77,821 research outputs found
Componentwise and Cartesian decompositions of linear relations
Let be a, not necessarily closed, linear relation in a Hilbert space
\sH with a multivalued part \mul A. An operator in \sH with \ran
B\perp\mul A^{**} is said to be an operator part of when A=B \hplus
(\{0\}\times \mul A), where the sum is componentwise (i.e. span of the
graphs). This decomposition provides a counterpart and an extension for the
notion of closability of (unbounded) operators to the setting of linear
relations. Existence and uniqueness criteria for the existence of an operator
part are established via the so-called canonical decomposition of . In
addition, conditions are developed for the decomposition to be orthogonal
(components defined in orthogonal subspaces of the underlying space). Such
orthogonal decompositions are shown to be valid for several classes of
relations. The relation is said to have a Cartesian decomposition if
A=U+\I V, where and are symmetric relations and the sum is
operatorwise. The connection between a Cartesian decomposition of and the
real and imaginary parts of is investigated
A note on the spectral mapping theorem of quantum walk models
We discuss the description of eigenspace of a quantum walk model with an
associating linear operator in abstract settings of quantum walk including
the Szegedy walk on graphs. In particular, we provide the spectral mapping
theorem of without the spectral decomposition of . Arguments in this
direction reveal the eigenspaces of characterized by the generalized
kernels of linear operators given by .Comment: 17 page
Localization of discrete time quantum walks on the glued trees
In this paper, we consider the time averaged distribution of discrete time
quantum walks on the glued trees. In order to analyse the walks on the glued
trees, we consider a reduction to the walks on path graphs. Using a spectral
analysis of the Jacobi matrices defined by the corresponding random walks on
the path graphs, we have spectral decomposition of the time evolution operator
of the quantum walks. We find significant contributions of the eigenvalues of the Jacobi matrices to the time averaged limit distribution of the
quantum walks. As a consequence we obtain lower bounds of the time averaged
distribution.Comment: 10 page
The Wold-type decomposition for -isometries
The aim of this paper is to study the Wold-type decomposition in the class of
-isometries. One of our main results establishes an equivalent condition for
an analytic -isometry to admit the Wold-type decomposition for . In
particular, we introduce the -kernel condition which we use to characterize
analytic -isometric operators which are unitarily equivalent to unilateral
operator valued weighted shifts for . As a result, we also show that
-isometric composition operators on directed graphs with one circuit
containing only one element are not unitarily equivalent to unilateral weighted
shifts. We also provide a characterization of -isometric unilateral operator
valued weighted shifts with positive and commuting weights
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