3,566 research outputs found
Binormal, Complex Symmetric Operators
In this paper, we describe necessary and sufficient conditions for a binormal
or complex symmetric operator to have the other property. Along the way, we
find connections to the Duggal and Aluthge transforms, and give further
properties of binormal, complex symmetric operators
Near invariance and symmetric operators
Let be a subspace of . We show that the operator of
multiplication by the independent variable has a simple symmetric regular
restriction to with deficiency indices if and only if is a nearly invariant subspace, with a meromorphic inner
function vanishing at . Here is unimodular, is an isometric
multiplier of into and is the Hardy space of the
upper half plane. Our proof uses the dilation theory of completely positive
maps
On the theory of self-adjoint extensions of symmetric operators and its applications to Quantum Physics
This is a series of 5 lectures around the common subject of the construction
of self-adjoint extensions of symmetric operators and its applications to
Quantum Physics. We will try to offer a brief account of some recent ideas in
the theory of self-adjoint extensions of symmetric operators on Hilbert spaces
and their applications to a few specific problems in Quantum Mechanics
Separability criterion for bipartite states and its generalization to multipartite systems
A group of symmetric operators are introduced to carry out the separability
criterion for bipartite and multipartite quantum states. Every symmetric
operator, represented by a symmetric matrix with only two nonzero elements, and
their arbitrary linear combinations are found to be entanglement witnesses. By
using these symmetric operators, Wootters' separability criterion for two-qubit
states can be generalized to bipartite and multipartite systems in arbitrary
dimensions.Comment: 5 page
Some Notes on Complex Symmetric Operators
In this paper we show that every conjugation on the Hardy-Hilbert space
is of type , where is an unitary operator and
, with . In the sequence, we extend this result for all separable Hilbert space
and we prove some properties of complex symmetry on .
Finally, we prove some relations of complex symmetry between the operators
and , where is the polar decomposition of
bounded operator on the separable
Hilbert space .Comment: 8 page
On Symmetric Operators in Noncommutative Geometry
In Noncommutative Geometry, as in quantum theory, classically real variables
are assumed to correspond to self-adjoint operators. We consider the relaxation
of the requirement of self-adjointness to mere symmetry for operators
which encode space-time information.Comment: 8 pages, LaTeX, to appear in Extended Proceedings I.S.I. Guccia
Workshop 98, Nova 199
A geometrical relation between symmetric operators and mutually unbiased operators
In this work we study the relation between the set of symmetric operators and
the set of mutually unbiased operators from finite plane geometry point of
view. Here symmetric operators are generalization of symmetric informationally
complete probability-operator measurements (SIC POMs), while mutually unbiased
operators are the operator generalization of mutually unbiased bases (MUB). We
also discuss the implication of this relation to the particular cases of rank-1
SIC POMs and MUB.Comment: comments are welcom
Phillips symmetric operators and their extensions
Let be a symmetric operator with equal defect numbers and let
be a set of unitary operators in a Hilbert space .
The operator is called -invariant if for all
. Phillips \cite{PH} constructed an example of
-invariant symmetric operator which has no
-invariant self-adjoint extensions. It was discovered that such
symmetric operator has a constant characteristic function \cite{KO}. For this
reason, each symmetric operator with constant characteristic function is
called a \emph{Phillips symmetric operator}
Averaged wave operators and complex-symmetric operators
We study the behaviour of sequences , where are
unitary operators, whose spectral measures are singular with respect to the
Lebesgue measure, and the commutator is small in a sense. The
conjecture about the weak averaged convergence of the difference to the zero operator is discussed and its connection
with complex-symmetric operators is established in a general situation. For a
model case where is the unitary operator of multiplication by on
, sufficient conditions for the convergence as in the Conjecture are
given in terms of kernels of integral operators.Comment: 13 page
Dense domains, symmetric operators and spectral triples
This article is about erroneous attempts to weaken the standard definition of
unbounded Kasparov module (or spectral triple). We present counterexamples to
claims in the literature that Fredholm modules can be obtained from these
weaker variations of spectral triple. Our counterexamples are constructed using
self-adjoint extensions of symmetric operators
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