We develop a new approach and present an independent solution to von Neumann's problem on the parametrization in explicit form of all nonnegative self-adjoint extensions of a densely defined nonnegative symmetric operator. Our formulas are based on the Friedrichs extension and also provide a description for closed sesquilinear forms associated with nonnegative self-adjoint extensions. All basic results of the well-known Kreȋn and BirmanVishik theory and its complementations are derived as consequences from our new formulas, including the parametrization (in the framework of von Neumann's classical formulas) for all canonical resolvents of nonnegative selfadjoint extensions. As an application all nonnegative quantum Hamiltonians corresponding to point-interactions in R 3 are described. Mathematics Subject Classification (2000). Primary 47A63, 47B25; Secondary 47B65.

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We develop direct and inverse spectral analysis for finite and semi-infinite non-self-adjoint Jacobi matrices with a rank-one imaginary part. It is shown that given a set of n not necessarily distinct nonreal numbers in the open upper (lower) half-plane uniquely determines an n × n Jacobi matrix with a rank-one imaginary part having those numbers as its eigenvalues counting algebraic multiplicity. Algorithms of reconstruction for such finite Jacobi matrices are presented. A new model complementing the well-known Livsic triangular model for bounded linear operators with a rank-one imaginary part is obtained. It turns out that the model operator is a non-self-adjoint Jacobi matrix. We show that any bounded, prime, nonself-adjoint linear operator with a rank-one imaginary part acting on some finite-dimensional (respectively separable infinite-dimensional Hilbert space) is unitarily equivalent to a finite (respectively semi-infinite) non-self-adjoint Jacobi matrix. This obtained theorem strengthens a classical result of Stone established for self-adjoint operators with simple spectrum. We establish the non-self-adjoint analogs of the Hochstadt and Gesztesy-Simon uniqueness theorems for finite Jacobi matrices with nonreal eigenvalues as well as an extension and refinement of these theorems for finite non-self-adjoint tri-diagonal matrices to the case of mixed eigenvalues, real and nonreal. A unique Jacobi matrix, unitarily equivalent to the operator of integration (F f )(x) = 2i l x f (t) dt in the Hilbert space L 2 [0, l] is found as well as spectral properties of its perturbations and connections with the well-known Bernoulli numbers. We also give the analytic characterization of the Weyl functions of dissipative Jacobi matrices with a rank-one imaginary part.

Passive linear systems τ = {A, B, C, D; H, M, N} have their transfer function Θτ (λ) = D + λC(I − λA) −1 B in the Schur class S(M, N). Using a parametrization of contractive block operators the transfer function Θτ (λ) is connected to the Sz.-Nagy-Foiaş characteristic function ΦA(λ) of the contraction A. This gives a new aspect and some explicit formulas for studying the interplay between the system τ and the functions Θτ (λ) and ΦA(λ). The method leads to some new results for linear passive discrete-time systems. Also new proofs for some known facts in the theory of these systems are obtained. Mathematics Subject Classification (2000). Primary 47A45, 47A48, 47A56, 47N70; Secondary 93B15.