Browsing by Author "Cojuhari, P."

Now showing 1 - 7 of 7

Open Access
Closed embeddings of Hilbert spaces
(2010) Cojuhari, P.; Gheondea, A.
Motivated by questions related to embeddings of homogeneous Sobolev spaces and to comparison of function spaces and operator ranges, we introduce the notion of closely embedded Hilbert spaces as an extension of that of continuous embedding of Hilbert spaces. We show that this notion is a special case of that of Hilbert spaces induced by unbounded positive selfadjoint operators that corresponds to kernel operators in the sense of L. Schwartz. Certain canonical representations and characterizations of uniqueness of closed embeddings are obtained. We exemplify these constructions by closed, but not continuous, embeddings of Hilbert spaces of holomorphic functions. An application to the closed embedding of a homogeneous Sobolev space on Rn in L2(Rn), based on the singular integral operator associated to the Riesz potential, and a comparison to the case of the singular integral operator associated to the Bessel potential are also presented. As a second application we show that a closed embedding of two operator ranges corresponds to absolute continuity, in the sense of T. Ando, of the corresponding kernel operators. © 2010 Elsevier Inc.
Open Access
Closely embedded Krein spaces and applications to Dirac operators
(Elsevier, 2011-04-15) Cojuhari, P.; Gheondea, A.
Motivated by energy space representation of Dirac operators, in the sense of K. Friedrichs, we recently introduced the notion of closely embedded Krein spaces. These spaces are associated to unbounded selfadjoint operators that play the role of kernel operators, in the sense of L Schwartz, and they are special representations of induced Krein spaces. In this article we present a canonical representation of closely embedded Krein spaces in terms of a generalization of the notion of operator range and obtain a characterization of uniqueness. When applied to Dirac operators, the results differ according to a mass or a massless particle in a dramatic way: in the case of a particle with a nontrivial mass we obtain a dual of a Sobolev type space and we have uniqueness, while in the case of a massless particle we obtain a dual of a homogenous Sobolev type space and we lose uniqueness. (C) 2010 Elsevier Inc. All rights reserved.
Open Access
Embeddings, operator ranges, and Dirac operators
(Springer Basel, 2011) Cojuhari, P.; Gheondea, A.
Motivated by energy space representation of Dirac operators, in the sense of K. Friedrichs, we recently introduced the notion of closely embedded Kreǐn spaces. These spaces are associated to unbounded selfadjoint operators that play the role of kernel operators, in the sense of L. Schwartz, and they are special representations of induced Kreǐn spaces. In this article we present a canonical representation of closely embedded Kreǐn spaces in terms of a generalization of the notion of operator range and obtain a characterization of uniqueness. When applied to Dirac operators, the results differ according to a mass or a massless particle in a dramatic way: in the case of a particle with a nontrivial mass we obtain a dual of a Sobolev type space and we have uniqueness, while in the case of a massless particle we obtain a dual of a homogenous Sobolev type space and we lose uniqueness. © 2010 Elsevier Inc.
Open Access
On generalised triplets of Hilbert spaces
(Editura Academiei Romane, 2020) Cojuhari, P.; Gheondea, Aurelian
We compare the concept of triplet of closely embedded Hilbert spaces with that of generalised triplet of Hilbert spaces in the sense of Berezanskii by showing when they coincide, when they are different, and when starting from one of them one can naturally produce the other one that essentially or fully coincides.
Open Access
On lifting of operators to Hilbert spaces induced by positive selfadjoint operators
(Academic Press, 2005) Cojuhari, P.; Gheondea, A.
We introduce the notion of induced Hilbert spaces for positive unbounded operators and show that the energy spaces associated to several classical boundary value problems for partial differential operators are relevant examples of this type. The main result is a generalization of the Krein-Reid lifting theorem to this unbounded case and we indicate how it provides estimates of the spectra of operators with respect to energy spaces. © 2004 Elsevier Inc. All rights reserved.
Open Access
Triplets of closely embedded Dirichlet type spaces on the unit polydisc
(Birkhaeuser Science, 2013) Cojuhari, P.; Gheondea, A.
We propose a general concept of triplet of Hilbert spaces with closed embeddings, instead of continuous ones, and we show how rather general weighted L2 spaces yield this kind of generalized triplets of Hilbert spaces for which the underlying spaces and operators can be explicitly calculated. Then we show that generalized triplets of Hilbert spaces with closed embeddings can be naturally associated to any pair of Dirichlet type spaces Dα(DN) of holomorphic functions on the unit polydisc DN and we explicitly calculate the associated operators in terms of reproducing kernels and radial derivative operators. We also point out a rigging of the Hardy space H2(DN) through a scale of Dirichlet type spaces and Bergman type spaces. © 2012 Springer Basel.
Open Access
Triplets of closely embedded Hilbert spaces
(Springer, 2014) Cojuhari, P.; Gheondea, A.
We obtain a general concept of triplet of Hilbert spaces with closed (unbounded) embeddings instead of continuous (bounded) ones. We provide a model and an abstract theorem as well for a triplet of closely embedded Hilbert spaces associated to positive selfadjoint operator H, that is called the Hamiltonian of the system, which is supposed to be one-to-one but may not have a bounded inverse. Existence and uniqueness results, as well as left-right symmetry, for these triplets of closely embedded Hilbert spaces are obtained. We motivate this abstract theory by a diversity of problems coming from homogeneous or weighted Sobolev spaces, Hilbert spaces of holomorphic functions, and weighted L2 spaces. An application to weak solutions for a Dirichlet problem associated to a class of degenerate elliptic partial differential equations is presented. In this way, we propose a general method of proving the existence of weak solutions that avoids coercivity conditions and Poincaré–Sobolev type inequalities. © 2014, Springer Basel.