Search results for "Polar decomposition"
showing 4 items of 4 documents
Unbounded Linear Operators in Hilbert Spaces
2002
In order to make this monograph self-contained, we summarize in this chapter some basic definitions and results for unbounded linear operators in a Hilbert space. In Section 1.1, we recall the definitions of C*-algebras and von Neumann algebras. In Section 1.2, we define and investigate the notion of closedness, the closure and the adjoint of an unbounded linear operator in a Hilbert space. Section 1.3 is devoted to the Cayley transform approach to the self-adjointness of a symmetric operator. Section 1.4 deals with the self-adjoint extendability of a symmetric operator with help of the deficiency spaces. In Section 1.5, we extend to unbounded self-adjoint operators the spectral theorem and…
Finite-Size and Illumination Conditions Effects in All-Dielectric Metasurfaces
2022
Dielectric metasurfaces have emerged as a promising alternative to their plasmonic counterparts due to lower ohmic losses, which hinder sensing applications and nonlinear frequency conversion, and their larger flexibility to shape the emission pattern in the visible regime. To date, the computational cost of full-wave numerical simulations has forced the exploitation of the Floquet theorem, which implies infinitely periodic structures, in designing such devices. In this work, we show the potential pitfalls of this approach when considering finite-size metasurfaces and beam-like illumination conditions, in contrast to the typical infinite plane-wave illumination compatible with the Floquet t…
Tomita—Takesaki Theory in Partial O*-Algebras
2002
This chapter is devoted to the development of the Tomita-Takesaki theory in partial O*-algebras. In Section 5.1, we introduce and investigate the notion of cyclic generalized vectors for a partial O*-algebra, generalizing that of cyclic vectors, and its commutants. Section 5.2 introduces the notion of a cyclic and separating system (M, λ, λ c ), which consists of a partial O*-algebra M, a cyclic generalized vector λ for M and the commutant λ c of λ. A cyclic and separating system (M, λ, λ c ) determines the cyclic and separating system ((M w ′ )′, λ cc , (λ cc ) c ) of the von Neumann algebra (M w ′ )′, and this makes it possible to develop the Tornita-Takesaki theory. Then λ can be extende…
Sweeping the Space of Admissible Quark Mass Matrices
2002
We propose a new and efficient method of reconstructing quark mass matrices from their eigenvalues and a complete set of mixing observables. By a combination of the principle of NNI (nearest neighbour interaction) bases which are known to cover the general case, and of the polar decomposition theorem that allows to convert arbitrary nonsingular matrices to triangular form, we achieve a parameterization where the remaining freedom is reduced to one complex parameter. While this parameter runs through the domain bounded by a circle with radius R determined by the up-quark masses around the origin in the complex plane one sweeps the space of all mass matrices compatible with the given set of d…