The connection between normalizable and spectral operators

A necessary and sufficient condition for an operator to be normalizable is given in terms of Dunford’s spectral theory. This paper is a part of the author’s Ph.D. thesis to be submitted to the Hebrew University. The author wishes to express his indebtedness to Professor S. R. Foguel for his guidance and kind encouragement.     

Time averaging for nonlinear parabolic operators

Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 41, No. 11, pp. 1483–1487, November, 1989.     

Certain averaging operators on Triebel-Lizorkin spaces

Abstract. In this article, we study the boundedness properties of the averaging operator S_t^γon Triebel-Lizorkin spaces■ for various p, q. As an application, we obtain the norm convergence rate for S_t^γ(f ) on Triebel-Lizorkin spaces and the relation between the smoothness imposed on functions and the rate of norm convergence of S_t^γis given.     

Spectral averaging for trace compatible operators

In this note the notions of trace compatible operators and infinitesimal spectral flow are introduced. We define the spectral shift function as the integral of infinitesimal spectral flow. It is proved that the spectral shift function thus defined is absolutely continuous and Kreĭn's formula is established. Some examples of trace compatible affine spaces of operators are given.

     

On a connection between powers of operators and fractional Cauchy problems

Many phenomena in mathematical physics and in the theory of stochastic processes are recently described through fractional evolution equations. We investigate a general framework for connections between ordinary non-homogeneous equations in Banach spaces and fractional Cauchy problems. When the underlying operator generates a strongly continuous semigroup, it is known, using a subordination argument, that the fractional evolution equation is well posed. In this case, we provide an explicit form of the solution involving special functions, one example being the Airy function.     

Some examples and the connection between cylindrical measures and measurable norms

In this paper, we construct an example of a cylindrical measure μ and a norm ‖·‖ such that ∥·∥ is μ-measurable by Dudley et al. and not μ-measurable by Gross and moreover is γ-measurable.     

A connection between operator topologies, polynomial interpolation, and synthesis of diagonal operators

In this paper we extend the work done by Deters and Seubert in [I. Deters, S. Seubert, Cyclic vectors of diagonal operators on the space of functions analytic on a disk, J. Math. Anal. Appl. 334 (2007) 1209-1219]. In particular, we shall improve upon a theorem in the aforementioned paper and make connections between the strong operator topology on H(B(0,1)), polynomial behavior on a sequence of points, and the synthesis of diagonal operators on H(B(0,1)).     

Some induced ordered weighted averaging operators and their use for solving group decision-making problems based on fuzzy preference relations

In [R.R. Yager, D.P. Filev, Operations for granular computing: Mixing words and numbers, in: Proceedings of the FUZZ-IEEE World Congress on Computational Intelligence, Anchorage, 1998, pp. 123–128] Yager and Filev introduced the Induced Ordered Weighted Averaging (IOWA) operator. In this paper, we provide some IOWA operators to aggregate fuzzy preference relations in group decision-making (GDM) problems. These IOWA operators when guided by fuzzy linguistic quantifiers allow the introduction of some semantics or meaning in the aggregation, and therefore allow for a better control over the aggregation stage developed in the resolution process of the GDM problems. In particular, we present the Importance IOWA (I-IOWA) operator, which applies the ordering of the argument values based upon the importance of the information sources; the Consistency IOWA (C-IOWA) operator, which applies the ordering of the argument values based upon the consistency of the information sources; and the Preference IOWA (P-IOWA) operator, which applies the ordering of the argument values based upon the relative preference values associated to each one of them. We provide a procedure to deal with ‘ties’ in respect to the ordering induced by the application of one of these IOWA operators; it consists of a sequential application of the above IOWA operators. We also present a selection process for GDM problems based on the concept of fuzzy majority and the above three IOWA operators. Finally, we analyse the reciprocity and consistency properties of the collective fuzzy preference relations obtained using IOWA operators.     

An averaging theorem inC *-Hilbert modules and operators without adjoint

Moscow State University. Translated from Funktsional'nyi Analiz i Ego Prilozheniya, Vol. 28, No. 3, pp. 88–92, July–September, 1994.     

regularity of averaging operators with higher fold singularities

In this paper, we give a sharp regularity result of averaging operators along curves in the plane with two-sided fold singularities.

     

The decompositions of rank-dependent poverty measures using ordered weighted averaging operators

This paper is concerned with rank-dependent poverty measures and shows that an ordered weighted averaging, hereafter OWA, operator underlies in the definition of these indices. The dual decomposition of an OWA operator into the self-dual core and the anti-self-dual remainder allows us to propose a decomposition for all the rank-dependent poverty measures in terms of incidence, intensity and inequality. In fact, in the poverty field, it is well known that every poverty index should be sensitive to the incidence of poverty, the intensity of poverty and the inequality among the poor individuals. However, the inequality among the poor can be analyzed in terms of either incomes or gaps of the distribution of the poor. And, depending on the side we focus on, contradictory results can be obtained. Nevertheless, the properties inherited by the proposed decompositions from the OWA operators oblige the inequality components to measure equally the inequality of income and inequality of gap overcoming one of the main drawbacks in poverty and inequality measurement. Finally, we provide an empirical illustration showing the appeal of our decompositions for some European Countries in 2005 and 2011.     

Two-parameter spectral averaging and localization for non-monotonic random Schrödinger operators

We prove exponential localization at all energies for two types of one-dimensional random Schrödinger operators: the Poisson model and the random displacement model. As opposed to Anderson-type models, these operators are not monotonic in the random parameters. Therefore the classical one-parameter version of spectral averaging, as used in localization proofs for Anderson models, breaks down. We use the new method of two-parameter spectral averaging and apply it to the Poisson as well as the displacement case. In addition, we apply results from inverse spectral theory, which show that two-parameter spectral averaging works for sufficiently many energies (all but a discrete set) to conclude localization at all energies.