Browsing by Subject "Friedrichs extension"

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Access status: Open Access ,
Frames and factorization of graph Laplacians
(2015) Jørgensen, Palle E.T.; Tian, Feng
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 $\mathscr{H}_{E}$ 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 $\mathscr{H}_{E}$ we characterize the Friedrichs extension of the $\mathscr{H}_{E}$-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 ($\mathscr{H}_{E}$, $\Delta$) where $\mathscr{H}_{E}$ in an energy Hilbert space, and $\Delta\left(=\Delta_{c}\right)$ 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 $\mathscr{H}_{E}$ consisting of dipoles. Now $\Delta$ is a well-defined semibounded Hermitian operator in both of the Hilbert $l^{2}\left(V\right)$ and $\mathscr{H}_{E}$. It is known to automatically be essentially selfadjoint as an $l^{2}\left(V\right)$-operator, but generally not as an $\mathscr{H}_{E}$ operator. Hence as an $\mathscr{H}_{E}$ operator it has a Friedrichs extension. In this paper we offer two results for the Friedrichs extension: a characterization and a factorization. The latter is via $l^{2}\left(V\right)$.
Access status: Open Access ,
Operators in divergence form and their Friedrichs and Kreĭn extensions
(2011) Arlinskij, Ûrij Moiseevič; Kovalev, Ûrij
For a densely defined nonnegative symmetric operator $\mathcal{A} = L_2^*L_1$ in a Hilbert space, constructed from a pair $L_1 \subset L_2$ of closed operators, we give expressions for the Friedrichs and Kreĭn nonnegative selfadjoint extensions. Some conditions for the equality $(L_2^* L_1)^* = L_1^* L_2$ are obtained. Applications to 1D nonnegative Hamiltonians, corresponding to point interactions, are given
Access status: Restricted ,
Przestrzenie energetyczne laplasjanów na grafach
(Data obrony: 2016-06-30) Koutny, Kamila
Wydział Matematyki Stosowanej
Access status: Restricted ,
Rozszerzenia samosprzężone półograniczonych operatorów symetrycznych
(Data obrony: 2018-07-27) Kustra, Marcin
Wydział Matematyki Stosowanej
Access status: Open Access ,
Towards theory of C-symmetries
(2017) Kužel', Sergìj Oleksandrovič; Sudìlovsʹka, Veronìka Igorìvna
The concept of $\mathcal{C}$-symmetry originally appeared in $\mathcal{PT}$-symmetric quantum mechanics is studied within the Krein spaces framework.
Access status: Restricted ,
Własności operatora C-symetrii w mechanice kwantowej
(Data obrony: 2016-06-30) Kamuda, Alan
Wydział Matematyki Stosowanej