RESULTADOS SOBRE OPERADORES LLENOS EN ESPACIOS DE HILBERT

Se prueba, entre otros, el siguiente resultado: Sea T:H→H un operador autoadjunto inyectivo, y K:H→H un operador de Riesz, tal que K∈Alglat(T)∩{T}'. Si K:H→H es lleno, entonces T:H→H es lleno. Palabras clave: Operador de Riesz, operador autoadjunto, operador lleno. Abstract It is proved here, among other results, the following: Let T:H→H be a self-adjoint injective operator, and K:H→H a Riesz operator, such that K∈Alglat(T)∩{T}'. If K:H→H is a full operator, then T:H→H is a full operator. Keywords: Riesz operator, self-adjoint operator, full operator. Resumo O siguiente resultado, entre outros, está provado: Seja T:H→H um operador autoadjunto limitado abaixo, e K:H→H um operador de Riesz, tal qual K∈AlglatT⋂{T}^'. Se K:H→H é um operador completo, então T:H→H é um operador completo. Palavras-chave: operador Riesz, operador autoadjunto completo.

Regularization of some classes of ultradistribution semigroups and sines

We systematically analyze regularization of different kinds of ultradistribution semigroups and sines, in general, with nondensely defined generators and contemplate several known results concerning the regularization of Gevrey type ultradistribution semigroups. We prove that, for every closed linear operator A which generates an ultradistribution semigroup (sine), there exists a bounded injective operator C such that A generates a global differentiable C-semigroup (C-cosine function) whose derivatives possess some expected properties of operator valued ultradifferentiable functions. With the help of regularized semigroups, we establish the new important characterizations of abstract Beurling spaces associated to nondensely defined generators of ultradistribution semigroups (sines). The study of regularization of ultradistribution sines also enables us to perceive significant ultradifferentiable properties of higher-order abstract Cauchy problems.

On Two-Parameter Regularized Semigroups and the Cauchy Problem

Suppose that is a Banach space and is an injective operator in , the space of all bounded linear operators on . In this note, a two-parameter -semigroup (regularized semigroup) of operators is introduced, and some of its properties are discussed. As an application we show that the existence and uniqueness of solution of the 2-abstract Cauchy problem , , , is closely related to the two-parameter -semigroups of operators.

SOLUTION OF THE MPC PROBLEM IN TERMS OF THE SZ.-NAGY–FOIAŞ DILATION THEORY

Motivated by physical problems, Misra, Prigogine and Courbage (MPC) studied the following problem: given a one-parameter unitary group {Ut} on a separable Hilbert space [Formula: see text], find a Hilbert space [Formula: see text], a contraction semigroup {Wt} on [Formula: see text] and an injective operator [Formula: see text] with dense range which intertwines the actions of {Ut} and {Wt} (ΛWt = Ut Λ). More precisely, they studied the case where [Formula: see text] is an L2-space over a probability space and both {Ut} and {Wt} are Markovian (i.e. positivity and identity preserving). MPC gave a sufficient condition for the existence of a solution of the above problem, the existence of a time operator associated to {Ut}. In this paper we prove that, using the Sz.-Nagy–Foiaş dilation theory, it is possible to give a constructive characterization of all the solutions of the MPC problem in the general context. This criterium allows one to construct a solution of the MPC problem for which no time operator exists. When specialized to L2-spaces and Markovian {Ut} and {Wt}, the present criterium is applied to address the so-called inverse problem of Statistical Mechanics, namely to characterize the intrinsically random dynamics {Ut}.