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The present paper is the first one in a series of two papers devoted to a unified approach to the problem of completeness of the generalized eigenvectors (the root vectors) for a specific class of linear non-selfadjoint unbounded differential operators. The list of the problems for which such operators are the dynamics generators includes the following: (a) initial boundary-value problem (IBVP) for a non-homogeneous string with both distributed and boundary damping; (b) IBVP for small vibrations of an ideal filament with dissipative boundary condition at one end and with a heavy load at the other end; (c) IBVP for a three-dimensional damped wave equation with spherically symmetric coefficients and both distributed and boundary damping. In the second paper of the series the following problems will be considered: (a) IBVP for a system of two coupled hyperbolic equations constituting a Timoshenko beam model with variable coefficients and boundary damping; (b) IBVP for a coupled Euler–Bernoulli and Timoshenko beam model with boundary energy dissipation (the model known in engineering literature as bending-torsion vibration model); (c) IBVP for a system of two Timoshenko beams coupled through linear Van der Waals forces. The model of (c) describes vibrational motion of a double-walled carbon nanotube. In all of the above cases, the result has been obtained by using Krien’s Theorem on completeness of root vectors of a dissipative operator with a nuclear imaginary part. © 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

In this paper, we study the determinant of perturbation connected with the dissipative operator L generated in L^2(I) by (1.1)-(1.5). Then using Livsic's theorem, we investigate the problem of completeness of the system of eigenfunctions and associated functions of L.

In this paper, we shall research on a class of thermo-viscous-elastic anisotropic dissipative material system equation. The analysis semi-group theory and two-dimensional singular integral operator identical relations are used in this paper; the equations denoted by some singular integral operator equalities are studied in two-dimensional round field. We shall push out the properties of the equations, the necessary and sufficient conditions of solvability. Furthermore, the formulas of their exponent operation are pushed out too.

In this paper we are interested in studying the dissipativity of degenerate mixed differential operators involving an interface point. We show that, under particular interface conditions, such operators generate analytic semigroups on an appropriate Hilbert space H . To illustrate the results an example is discussed.

This paper considers observer and damping feedback design for control affine systems that admit an approximate dissipative Hamiltonian realization. The approach consists in constructing a locally-defined dissipative potential function by homotopy decomposition of a differential one-form associated to the system drift vector field. Under a condition on the obtained potential function convexity, a coordinate transformation is then defined such that the original system can be approximated by a dissipative Hamiltonian system with a constant structure. Using the approximate dissipative Hamiltonian realization, damping control and observer design are then carried, and the limitations associated to the approximation are discussed. Application to a predator-prey system is presented to illustrate the construction.

International audience; We study the semiclassical measures for the solution of a dissipative Helmholtz equation with a source term concentrated on a bounded submanifold. The potential is not assumed to be non-trapping, but trapped trajectories have to go through the region where the absorption coefficient is positive. In that case, the solution is microlocally written around any point away from the source as a sum (finite or infinite) of lagragian distributions. Moreover we prove and use the fact that the outgoing solution of the dissipative Helmholtz equation is microlocally zero in the incoming region.

The purpose of this paper is to study the existence of fixed points for the sum of two nonlinear operators in the framework of real Banach spaces. Later on, we give some examples of applications of this type of results (© 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

We study the topological structure of singular (in the sense of the Feigenbaum-Sharkovskii-Magnitskii theory) attractors of nonlinear dissipative systems of differential equations. We show that any such attractor is a stable nonperiodic trajectory lying on a two-dimensional infinitely folded heteroclinic separatrix manifold generated by the unstable two-dimensional invariant manifold of the original singular cycle as the bifurcation parameter of the system varies. The results obtained for two-dimensional nonautonomous and three-dimensional autonomous dissipative systems are generalized to autonomous multi- and infinite-dimensional dissipative systems as well as to conservative (in particular, Hamiltonian) systems.

We provide a criterion giving a formula for the directional (or contingent) subdifferential of the difference of two convex functions. We even extend it to the difference of two approximately starshaped functions. Our analysis relies on a notion of approximate monotonicity for operators which is much less demanding than the usual one.

Adapting Mourre's commutator method to the dissipative setting, we prove a limiting absorption principle for a class of abstract dissipative operators. A consequence is the resolvent estimates for the high frequency Helmholtz equation when trapped trajectories meet the set where the imaginary part of the potential is non-zero. We also give the resolvent estimates in Besov spaces.

In this paper, we determine the normalization factor of photo-added squeezed vacuum state (PASVS) as a Legendre polynomial of the squeezing parameter by virtue of principle of mathematical induction, and derive the normally ordered density operator of PASVS in dissipative channel. In addition, we also give the explicit analytical expression of Husimi function in dissipative channel and study its behavior with evolution time graphically.

A class of infinite-dimensional dissipative dynamical systems is defined for which the slow invariant manifolds can be calculated. Large-time behavior of the evolution of such systems is studied.

Degenerate parametric amplification accompanied by dissipation is analyzed within the canonical operator formalism for quantum dissipative systems named non-equilibrium thermo field dynamics. The vacuum of the system is subject to both dissipation and breaking of phase symmetry due to squeezing. The annihilation-creation operators for the vacuum are derived and the structure of the vacuum is examined. The effects of dissipation on squeezing and uncertainty relation are estimated.

In this paper we prove the analyticity of the semigroups generated by some singular differential matrix operators of the form 2 2 () =, dB x d x dx dx   in the Banach space ([0, ], ( )), d CM C  with suitable boundary con

A method is presented for constructing point vortex models in the plane that dissipate the Hamiltonian function at any prescribed rate and yet conserve the level sets of the invariants of the Hamiltonian model arising from the SE (2) symmetries. The method is purely geometric in that it uses the level sets of the Hamiltonian and the invariants to construct the dissipative field and is based on elementary classical geometry in ℝ3. Extension to higher-dimensional spaces, such as the point vortex phase space, is done using exterior algebra. The method is in fact general enough to apply to any smooth finite-dimensional system with conserved quantities, and, for certain special cases, the dissipative vector field constructed can be associated with an appropriately defined double Nambu–Poisson bracket. The most interesting feature of this method is that it allows for an infinite sequence of such dissipative vector fields to be constructed by repeated application of a symmetric linear operator (matrix) at each point of the intersection of the level sets.

In the Redfield theory framework, we consider the problem of the vibrational dynamics in dissipative systems. We decompose the Hamiltonian of interaction of the observed system with a thermal bath into terms that are products of system transition operators and bath transition operators. Using the decomposition, we construct a correlation function matrix containing all the information about the interaction of the system with the bath and obtain a new operator form of the Redfield equation. We consider the procedure for factoring the interaction operator and constructing the correlation function matrix. Using the diagonalization of the obtained matrix, we give correlated dissipation operators, whose introduction simplifies the form of the Redfield equations. We show that for several problems in which fundamental transition frequencies can be chosen, this procedure significantly reduces the dimensionality of the dissipative dynamics problem.

Using a version of Erhling's inequality for L p -Sobolev spaces H s,p on ℝ n , −∞ < s < +∞, 1 ≤ p < ∞ given by Wong in his paper [M.W. Wong, Erhling's inequality and pseudo-differential operators on L p (ℝ n ), Cubo 8 (2006), pp. 97–108]. we establish an analogue of Agmon–Douglis–Nirenberg inequality for M-hypoelliptic pseudo-differential operators perturbed by singular potentials on L p (ℝ n ), 1 < p < ∞. We also state and prove some facts concerning the essential spectra of M-hypoelliptic pseudo-differential operators T σ on L p (ℝ n ), 1 < p < ∞, perturbed by the operators of the form . A self-adjointness result is also proved for such perturbations of M-hypoelliptic pseudo-differential operators on L 2(ℝ n ) whose symbols are independent of x in ℝ n . Moreover, as in the work [M.W. Wong, Erhling's inequality and pseudo-differential operators on L p (ℝ n ), Cubo 8 (2006), pp. 97–108] by Wong an application to strongly continuous semigroups of contractions generated by M-hypoelliptic pseudo-differential...

We study a system of two first-order differential equations arising in averaging nonlinear systems over fast single-frequency oscillations. We consider the situation where the original system contains weak dissipative terms. We construct the asymptotic form of a two-parameter solution with an unbounded increasing amplitude. This result gives a key for understanding autoresonance in weak dissipative systems as a phenomenon of significant increase in the forced nonlinear oscillation initiated by a small external pumping.

Functional models are constructed for commutative systems of bounded linear non-self-adjoint operators which do not contain dissipative operators (which means that is not a dissipative operator for any , ). A significant role is played here by the de Branges transform and the function classes occurring in this context. Classes of commutative systems of operators for which such a construction is possible are distinguished. Realizations of functional models in special spaces of meromorphic functions on Riemann surfaces are found, which lead to reasonable analogues of de Branges spaces on these Riemann surfaces. It turns out that the functions and determining the order of growth in de Branges spaces on Riemann surfaces coincide with the well-known Baker-Akhiezer functions. Bibliography: 11 titles.

We consider the operator function L(α, θ) = A(α) − θR of two complex arguments, where A(α) is an analytic operator function defined in some neighborhood of a real point α0 ∈ ℝ and ranging in the space of bounded operators in a Hilbert space ℋ. We assume that A(α) is a dissipative operator for real α in a small neighborhood of the point α0 and R is a bounded positive operator; moreover, the point α0 is a normal eigenvalue of the operator function L(α, θ0) for some θ0 ∈ ℝ, and the number θ0 is a normal eigenvalue of the operator function L(α0θ). We obtain analogs and generalizations of well-known results for self-adjoint operator functions A(α) on the decomposition of α- and θ-eigenvalues in neighborhoods of the points α0 and θ0, respectively, and on the representation of the corresponding eigenfunctions by series.

In the current paper, we derive a rigorous convergence analysis for a broad range of splitting schemes applied to abstract nonlinear evolution equations, including the Lie and Peaceman-Rachford splittings. The analysis is in particular applicable to (possibly degenerate) quasilinear parabolic problems and their dimension splittings. The abstract framework is based on the theory of maximal dissipative operators, and we both give a summary of the used theory and some extensions of the classical results. The derived convergence results are illustrated by numerical experiments.

A class of infinite-dimensional dissipative dynamical systems is defined for which there exists a unique equilibrium point, and the rate of convergence to this point of the trajectories of a dynamical system from the above class is exponential. All the trajectories of the system converge to this point as t → +∞, no matter what the initial conditions are. This class consists of strongly dissipative systems. An example of such systems is provided by passive systems in network theory (see, e.g., MR0601947 (83m:45002)).

We analyze the motion of a particle governed by a generalized Langevin equation with fractional derivative, nonlocal dissipative force and linear external force. We consider the dissipative memory kernel given by the exponential and power-law functions. For these cases, one can obtain exact results for the relaxation function. The long-time behavior of this function is also investigated. Our results show that the inertial term has no significant influence on the long-time behavior.

We construct an example of a real Banach space whose group of surjective isometries has no uniformly continuous one-parameter semigroups, but the group of surjective isometries of its dual contains infinitely many of them. Other examples concerning numerical index, hermitian operators and dissipative operators are also shown., To appear in J. Funct. Anal

We show that stationary statistical properties for uniformlydissipative dynamical systems are upper semi-continuous underregular perturbation and a special type of singular perturbationin time of relaxation type. The results presented are applicableto many physical systems such as the singular limit of infinitePrandtl-Darcy number in the Darcy-Boussinesq system forconvection in porous media, or the large Prandtl asymptotics forthe Boussinesq system.

We analyze the motion of a particle governed by a generalized Langevin equation with nonlocal dissipative force, linear external force and a constant load force. We consider the dissipative memory kernel consisting of two terms. One of them is described by the Dirac delta function which represents a local friction, whereas for the second one we consider two types: the exponential and power-law functions which represent nonlocal dissipative forces. For these cases, one can obtain exact results for the relaxation function. Then, we obtain the first moments and variances of the displacement and velocity. The long-time behaviors of these quantities are also investigated.

We consider a class of abstract evolution systems governed by continuous perturbations of m -dissipative operators and we prove a necessary and sufficient condition for a given locally closed set to be C 0 -viable. We conclude with a comparison result for a fully nonlinear reaction–diffusion system.

We prove the well-posed solvability in the strong sense of the boundary value Problems $$ \begin{gathered} ( - 1)\frac{{_m d^{2m + 1} u}} {{dt^{2m + 1} }} + \sum\limits_{k = 0}^{m - 1} {\frac{{d^{k + 1} }} {{dt^{k + 1} }}} A_{2k + 1} (t)\frac{{d^k u}} {{dt^k }} + \sum\limits_{k = 1}^m {\frac{{d^k }} {{dt^k }}} A_{2k} (t)\frac{{d^k u}} {{dt^k }} + \lambda _m A_0 (t)u = f, \hfill \\ t \in ]0,t[,\lambda _m \geqslant 1, \hfill \\ {{d^i u} \mathord{\left/ {\vphantom {{d^i u} {dt^i }}} \right. \kern-\nulldelimiterspace} {dt^i }}|_{t = 0} = {{d^j u} \mathord{\left/ {\vphantom {{d^j u} {dt^j }}} \right. \kern-\nulldelimiterspace} {dt^j }}|_{t = T} = 0,i = 0,...,m,j = 0,...,m - 1,m = 0,1,..., \hfill \\ \end{gathered} $$ where the unbounded operators A s (t), s > 0, in a Hilbert space H have domains D(A s (t)) depending on t, are subordinate to the powers A 1−(s−1)/2m (t) of some self-adjoint operators A(t) ≥ 0 in H, are [(s+1)/2] times differentiable with respect to t, and satisfy some inequalities. In the space H, the maximally accretive operators A 0(t) and the symmetric operators A s (t), s > 0, are approximated by smooth maximally dissipative operators B(t) in such a way that $$ \begin{gathered} \mathop {lim}\limits_{\varepsilon \to 0} Re(A_0 (t)B_\varepsilon ^{ - 1} (t)(B_\varepsilon ^{ - 1} (t))^ * u,u)_H = Re(A_0 (t)u,u)_H \geqslant c(A(t)u,u)_H \hfill \\ \forall u \in D(A_0 (t)),c > 0, \hfill \\ \end{gathered} $$ , where the smoothing operators are defined by $$ B_\varepsilon ^{ - 1} (t) = (I - \varepsilon B(t))^{ - 1} ,(B_\varepsilon ^{ - 1} (t)) * = (I - \varepsilon B^ * (t))^{ - 1} ,\varepsilon > 0. $$ .

We present a quantum theory of decoherence for a mesoscopic dissipative circuit. Using Kirchhoff laws of classical circuit and quantum theory, a Hamitonian of the circuit is derived. The dissipative circuit elements (external impedance, shunt resistors) are described using the Caldeira–Leggett model. The master equation for the dissipative circuit in the Born–Markov approximation is derived and brought into the well-known Bloch–Redfield formalism. Then, we study decoherence properties of multilevel dissipative quantum circuit. At a very low temperature, energy relaxation time and decoherence time of the circuit are give when adopting two-level state approximation.

In this paper we consider a dissipative Schrodinger boundary value problem in the limit-circle case with the spectral parameter in the boundary condition. The approach is based on the use of the maximal dissipative operator, and the spectral analyzes of this operator is adequate for the boundary value problem to be solved. We construct a self-adjoint dilation of the maximal dissipative operator and its incoming and outgoing spectral representations, which make it possible to determine the scattering matrix of the dilation. We also construct a functional model of the maximal dissipative operator and define its characteristic function in terms of solutions of the corresponding Schrodinger equation. We prove theorems on the completeness of the system of eigenvectors and associated vectors of the maximal dissipative operator and the Schrodinger boundary value problem. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

We construct the symmetric functional model of an arbitrary closed operator with non-empty resolvent set acting on a separable Hilbert space. The construction is based on the explicit form of the Sz.-Nagy-Foias model of a closed dissipative operator, the Potapov-Ginzburg transform of characteristic functions, and certain resolvent identities. All considerations are carried out under minimal assumptions, and obtained results are directly applicable to problems typically arising in mathematical physics. Explicit formulae for all the objects participating in the model construction are provided.

We consider the cubic nonlinear Schrodinger equations of derivative type with small initial data. We present a structural condition on the nonlinear terms under which the corresponding Cauchy problem has a dissipative nature.

It has been found that Markovian quantum dissipative processes, described by the Lindblad equation, may have attractive steady-state manifolds, in which dissipation and decoherence can play a positive role to quantum information processing. In this article, we show that such attractive steady-state manifolds with the same positive features as in Markovian dissipative processes can exist in the non-Markovian dissipative processes. Our finding indicates that the dissipation-assisted schemes implemented in the Markovian systems can be directly generalized to the non-Markovian systems., 5 pages, no figure

Whether or not the solution to 2D resistive MHD equations is globally smooth remains open. This paper establishes the global regularity of solutions to the 2D almost resistive MHD equations, which require the dissipative operators $\mathcal{L}$ weaker than any power of the fractional Laplacian. The result is an improvement of the one of Fan et al. (Global Cauchy problem of 2D generalized MHD equations, Monatsh. Math., 175 (2014), pp. 127-131) which ask for $\alpha>0, \beta=1$., Comment: 16 pages

In this article, we prove that data assimilation by feedback nudging can be achieved for the three-dimensional quasi-geostrophic equation in a simplified scenario using only large spatial scale observables on the dynamical boundary. On this boundary, a scalar unknown (buoyancy or surface temperature of the fluid) satisfies the surface quasi-geostrophic equation. The feedback nudging is done on this two-dimensional model, yet ultimately synchronizes the streamfunction of the three-dimensional flow. The main analytical difficulties are due to the presence of a nonlocal dissipative operator in the surface quasi-geostrophic equation. This is overcome by exploiting a suitable partition of unity, the modulus of continuity characterization of Sobolev space norms, and the Littlewood-Paley decomposition to ultimately establish various boundedness and approximation-of-identity properties for the observation operators., Comment: 28 pages, referee comments incorporated, references added, abstract and introduction modified, main theorems cover full subcritical range of dissipation, certain boundedness properties of observation operators extended

We consider the initial value problem for the nonlinear dissipative Schrödinger equations with a gauge invariant nonlinearityλup-1uof orderpnp-1/2pRe λas in the previous works.

In an isolated single-particle quantum system a spatial disorder can induce Anderson localization. Being a result of interference, this phenomenon is expected to be fragile in the face of dissipation. Here we show that dissipation can drive a disordered system into a steady state with tunable localization properties. This can be achieved with a set of identical dissipative operators, each one acting non-trivially only on a pair of neighboring sites. Operators are parametrized by a uniform phase, which controls selection of Anderson modes contributing to the state. On the microscopic level, quantum trajectories of a system in a localized steady regime exhibit intermittent dynamics consisting of long-time sticking events near selected modes interrupted by jumps between them., Comment: 5 pages, 5 figures

Quantum systems which interact with their environment are often modeled by maximal dissipative operators or so-called Pseudo-Hamiltonians. In this paper the scattering theory for such open systems is considered. First it is assumed that a single maximal dissipative operator A D in a Hilbert space ${\mathfrak H}$ is used to describe an open quantum system. In this case the minimal self-adjoint dilation $\widetilde K$ of A D can be regarded as the Hamiltonian of a closed system which contains the open system $\{A_{\!D},{\mathfrak H}\}$ , but since $\widetilde K$ is necessarily not semibounded from below, this model is difficult to interpret from a physical point of view. In the second part of the paper an open quantum system is modeled with a family {A(μ)} of maximal dissipative operators depending on energy μ, and it is shown that the open system can be embedded into a closed system where the Hamiltonian is semibounded. Surprisingly it turns out that the corresponding scattering matrix can be completely recovered from scattering matrices of single pseudo-Hamiltonians as in the first part of the paper. The general results are applied to a class of Sturm–Liouville operators arising in dissipative and quantum transmitting Schrodinger–Poisson systems.

We show that a large class of dissipative systems can be brought to a canonical form by introducing complex co-ordinates in phase space and a complex-valued hamiltonian. A naive canonical quantization of these systems lead to non-hermitean hamiltonian operators. The excited states are unstable and decay to the ground state. We also compute the tunneling amplitude across a potential barrier by solving the dissipative version of the Schrodinger equation. We then generalize the formalism to cases where the configuration space is a curved Riemannian manifold.

In this paper we study a class of backward stochastic differential equations (BSDEs) of the form dY(t)= -AY(t)dt -f_0(t,Y(t))dt -f_1(t,Y(t),Z(t))dt + Z(t)dW(t) on the interval [0,T], with given final condition at time T, in an infinite dimensional Hilbert space H. The unbounded operator A is sectorial and dissipative and the nonlinearity f_0(t,y) is dissipative and defined for y only taking values in a subspace of H. A typical example is provided by the so-called polynomial nonlinearities. Applications are given to stochastic partial differential equations and spin systems., Comment: 22 pages

Let $$ j_{mm} = \left(\matrix{{I_m}&{0}\cr 0&{-I_m}\cr}\right), \quad {\mathcal{J}}_{m} = \left(\matrix{{0}&{-iI_m}\cr {iI_m}&{0}\cr}\right), $$ Im is the identity matrix of order m. Let W(λ) be an entire matrix valued function of order 2m, W(0) = I2m, the values of W(λ) are jmm-unitary at the imaginary axis and strictly jmm-expansive in the open right half-plane. The blocks of order m of the matrix W(λ) with appropriate signs are treated as coefficients of algebraic Riccati equation. It is proved that for any λ with positive real part this equation has a unique contractive solution θ(λ). The matrix valued function θ(λ) can be represented in a form θ(λ) = θA(iλ) where θA(μ) is the characteristic function of some maximal dissipative operator A. This operator is in a natural way constructed starting from the Hamiltonian system of the form $$ \frac{dx(\tau)}{d \tau} = i \mathcal{J}_{m}K(\tau)x(\tau), \quad \tau \in [0;+\infty) $$ with periodic coefficients.

We prove that a dissipative operator in the Krein space has a maximal nonnegative invariant subspace provided that the operator admits matrix representation with respect to the canonical decomposition of the space and the upper right operator in this representation is compact relative to the lower right operator. Under the additional assumption that the upper and lower left operators are bounded (the so-called Langer condition), this result was proved (in increasing order of generality) by Pontryagin, Krein, Langer, and Azizov. We relax the Langer condition essentially and prove under the new assumptions that a maximal dissipative operator in the Krein space has a maximal nonnegative invariant subspace such that the spectrum of its restriction to this subspace lies in the left half-plane. Sufficient conditions are found for this restriction to be the generator of a holomorphic semigroup or a C 0-semigroup.

In this paper we consider the nonselfadjoint (dissipative) Schrodinger boundary value problem in the limit-circle case with an eigenparameter in the boundary condition. Since the boundary conditions are nonselfadjoint, the approach is based on the use of the maximal dissipative operator, and the spectral analysis of this operator is adequate for the boundary value problem. We construct a selfadjoint dilation of the maximal dissipative operator and its incoming and outgoing spectral representations, which make it possible to determine the scattering matrix of the dilation. We construct a functional model of the maximal dissipative operator and define its characteristic function in terms of solutions of the corresponding Schrodinger equation. Theorems on the completeness of the system of eigenvectors and the associated vectors of the maximal dissipative operator and the Schrodinger boundary value problem are given.

We construct the modified wave operator for the nonlinear Schrodinger type equations $u_{t}-\frac{i}{\beta }\| partial _{x} |^{\beta }u=i\lambda \ |u| ^{\rho -1}u,$ for $\( t,x ) \in \mathbf{R}\times \mathbf{R.}$ We find the solutions in the neighborhood of suitable approximate solutions provided that $\beta \geq 2$, $\Im \lambda >0$ and $\rho

In this paper a triangular model of a class of unbounded non-selfadjoint K r -operators A presented as a coupling of dissipative and anti-dissipative operators in a Hilbert space with real absolutely continuous spectra and with different domains of A and A * is considered. The asymptotic behaviour of the corresponding non-dissipative processes T t f = e itA f, generated from the semigroups T t with generators iA, as t → ± ∞ are obtained. The strong wave operators, the scattering operator for the couple (A*, A) and the similarity of A and the operator of multiplication by the independent variable are obtained explicitly. The considerations are based on the triangular models and characteristic functions of A. Kuzhel for unbounded operators and the limit values of the multiplicative integrals, describing the characteristic function of the considered model.

In this paper, non-self-adjoint Sturm-Liuville operators in Weyl’s limit-circle case are studied. We first determine all the non-self-adjoint boundary conditions yielding dissipative operators for each allowed Sturm-Liouville differential expression. Then, using the characteristic determinant, we prove the completeness of the system of eigenfunctions and associated functions for these dissipative operators.

We present a generalization of Koplienko–Neidhardt trace formula for pairs of Hilbert space operators (T , V ) with T contractive and V unitary such that T – V is a Hilbert–Schmidt operator. We extend the result to pairs of contractions and then, via Cayley transform, to pairs of maximal dissipative operators. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)