7 research outputs found
Frames and Factorization of Graph Laplacians
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 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 we
characterize the Friedrichs extension of the -graph Laplacian.
We consider infinite connected network-graphs , for
vertices, and \emph{E} for edges. To every conductance function on the
edges of , there is an associated pair
where in an energy
Hilbert space, and is the -Graph Laplacian;
both depending on the choice of conductance function . When a conductance
function is given, there is a current-induced orientation on the set of edges
and an associated natural Parseval frame in consisting of
dipoles. Now is a well-defined semibounded Hermitian operator in both
of the Hilbert and . It is known to
automatically be essentially selfadjoint as an -operator,
but generally not as an operator. Hence as an
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 .Comment: 39 pages, 12 figure