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1

Frames and factorization of graph Laplacians

Jorgensen P., Tian F.

Opuscula Mathematica

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2015

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Vol. 35, no. 3

293--332

EN

Using functions from electrical networks (graphs with resistors assigned to edges), we prove existence (with explicit formulas) of a canonical Parseval frame in the energy Hilbert space [formula] of a prescribed infinite (or finite) network. Outside degenerate cases, our Parseval frame is not an orthonormal basis. We apply our frame to prove a number of explicit results: With our Parseval frame and related closable operators in [formula] we characterize the Priedrichs extension of the [formula]-graph Laplacian. We consider infinite connected network-graphs G = (V, E), V for vertices, and E for edges. To every conductance function c on the edges E of G, there is an associated pair [formula] where [formula] in an energy Hilbert space, and Δ (=Δc) is the c-graph Laplacian; both depending on the choice of conductance function c. When a conductance function is given, there is a current-induced orientation on the set of edges and an associated natural Parseval frame in [formula] consisting of dipoles. Now Δ is a well-defined semibounded Hermitian operator in both of the Hilbert [formula] and [formula]. It is known to automatically be essentially selfadjoint as an [formula]-operator, but generally not as an [formula] operator. Hence as an [formula] operator it has a Friedrichs extension. In this paper we offer two results for the Priedrichs extension: a characterization and a factorization. The latter is via [formula].

2

On the closedness, the self-adjointness and the normality of the product of two unbounded operators

Mortad M. H.

Demonstratio Mathematica

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2012

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Vol. 45, nr 1

161-167

EN

Let A and B be two unbounded densely defined operators on a Hilbert space H . The purpose of this work is to give simple conditions that make the product AB closed, self-adjoint and normal provided the two operators are so.

3

Operators in Banach modules and the Weyl functional calculus for generators of n-parameter C0-groups

Malejki M.

Opuscula Mathematica

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2001

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Vol. 21

65-92

EN

In the paper, I am going to work with unbounded operators in special Banach modules over real Clifford algebras. I propose the definition of n-spectrum for these operators. The mentioned definition is similar to one considered by A. Mclntosh and A. Pryde for bounded operators. The main aim of this work is to describe the support of the Weyl functional calculus T(A) for several operators (formulas for T(A) is according to Anderson) using the Clifford algebras methods. A = (A1,... , Am) is a tuple of operators in a Banach space. The operators Aj are not necessarily bounded but it will be assumed that the tuple A is a generator of the m-parameter C0-group, which satisfies a polynomial growth condition. The final result is the formula suppT(A) = σ(n)(A) ∩ Rm, where σ(n)(A) is the n-spectrum for an operator associated to the tuple A and n ≤ m is an integer.

PL

W tej pracy zajmuję się badaniem nieograniczonych operatorów działających w pewnych modułach Banacha nad algebrami Clifforda. Proponuję tu definicję pewnego rodzaju widma dla takich operatorów, zwanego n-widmem. Definicja ta jest wzorowana na pracy A. Mclntosha i A. Pryde'a, którzy postawili ją dla operatorów ograniczonych. Głównym celem tej pracy jest jednak opisanie nośnika dla rachunku funkcyjnego Weyla T(A) (dla abstrakcyjnych operatorów - wprowadzonego przez W. O. Andersona) dla układów operatorów. Udaje się to wykonać dzięki wykorzystaniu analizy cliffordowskiej. Układ A = (A1,... , Am) jest generatorem m-parametrowej silnie ciągłej grupy operatorów w przestrzeni Banacha. Będę zakładać, że wzrost tej grupy jest ograniczony wielomianowo. Głównym rezultatem tej pracy jest wykazanie równości suppT(A) = σ(n)(A) ∩Rm, gdzie σ(n)(A) jest wspomnianym n-widmem dla pewnego operatora związanego z układem A, natomiast n ≤ m jest pewną liczbą naturalną.

4

On normal extensions of unobounded operators

Ota S.

Bulletin of the Polish Academy of Sciences. Mathematics

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1998

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Vol. 46, no 3

291-301

EN

Subnormality of unbounded operators in a Hilbert space is studied. It is shown that a closed subnormal operator, unlike a closed symmetric operator, has not necesarily a normal extension of the second kind (in terms of Naimark). In connection with this, the uniqueness of the normal extension to the same Hilbert space is discussed.