196,925 research outputs found
Ternary structures in Hilbert spaces
No full textPhDTernary structures in Hilbert spaces arose in the study of in nite dimensional
manifolds in di erential geometry. In this thesis, we develop a structure theory
of Hilbert ternary algebras and Jordan Hilbert triples which are Hilbert spaces
equipped with a ternary product. We obtain several new results on the classi -
cation of these structures. Some results have been published in [2].
A Hilbert ternary algebra is a real Hilbert space (V; h ; i) equipped with a
ternary product [ ; ; ] satisfying h[a; b; x]; yi = hx; [b; a; y]i for a; b; x and y in V .
A Jordan Hilbert triple is a real Hilbert space in which the ternary product f ; ; g
is a Jordan triple product. It is called a JH-triple if the identity hfa; b; xg; xi =
hx; fb; a; xgi holds in V . JH-triples correspond to a class of Lie algebras which
play an important role in symmetric Riemannian manifolds.
We begin by proving new structure results on ideals, centralizers and derivations
of Hilbert ternary algebras. We characterize primitive tripotents in Hilbert
ternary algebras and use them as coordinates to classify abelian Hilbert ternary
algebras. We show that they are direct sums of simple ones, and each simple
abelian Hilbert ternary algebra is ternary isomorphic to the algebra C2(H;K) of
Hilbert-Schmidt operators between real, complex or quaternion Hilbert spaces H
and K.
Further, we describe completely the ternary isomorphisms and ternary antiisomorphisms
between abelian Hilbert ternary algebras. We show that each
ternary isomorphism between simple algebras C2(H;K) and C2(H0;K0) is of
the form (x) = Jxj where j : H0 ! H and J : K ! K0 are isometries. A
ternary anti-isomorphism is of the form (x) = Jx j where j : H0 ! K and
J : H ! K0 are isometries.
The structures of Hilbert ternary algebras and JH-triples are closely related.
Indeed, we show that each JH-triple (V; f ; ; g) admits a decomposi-
6
tion V = Vs
L
V ?
s where (Vs; f ; ; g) is a Hilbert ternary algebra which is usually
nonabelian and unless V = Vs, the orthogonal complement V ?
s is always
a nonabelian Hilbert ternary algebra in the derived ternary product [a; b; c]d =
fa; b; cg fb; a; cg. Hence JH-triples provide important examples of nonabelian
Hilbert ternary algebras. We determine exactly when Vs and V ?
s are Jordan
triple ideals of V . We show, in each dimension at least 2, there is a JH-triple
(V; f ; ; g) for which V 6= Vs, equivalently, (V; f ; ; g) is not a Hilbert ternary
algebra.
Bragg grating based integrated photonic Hilbert transformers
No full textPlanar Bragg grating based photonic Hilbert transformers are experimentally demonstrated in this work. Planar Bragg gratings are utilized to implement a general Hilbert transform, a fractional order Hilbert transform and terahertz bandwidth Hilbert transforms, using a direct UV grating writing technique in a silica-on-silicon platform. The design, fabrication and integration of the proposed devices are discussed and presented. The photonic Hilbert transformers could be further monolithically integrated with flat top reflectors and interferometric structures, implementing all-optical single-sideband filter
The Field of Norms Functor and the Hilbert Symbol
No full textThe classical Hilbert symbol of a higher local field containing a primitive -th root of unity is a pairing , describing Kummer extensions of exponent . In this thesis we define a generalised Hilbert symbol and prove a formula for it. Our approach has several ingredients.
The field of norms functor of Scholl associates to any strictly deeply ramified tower a field of characteristic . Separable extensions of correspond functorially to extensions of , giving rise to .
We define morphisms which are compatible with the norms for every . Using these, we show that field of norms functor commutes with the reciprocity maps and constructed by Fesenko.
Imitating Fontaine's approach, we obtain an invariant form of Parshin's formula for the Witt pairing in characteristic . The `main lemma' relates Kummer extensions of and Witt extensions of , allowing us to derive a formula for the generalised Hilbert symbol , where is the -adic completion of
Some inequalities for convex functions of selfadjoint operators in Hilbert spaces
No full textSome inequalities for convex functions of selfadjoint operators in Hilbert
spaces under suitable assumptions for the involved operators are given. Applications
for particular cases of interest are also provided
Phase controlled integrated interferometric single-sideband filter based on planar Bragg gratings implementing photonic Hilbert transform
No full textThe first monolithically integrated all-optical single-sideband filter based on photonic Hilbert transform and planar Bragg gratings is proposed and experimentally demonstrated. Single-sideband suppression of 12 dB at 6 GHz and sideband switching are achieved via thermal tuning. An X-coupler, photonic Hilbert transformer, flat top reflector and a micro heater are incorporated in a single silicon-on-silica substrate. The device can be thermally tuned by the micro heater on top of the channel waveguide. The device is fabricated using a combination of direct UV grating writing technology and photolithography
Brussels-Austin Nonequilibrium Statistical Mechanics in the Later Years: Large Poincaré Systems and Rigged Hilbert Space
No full textThis second part of a two-part essay discusses recent developments in the Brussels-Austin Group after the mid 1980s. The fundamental concerns are the same as in their similarity transformation approach (see Part I), but the contemporary approach utilizes rigged Hilbert space (whereas the older approach used Hilbert space). While the emphasis on nonequilibrium statistical mechanics remains the same, the use of similarity transformations shifts to the background. In its place arose an interest in the physical features of large Poincaré systems, nonlinear dynamics and the mathematical tools necessary to analyze them
Operadores lineares em espaços de Hilbert e aplicações
No full textDissertação (mestrado) - Universidade Federal de Santa Catarina, Centro de Ciências Físicas e Matemáticas. Programa de Pós-graduação em Matemática e Computação CientíficaNesta disserta¸c#ao n´os estudamos propriedades gerais de operadores lineares em espacos de Hilbert e aplicacoes. Em particular, o problema de existencia e unicidade de extensoes autoadjuntas de um operador linear e considerado. Varios exemplos importantes sao trabalhados em detalhe: os operadores de multiplicacao e os operadores diferenciais de Laplace e Schrodinger
Some combinatorial identities related to commuting varieties and Hilbert schemes
No full textIn this article we explore some of the combinatorial consequences of recent results relating the isospectral commuting variety and the Hilbert scheme of points in the plane
wsc0/hilbert: Old Faithful
No full text<p>This initial release provides a new method of computing a Hilbert transform in the form of a windowing method on top of a frequency domain rotation.</p>
Inequalities for some Functionals Associated with Bounded Linear Operators in Hilbert Spaces
No full textSome inequalities between the operator norm, numerical radius
and functional are established. New upper bounds for the nonnegative
quantity that complement some recent results of the author are given as well
- …
