On dissipative stochastic equations in a Hilbert space

Summary A general existence and uniqueness theorem for solutions of linear dissipative stochastic differential equation in a Hilbert space is proved. The dual equation is introduced and the duality relation is established. Proofs take inspirations from quantum stochastic calculus, however without using it. Solutions of both equations provide classical stochastic representation for a quantum dynamical semigroup, describing quantum Markovian evolution. The problem of the mean-square norm conservation, closely related to the unitality (non-explosion) of the quantum dynamical semigroup, is considered and a hyperdissipativity condition, ensuring such conservation, is discussed. Comments are given on the existence of solutions of a nonlinear stochastic differential equation, introduced and discussed recently in physical literature in connection with continuous quantum measurement processes.     

Limit-ill-posed equations in a Hilbert space

The behavior of the solution of a limit-ill-posed problem on fixed compacta is investigated for integral operators, acting in a Hilbert space.Translated from Ukrainskii Matematicheskii Zhurnal, vol. 43, No. 2, pp. 241–247, February, 1991.     

Continuous dependence on initial geometry for a class of abstract equations in Hilbert space

Conditions are given under which members of a class of uniformly bounded solutions to the Cauchy problem associated with equations of the form Mu_tt ? Nu = F, in Hilbert space, depend continuously on perturbations of the initial geometry; our results generalize a similar theorem of Knops and Payne for classical solutions to initial-boundary value problems in linear elastodynamics.     

Remarks on the algebraic Riccati equation in Hilbert space

Under certain assumptions the existence and uniqueness of the solution of the inner product algebraic Riccati equation of optimal control in Hilbert space are proved. Some related problems such as stabilizability and exact controllability of control systems are investigated also.     

On the exact degree of complexity of a class of operator equations of the second kind in a Hilbert space

The exact exponent of complexity is found for approximate solutions of a certain class of operator equations in a Hilbert space. A method for information setup and the algorithm for realization of this optimal degree are presented. As a consequence, we find the exact exponent of complexity for approximate solutions of Fredholm integral equations of the second kind whose kernels and free terms include square integrable -derivatives.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 46, No. 7, pp. 893–903, July, 1994.     

Nonlinear Volterra equations on a Hilbert space

We consider equations of the form, u(t) = ? ∝₀^tA(t ? τ)g(u(τ)) dτ + ?(t) (I) on a Hubert space $\text{H}$. A(t) is a family of bounded, linear operators on $\text{H}$ while g is a transformation on $\text{D}$_g ?$\text{H}$ which can be nonlinear and unbounded. We give conditions on A and g which yield stability and asymptotic stability of solutions of (I). It is shown, in particular, that linear combinations with positive coefficients of the operators e^Mt and ?e^Mtsin Mt where M is a bounded, negative self-adjoint operator on $\text{H}$ satisfy these conditions. This is shown to yield stability results for differential equations of the form, $\text{Q (}\text{d}\text{dt}\text{) u = ? P (}\text{d}\text{dt}\text{) g(u(t)) + χ(t)}$, on $\text{H}$.     

On the asymptotic behavior of solutions of integro-differential equations in a Hilbert space

We consider integro-differential equations that are an abstract form of the well-known Gurtin-Pipkin equation. We obtain representations of strong solutions of these equations in the form of series in the exponentials corresponding to points of spectrum of the symbols of such equations.     

Semilinear integro-differential equations in Hilbert space

Existence, uniqueness and asymptotic behavior of solutions to integro-differential equations involving monotone operators are discussed.     

On the injectivity of a class of Wiener-Hopf integral operators in a Hilbert space

A class of Wiener-Hopf integral operators, with kernels vanishing along the positive real axis, is obtained from considering weighted transaxial line-integrals of rotationally symmetric functions defined on ². An analysis of these operators is given when acting in, the Hilbert space L²(⁺). A necessary and sufficient condition for injectivity is established and inversion formulas are provided in some cases. A specific operator falling into this class, the so-called incomplete Abel transform., is presented and an inversion formula is given. This inversion formula makes precise a formal result previously established in Dallaset al. [J. Opt. Soc. Am. A4, 2039 (1987)] and it is also shown to be consistent with an inversion formula derived by Hansen [J. Opt. Soc. Am. A9, 2126 (1992)].This research was supported by NIH/NCI Grant R01 CA49261     

On the Riccati equations

In this article we give, in terms of so-called Berezin symbols, some necessary conditions for the solvability of the Riccati equation XAX+XB-CX-D=0XAX+XB-CX-D=0 on the set T{\cal T} of all Toeplitz operators on the Hardy space H²(\Bbb D)H^2({\Bbb D}) .     

On the Riccati equations

In this article we give, in terms of so-called Berezin symbols, some necessary conditions for the solvability of the Riccati equation on the set of all Toeplitz operators on the Hardy space . Author’s address: Department of Technical Programs, Isparta (MYO) Vocational School, Suleyman Demirel University, 32260 Isparta, Turkey