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.     

On a class of Riccati equations in a Hilbert space

We establish existence and uniqueness of solutions of a class of Riccati equations in Hilbert space ocurring in filtering problems for distributed parameter systems using point sensors.     

On operators generating conditional bases in a Hilbert space

A spectral characterization is given to the linear operators which in a Hilbert space transform some complete orthonormal system into a conditional basis.Translated from Matematicheskie Zametki, Vol. 12, No, 1, pp. 73–84, July, 1972.     

On the eigenvalues of compact operators in a Hilbert space

Summary This paper contains several remarks concerning the behavior of the eigenvalues of compact operators in a Hilbert space. The results are related to the theorem for a finite matrix obtained as Theorem 1 in my previous paper[5].This work was partially supported by a research grant of the Sakkokai Foundation.     

On interpolation approximation of differentiable operators in a Hilbert space

In a Hilbert space, we construct an interpolation approximation of the Taylor polynomial for differentiable operators. By using this approximation, we obtain estimates of accuracy for analytic operators that strengthen previously known results and for operators containing finitely many Fréchet derivatives. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 58, No. 4, pp. 554–563, April, 2006.     

On the definition of Hilbert space

The projection theorem expresses a central feature of classical Hilbert space. Do other infinite dimensional sesquilinear spaces share this property? We show here that this is not the case for several prominent candidates; in particular Kalish's p-adic Hilbert spaces, Springers non archimedean normed spaces, the positive definite spaces over ordered fields. This yields interesting characterizations of classical Hilbert space.     

On complementary subspaces of Hilbert space

Every pair of non-trivial topologically complementary subspaces of a Hilbert space is unitarily equivalent to a pair of the form on a Hilbert space . Here is possibly , is a positive injective contraction and denotes the graph of . For such a pair the following are equivalent: (i) is similar to a pair in generic position; (ii) and have a common algebraic complement; (iii) is similar to for some operators on a Hilbert space. These conditions need not be satisfied. A second example is given (the first due to T. Kato), involving only compact operators, of a double triangle subspace lattice which is not similar to any operator double triangle.

     

Sampling in a Hilbert space

An analog of the Whittaker-Shannon-Kotel'nikov sampling theorem is derived for functions with values in a separable Hilbert space. The proof uses the concept of frames and frame operators in a Hilbert space. One of the consequences of this theorem is that it allows us to derive sampling theorems associated with boundary-value problems and some homogeneous integral equations, which in turn gives us a generalization of another sampling theorem by Kramer.