David Zarnadze

The well-known A. Grothendieck's theorem on a homomorphism between locally convex spaces is generalized to the case of topologies which are incompatible with dualities. On the basis of this theorem, necessary and sufficient conditions are obtained for a weak homomorphism (resp. its adjoint operator, resp. its double adjoint operator) to be again a homomorphism in various topologies of the initial (resp. dual, resp. bidual) spaces. Some new classes of pairs of locally convex spaces satisfying these conditions are established. The results obtained have enabled us to reveal new properties of frequently encountered homomorphisms and weakly open operators, as well as to strengthen and generalize some well-known results.

The paper gives an extension of the fundamental principles of selfadjoint operators in Fréchet-Hilbert spaces, countable-Hilbert and nuclear Fréchet spaces. Generalizations of the well known theorems of von Neumann, Hellinger-Toeplitz, Friedrichs and Ritz are obtained. Definitions of generalized central and generalized spline algorithms are given.

Examples of selfadjoint and positive definite elliptic differential operators satisfying the above conditions are given. The validity of theoretical results in the case of a harmonic oscillator operator is confirmed by numerical calculations.

Let H be a Hilbert space and K:H→H be a compact, selfadjoint and one-to one operator, having everywhere dense range. From the elements of the space H on which the inverse to K operator K^(-1) can be applied an infinite number of times, a Frechet space D(K^(-∞)) and an acting from this space in H operator F_(∞ ) are constructed. Instead of the equation Ku=f we consider the equation K_∞ u=f which has in D(K^(-∞)) an unique and stable solution. It is proved that the sequence of approximation solutions constructed by Ritz’s extended method converges to exact solution in the space D(K^(-∞)) and also in the energetic space of the operator F_(∞ ) . 

In the present paper the notion of spline is generalized for the case where not one set of problem elements, but a decreasing sequence of problem elements sets on a linear space is given. The generalized interpolation spline realizes a minimum not only of the metric, but also of the corresponding Minkowski functional. The necessary and sufficient condition for the existence of generalized spline for arbitrary nonadaptive information of cardinality 1 is given.

The worst case setting of linear problems, when the error is measured with the help of a metric, is studied. The notions of generalized spline and generalized central algorithms are introduced. Some conditions for a generalized spline algorithm to be linear and generalized central are given. The obtained results are applied to operator equations with positive operators in some Hilbert spaces. Examples of strong degenerated elliptic and their inverse operators satisfying the conditions appearing in the obtained theorems, are given.

The worst case setting of linear problems, when the error is measured with the help of a metric, is studied. In [5], a linear generalized central spline algorithm is constructed for the approximate calculation of solution operators of equations, containing positive differential and integral operators. In this paper an algorithm is constructed for same form equations with the operators admitting a singular value decomposition. In particular, a linear generalized central spline algorithm for computerized tomography problem is constructed and studied.

The worst case setting of linear problems, when the error is measured with the help of a metric, is studied. After transferring of ill-posed problem of computerized tomography from Hilbert space to some Frechet-Hilbert space, this problem become well-posed. In this paper for transferred computerized tomography problem a linear generalized central spline algorithms is constructed and studied. First algorithm is based on singular value decomposition (SVD) of the Radon operator introduced by A. K. Louis and Ditz ]. Second is based on a SVD introduced by M. Davison . The used for inversion of Radon operator spaces, norms, metric, operators and approximate methods are original, quite different from the classical and was not considered up to now.

The ill-posed equation Ku = f is considered, where K : H → H is a linear compact selfadjoint injective positive operator and H is a Hilbert space. The Hilbert space D(K−n) of n-orbits of the operator K−1 is introduced taking into account the topology. We transfer the considered equation in this space and construct a linear central spline algorithm for approximate solution of transferred equation (Theorem 1). It is proved that the projective limit of the sequence of n-orbits spaces is the space of all orbits D(K−∞) in which the transferred equation becomes well posed.

In this article some results are collected about finite orbits and orbits of observable operators at the states of quantum mechanical systems, orbital Hilbert spaces of finite orbits and Frechet-Hilbert spaces of all orbits, orbital operators acting in the Hilbert space of finite orbits and in the Frechet-Hilbert space of all orbits. Moreover, the problem of the approximate solution of equations containing orbital operators in the Hilbert space of finite orbits and in the Frechet-Hilbert space of all orbits is considered. The creation of operators orbits, orbital spaces, orbital operators, we call as orbitization of quantum mechanics or quantum mechanics with orbital operators and the totality of the results obtained as orbital quantum mechanics. 

We fix a continuous linear operator A:H→MA:H→M acting between the Hilbert spaces H and M that admits a singular value decomposition (SVD). We consider the following ill-posed problem: for an element f∈Mf∈M , find u∈Hu∈H such that Au=fA⁢u=f and a generalized solution in the sense of Moore–Penrose u is sought that satisfies the equation A∗Au=A∗fA*⁢A⁢u=A*⁢f . Moreover, we fix an integer n∈N0={0,1,2,…}n∈ℕ0={0,1,2,…} and transfer this equation to a special Hilbert space D((A∗A)−n)D⁢((A*⁢A)-n) of n-orbits. For an approximate solution of this equation in the case of a nonadaptive information on the right-hand side f, a linear spline algorithm is constructed. The specificity of the considered norm is that the approximate solution is the truncated singular value decomposition (TSVD) and does not depend on n. In the case n=0n=0 , the space D((A∗A)−n)D⁢((A*⁢A)-n) coincides with H and we obtain the results for the latter space. In the limiting case of the Fréchet–Hilbert space of all orbits D((A∗A)−∞)D⁢((A*⁢A)-∞) , the equation A∗Au=A∗fA*⁢A⁢u=A*⁢f becomes well-posed and was considered in [D. Ugulava and D. Zarnadze, On a linear generalized central spline algorithm of computerized tomography, Proc. A. Razmadze Math. Inst. 168 2015, 129–148]. It is also noted that the space D((A∗A)−∞)D⁢((A*⁢A)-∞) is the projective limit of the sequence of Hilbert spaces {D((A∗A)−n)}{D⁢((A*⁢A)-n)} . The application of the obtained results for the computerized tomography problem, i.e., for the inversion of the Radon transform Rℜ with the SVD of Louis [A. K. Louis, Orthogonal function series expansions and the null space of the Radon transform, SIAM J. Math. Anal. 15 1984, 3, 621–633] in the space D((R∗R)−n))D((ℜ*ℜ)-n)) is given.

The equation Au=f with a linear symmetric positive definite operator A:D(A)⊂H→H

 having a discrete spectrum and dense image in a complex Hilbert space H is considered. This equation is transferred into the Hilbert space of finite orbits D(An)

 as well as into the Fréchet space of all orbits D(A∞), that is, the projective limit of the sequence of spaces {D(An)}. For an approximate solution of the inverse of A, linear spline central algorithms in these spaces are constructed. The convergence of the sequence of approximate solutions to the exact solution is proved. The obtained results are applied to the quantum harmonic oscillator operator Au(t)=−u″(t)+t2u(t), t∈R, in the Hilbert space of finite orbits D(An), and in the Fréchet space of all orbits D(A∞) that in this case coincides with the Schwartz space of rapidly decreasing functions. Some quantum mechanical interpretations of obtained results are also given.