ 1} be an arbitrary sequence of real numbers. Conditions for the existence of a pure singular per-turbationà of A solving the inverse eigenvalue problemÃψ k = λ k ψ k , k ≥ 1 are given.

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A singular rank one perturbation A α = A + α ϕ, · ϕ of a selfadjoint operator A in a Hilbert space H is considered, where 0 = α ∈ R∪∞ and ϕ ∈ H −2 but ϕ / ∈ H −1 , with H s , s ∈ R, the usual A−scale of Hilbert spaces. A modified version of the Aronszajn-Krein formula is given. It has the form F α (z) = F (z)−α 1+αF (z) where F α denotes the regularized Borel transform of the scalar spectral measure of A α associated with ϕ. Using this formula we develop a variant of the well known Aronszajn-Donoghue spectral theory for a general rank one perturbation of the H −2 class. Mathematics Subject Classification (2000). Primary 47A10; Secondary 47A55.

We prove an abstract theorem on the preservation of the absolutely continuous spectrum for block operator matrices.