427,873 research outputs found
The Shilov boundary of an operator space - and the characterization theorems
We study operator spaces, operator algebras, and operator modules, from the
point of view of the `noncommutative Shilov boundary'. In this attempt to
utilize some `noncommutative Choquet theory', we find that Hilbert
Cmodules and their properties, which we studied earlier in the operator
space framework, replace certain topological tools. We introduce certain
multiplier operator algebras and Calgebras of an operator space, which
generalize the algebras of adjointable operators on a Cmodule, and the
`imprimitivity Calgebra'. It also generalizes a classical Banach space
notion. This multiplier algebra plays a key role here. As applications of this
perspective, we unify, and strengthen several theorems characterizing operator
algebras and modules, in a way that seems to give more information than other
current proofs. We also include some general notes on the `commutative case' of
some of the topics we discuss, coming in part from joint work with Christian Le
Merdy, about `function modules'.Comment: This is the final revised versio
A Rule-Based Logic for Quantum Information
In the present article, we explore a new approach for the study of
orthomodular lattices, where we replace the problematic conjunction by a binary
operator, called the Sasaki projection. We present a characterization of
orthomodular lattices based on the use of an algebraic version of the Sasaki
projection operator (together with orthocomplementation) rather than on the
conjunction. We then define of a new logic, which we call Sasaki Orthologic,
which is closely related to quantum logic, and provide a rule-based definition
of this logic
Reconstruction algorithms for a class of restricted ray transforms without added singularities
Let and denote a restricted ray transform along curves and a
corresponding backprojection operator, respectively. Theoretical analysis of
reconstruction from the data is usually based on a study of the
composition , where is some local operator (usually a derivative).
If is chosen appropriately, then is a Fourier Integral Operator
(FIO) with singular symbol. The singularity of the symbol leads to the
appearance of artifacts (added singularities) that can be as strong as the
original (or, useful) singularities. By choosing in a special way one can
reduce the strength of added singularities, but it is impossible to get rid of
them completely.
In the paper we follow a similar approach, but make two changes. First, we
replace with a nonlocal operator that integrates along a
curve in the data space. The result resembles the generalized
Radon transform of . The function is defined on pairs
, where is an open set
containing the support of , and is the unit sphere in .
Second, we replace with a backprojection operator that integrates
with respect to over . It turns out that if and
are appropriately selected, then the composition is an
elliptic pseudodifferential operator of order zero with principal symbol 1.
Thus, we obtain an approximate reconstruction formula that recovers all the
singularities correctly and does not produce artifacts. The advantage of our
approach is that by inserting we get access to the frequency
variable . In particular, we can incorporate suitable cut-offs in
to eliminate bad directions , which lead to added singularities
Isoperimetric inequalities for the logarithmic potential operator
In this paper we prove that the disc is a maximiser of the Schatten -norm
of the logarithmic potential operator among all domains of a given measure in
, for all even integers . We also show that the
equilateral triangle has the largest Schatten -norm among all triangles of a
given area. For the logarithmic potential operator on bounded open or
triangular domains, we also obtain analogies of the Rayleigh-Faber-Krahn or
P{\'o}lya inequalities, respectively. The logarithmic potential operator can be
related to a nonlocal boundary value problem for the Laplacian, so we obtain
isoperimetric inequalities for its eigenvalues as well.Comment: revised version with corrected formulations and arguments; to replace
the previous versio
On the H^1-L^1 boundedness of operators
We prove that if q is in (1,\infty), Y is a Banach space and T is a linear
operator defined on the space of finite linear combinations of (1,q)-atoms in
R^n which is uniformly bounded on (1,q)-atoms, then T admits a unique
continuous extension to a bounded linear operator from H^1(R^n) to Y. We show
that the same is true if we replace (1,q)-atoms with continuous
(1,\infty)-atoms. This is known to be false for (1,\infty)-atoms.Comment: This paper will appear in Proceedings of the American Mathematical
Societ
Koopman operator-based model reduction for switched-system control of PDEs
We present a new framework for optimal and feedback control of PDEs using
Koopman operator-based reduced order models (K-ROMs). The Koopman operator is a
linear but infinite-dimensional operator which describes the dynamics of
observables. A numerical approximation of the Koopman operator therefore yields
a linear system for the observation of an autonomous dynamical system. In our
approach, by introducing a finite number of constant controls, the dynamic
control system is transformed into a set of autonomous systems and the
corresponding optimal control problem into a switching time optimization
problem. This allows us to replace each of these systems by a K-ROM which can
be solved orders of magnitude faster. By this approach, a nonlinear
infinite-dimensional control problem is transformed into a low-dimensional
linear problem. In situations where the Koopman operator can be computed
exactly using Extended Dynamic Mode Decomposition (EDMD), the proposed approach
yields optimal control inputs. Furthermore, a recent convergence result for
EDMD suggests that the approach can be applied to more complex dynamics as
well. To illustrate the results, we consider the 1D Burgers equation and the 2D
Navier--Stokes equations. The numerical experiments show remarkable performance
concerning both solution times and accuracy.Comment: arXiv admin note: text overlap with arXiv:1801.0641
- …