Proving the Equality of the Spaces Q_b^r (A),Q_b^l (A) and BL(X) where X is a Complex Banach Space

Cabrera and Mohammed proved that the right and left bounded algebras of quotients and of norm ideal on a Hilbert space are equal to Banach algebra of all bounded linear operators on . In this paper, we prove that where is a norm ideal on a complex Banach space .

COMPETITION OF REPRESENTATION OF VALUE ALTERNATIVES TO SOCIO-POLITICAL REALITY OF UKRAINE

The article presents an analysis of the competition between the imaginations of value alternatives relative to the socio-political reality of Ukraine in the context of the archetypal approach. The study of this problem is based on the works of French sociologists — Gilbert Durand and Michel Maffesoli, as well as developments of scientists of the Ukrainian school of archetypics. According to the changes in the psychosocial nature of modern society and man, the expediency of applying the archetypal approach to understanding social and political phenomena, processes, and also the characters and types of managers is proved. Archetypes, manifested through imagination, symbols and images, provide an opportunity to see the diversity of socio-political life holistically, without dominating one or other of its sides.It is noted that a social system of any scale presupposes the existence of a system of certain values shared by the majority of society, since it is in them that the answers that this or that society gives to fundamental worldview problems are contained. Thus, in the epoch of postmodern, the valuable semantic structure of plurality with a priority of self-realization of the personality and expansion of the sphere of its individual choice is characteristic for this time. At the same time, a significant role in this process is played by symbolic capital — prestige, reputation, image, which, in the main, is modeled and supported by virtual reality and igroizization. In such conditions, people not only define varieties of ideal values, converting them into each other, but retain them in the social imagination as an imperative, norm, ideal. It has also been confirmed that any accumulation must be used, since the multiple values of social imagination, realized in the psychosocial concepts of people and society, lead to the development of public opinion and in solidarization. It is established that the competition of multiple value representations is a personinternalized idea that has a transpersonal character that influences the creation of a socio-political reality in Ukraine.

Some Estimates of Certain Subnormal and Hyponormal Derivations

We prove that if and are subnormal operators and is a bounded linear operator such that is a Hilbert-Schmidt operator, then is also a Hilbert-Schmidt operator and for belongs to a certain class of functions. Furthermore, we investigate the similar problem in the case that , are hyponormal operators and is such that belongs to a norm ideal , and we prove that and for being in a certain class of functions.

On the Uniqueness of Wave Operators Associated with Non-Trace Class Perturbations

Voiculescu has previously established the uniqueness of the wave operator for the problem of -perturbation of commuting tuples of self-adjoint operators in the case where the norm ideal has the property , where {Pn} is any sequence of orthogonal projections with rank(Pn) = n. We prove that the same uniqueness result holds true so long as is not the trace class. (It is well known that there is no such uniqueness in the case of trace-class perturbation.)

On numerical ranges of generalized derivations and related properties

This paper is concerned with the numerical range and some related properties of the operator Δ/ S: T → AT – TB(T∈S), where A, B are (bounded linear) operators on the normed linear spaces X and Y. respectively, and S is a linear subspace of the space ℒ (Y, X) of all operators from Y to X. S is assumed to contain all finite operators, to be invariant under Δ, and to be suitably normed (not necessarily with the operator norm). Then the algebra numerical range of Δ/ S is equal to the difference of the algebra numerical ranges of A and B. When X = Y and S = ℒ (X), Δ is Hermitian (resp. normal) in ℒ (ℒ(X)) if and only if A–λ and B–λ are Hermitian (resp. normal) in ℒ(X)for some scalar λ;if X: = H is a Hilbert space and if S is a C *-algebra or a minimal norm ideal in ℒ(H)then any Hermitian (resp. normal) operator in S is of the form Δ/ S for some Hermitian (resp. normal) operators A and B. AT = TB implies A*T = TB* are hyponormal operators on the Hilbert spaces H1 and H2, respectively, and T is a Hilbert-Schmidt operator from H2 to H1.