Weyl's Theorem Research Articles

Quantization in fibering polarizations, Mabuchi rays and geometric Peter–Weyl theorem

In this paper we use techniques of geometric quantization to give a geometric interpretation of the Peter–Weyl theorem. We present a novel approach to half-form corrected geometric quantization in a specific type of non-Kähler polarizations and study one important class of examples, namely cotangent bundles of compact connected Lie groups K. Our main results state that this canonically defined polarization occurs in the geodesic boundary of the space of K×K-invariant Kähler polarizations equipped with Mabuchi's metric, and that its half-form corrected quantization is isomorphic to the Kähler case. An important role is played by invariance of the limit polarization under a torus action.Unitary parallel transport on the bundle of quantum states along a specific Mabuchi geodesic, given by the coherent state transform of Hall, relates the non-commutative Fourier transform for K with the Borel–Weil description of irreducible representations of K.

Weyl's Theorem for Algebraically (\(\wp, \wr, \varrho\))-Paranormal and Algebraically (\(\wp, \wr, \varrho\))-*-Paranormal Operators

Present L be an algebraically (\(\wp, \wr, \varrho\))-Paranormal and algebraically (\(\wp, \wr, \varrho\))-*-Paranormal operators on \(L^2\) space. We examine Weyl's theorem, a-Browder's theorem and spectral mapping theorem holds for weyl's spectrum of L and essential approximate point spectrum of L.

The Peter–Weyl theorem & harmonic analysis on S^n

For finite groups, the Artin–Wedderburn theorem gives a precise decomposition of the algebra of all C-valued functions into matrix algebras. Specialized to the case of cyclic groups, this produces the classical discrete Fourier transform. In this paper, we endeavor to extend these techniques to compact topological groups, proving similar harmonic decompositions on S1, S2, and S3.

ANALOGUES OF JACOBI AND WEYL THEOREMS FOR INFINITE-DIMENSIONAL TORI

Generalizations of the Jacobi and Weyl theorems on finitedimensional linear flows to the case of linear flows on infinite-dimensional tori are presented. Conditions for periodicity, non-wandering, ergodicity and transitivity of trajectories of an infinite-dimensional linear flow are obtained. It is shown that for infinite-dimensional linear flows there is a new type of trajectories that is absent in the finite-dimensional case

Polynomial stability of piezoelectric beams with magnetic effect and tip body

AbstractIn this paper, we consider a dissipative system of one‐dimensional piezoelectric beam with magnetic effect and a tip load at the free end of the beam, which is modeled as a special form of double boundary dissipation. Our main aim is to study the well‐posedness and asymptotic behavior of this system. By introducing two functions defined on the right boundary, we first transform the original problem into a new abstract form, so as to show the well‐posedness of the system by using Lumer–Phillips theorem. We then divide the original system into a conservative system and an auxiliary system, and show that the auxiliary problem generates a compact operator. With the help of Weyl's theorem, we obtain that the system is not exponentially stable. Moreover, we prove the polynomial stability of the system by using a result of Borichev and Tomilov (Math. Ann. 347 (2010), 455–478).

The double Gelfand–Cetlin system, invariance of polarization, and the Peter–Weyl theorem

The bundle map T⁎U(n)⟶U(n) provides a real polarization of the cotangent bundle T⁎U(n), and yields the geometric quantization Q1(T⁎U(n))=L2(U(n)). We use the Gelfand–Cetlin systems of Guillemin and Sternberg to show that T⁎U(n) has a different real polarization with geometric quantization Q2(T⁎U(n))=⨁αVα⊗Vα⁎, where the sum is over all dominant integral weights α of U(n). The Peter–Weyl theorem, which states that these two quantizations are isomorphic, may therefore be interpreted as an instance of “invariance of polarization” in geometric quantization.

Totally (m, n)-paranormal operators

In this paper, we investigate the class of totally (m, n)-paranormal operators on Hilbert space and prove some additional properties of totally (m, n)-paranormal operators on Hilbert space. Also, we show that Weyl's theorem holds for operators in this class. Moreover, we also show that for totally (m, n)-paranormal operators, the Weyl spectrum satisfies the spectral mapping theorem.

Theorems of Carathéodory, Minkowski–Weyl, and Gordan up to Symmetry

In this paper we extend three classical and fundamental results in polyhedral geometry, namely, Carathéodory’s theorem, the Minkowski–Weyl theorem, and Gordan’s lemma, to infinite dimensional spaces, in which considered cones and monoids are invariant under actions of symmetric groups.

The Supergeometric Algebra

Spinors are central to physics: all matter (fermions) is made of spinors, and all forces arise from symmetries of spinors. It is common to consider the geometric (Clifford) algebra as the fundamental edifice from which spinors emerge. This paper advocates the alternative view that spinors are more fundamental than the geometric algebra. The algebra consisting of linear combinations of scalars, column spinors, row spinors, multivectors, and their various products, can be termed the supergeometric algebra. The inner product of a row spinor with a column spinor yields a scalar, while the outer product of a column spinor with a row spinor yields a multivector, in accordance with the Brauer–Weyl (Am J Math 57: 425–449, 1935, https://doi.org/10.2307/2371218 ) theorem. Prohibiting the product of a row spinor with a row spinor, or a column spinor with a column spinor, reproduces the exclusion principle. The fact that the index of a spinor is a bitcode is highlighted.

Properties (RB) and (gRB) for Bounded Linear Operators

In this article, we consider pseudo invertible operators for study of the relationship between the space of relatively regular operators and some generalizations of Weyl and Browder theorems. By using the analysis and representation of pseudo invertible operators, some new properties in connection with Browder’s type theorems, were presented for bounded linear operators T ∈ B(X). These properties, which we refer to as property (RB), imply that All poles of the resolvent of T of finite rank in the typical spectrum are precisely those places of the spectrum for which a reasonably regular operator with its pseudo inverse operator is surjective. (γI – T)† ∈ SU(X), In the usual spectrum, the set of all poles of the resolvent is exactly those points of the spectrum for which we call property (gRB). γI – T is a B-relatively regular operator with its pseudo inverse operator is surjective (γI – T)† ∈ SU(X). In addition, several sufficient and necessary conditions for which properties (RB) and (gRB) hold are given.

An overview of generalised Kac-Moody algebras on compact real manifolds

A generalised notion of Kac-Moody algebra is defined using smooth maps from a compact real manifold M to a finite-dimensional Lie group, by means of complete orthonormal bases for a Hermitian inner product on the manifold and a Fourier expansion. The Peter–Weyl theorem for the case of manifolds related to compact Lie groups and coset spaces is discussed, and appropriate Hilbert bases for the space L2(M) of square-integrable functions are constructed. It is shown that such bases are characterised by the representation theory of the compact Lie group, from which a complete set of labelling operator is obtained. The existence of central extensions of generalised Kac-Moody algebras is analysed using a duality property of Hermitian operators on the manifold, and the corresponding root systems are constructed. Several applications of physically relevant compact groups and coset spaces are discussed.

Explicit Kronecker–Weyl theorems and applications to prime number races

We prove explicit versions of the Kronecker–Weyl theorems, both in a discrete and a continuous settings, without any linear independence hypothesis. As an application, we propose an alternative approach to problems concerning asymptotic densities in prime number races, over number fields and over function fields in one variable over finite fields, in the language of random variables. Our approach allows us to prove new results on the existence and positivity of some of those densities, which, in the case of races over function fields, do not require any linear independence hypothesis.

Conformal Nets V: Dualizability

We prove that finite-index conformal nets are fully dualizable objects in the 3-category of conformal nets. Therefore, assuming the cobordism hypothesis applies, there exists a local framed topological field theory whose value on the point is any finite-index conformal net. Along the way, we prove a Peter–Weyl theorem for defects between conformal nets, namely that the annular sector of a finite defect is the sum of every sector tensored with its dual.

Spectral theory of B-Weyl elements and the generalized Weyl's theorem in primitive C*-algebra

Let $\mathcal{A}$ be a unital primitive C*-algebra. This paper studies the spectral theories of B-Weyl elements and B-Browder elements in $\mathcal{A}$, including the spectral mapping theorem and a characterization of B-Weyl spectrum. In addition, we characterize the generalized Weyl's theorem and the generalized Browder's theorem for an element $a\in\mathcal{A}$ and $f(a)$, where $f$ is a complex-valued function analytic on a neighborhood of $\sigma(a)$. What's more, the perturbations of the generalized Weyl's theorem under the socle of $\mathcal{A}$ and quasinilpotent element are illustrated.

Algebraic extension of $\mathcal{A}^{*}_{n}$ operator

$T\in L(H_{1}\oplus H_{2})$ is said to be an algebraic extension of a $\mathcal{A}^{*}_{n}$ operator if $$ T = \begin{pmatrix} T_{1} & T_{2} \\O & T_{3} \end{pmatrix} $$ is an operator matrix on $H_{1}\oplus H_{2}$, where $T_{1}$ is a $\mathcal{A}^{*}_{n}$ operator and $T_{3}$ is a algebraic.In this paper, we study basic and spectral properties of an algebraic extension of a $\mathcal{A}^{*}_{n}$ operator. We show that every algebraic extension of a $\mathcal{A}^{*}_{n}$ operator has SVEP, is polaroid and satisfies Weyl's theorem.

Generalized derivations and Hom-Lie algebra structures on

The purpose of this paper is to show that there are Hom-Lie algebra structures on where D is a special type of generalized derivation of and is an algebraically closed field of characteristic zero. We study the representation theory of Hom-Lie algebras within the appropriate category and prove that any finite dimensional representation of a Hom-Lie algebra of the form is completely reducible, analogously to the well-known theorem of Weyl from the classical Lie theory. We apply this result to characterize the non-solvable Lie algebras of this type having an invertible generalized derivation D.

Universal Approximations of Invariant Maps by Neural Networks

We describe generalizations of the universal approximation theorem for neural networks to maps invariant or equivariant with respect to linear representations of groups. Our goal is to establish network-like computational models that are both invariant/equivariant and provably complete in the sense of their ability to approximate any continuous invariant/equivariant map. Our contribution is three-fold. First, in the general case of compact groups we propose a construction of a complete invariant/equivariant network using an intermediate polynomial layer. We invoke classical theorems of Hilbert and Weyl to justify and simplify this construction; in particular, we describe an explicit complete ansatz for approximation of permutation-invariant maps. Second, we consider groups of translations and prove several versions of the universal approximation theorem for convolutional networks in the limit of continuous signals on euclidean spaces. Finally, we consider 2D signal transformations equivariant with respect to the group SE(2) of rigid euclidean motions. In this case we introduce the “charge–conserving convnet”—a convnet-like computational model based on the decomposition of the feature space into isotypic representations of SO(2). We prove this model to be a universal approximator for continuous SE(2)—equivariant signal transformations.

Lorentz- and permutation-invariants of particles

Two theorems of Weyl tell us that the algebra of Lorentz- (and parity-) invariant polynomials in the momenta of n particles are generated by the dot products and that the redundancies which arise when n exceeds the spacetime dimension d are generated by the (d + 1)-minors of the n × n matrix of dot products. We extend the first theorem to include the action of an arbitrary permutation group P ⊂ S n on the particles, to take account of the quantum-field-theoretic fact that particles can be indistinguishable. Doing so provides a convenient set of variables for describing scattering processes involving identical particles, such as pp → jjj, for which we provide an explicit minimal set of Lorentz- and permutation-invariant generators. Additionally, we use the Cohen–Macaulay structure of the Lorentz-invariant algebra to provide a more direct characterisation in terms of a Hironaka decomposition. Among the benefits of this approach is that it can be generalized straightforwardly to when parity is not a symmetry and to cases where a permutation group acts on the particles. In the first non-trivial case, n = d + 1, we give a homogeneous system of parameters that is valid for the action of an arbitrary permutation symmetry and make a conjecture for the full Hironaka decomposition in the case without permutation symmetry. An appendix gives formulæ for the computation of the relevant Hilbert series for d ⩽ 4.

Weyl's theorem for commuting tuples of paranormal and $\ast $-paranormal operators

We show that a commuting pair $T=(T_1,T_2)$ of $\ast $-paranormal operators $T_1$ and $T_2$ with quasitriangular property satisfies Weyl’s theorem-I, that is, $$ \sigma _{\rm T}(T)\setminus \sigma _{\rm T_W}(T)=\pi _{00}(T) $$ and a commuting pair of para

Tensor product and variants of Weyl's type theorem for p - w -hyponormal operators

A Hilbert space operator $T$ is said to be $p$-$w$-hyponormal with $0 < p\leq 1$ if $|\widetilde{T}|^p\geq |T|^p\geq |\widetilde{T}^{*}|^p$, where $\widetilde{T}$ is the Aluthge transform. In this paper we prove basic properties of these operators. Using these results, we also prove that if $P$ is a Riesz idempotent for a non-zero isolated point $\lambda$ of the spectrum of $T$, then $P$ is self-adjoint. Among other things, we prove these operators are finitely ascensive and that, for non-zero $p$-$w$-hyponormal $T$ and $S$, $T\otimes S$ is $p$-$w$-hyponormal if and only if $T$ and $S$ are $p$-$w$-hyponormal. Moreover, it is shown that property $(gt)$ holds for $f(T)$, where $f\in H_{nc}(\sigma(T)).$ <br><br> Оператор $T$ у гільбертовім просторі називається $p$-$w$-гіпонормальним, де $0 < p\leq 1$, якщо $ |\widetilde{T}|^p\geq |T|^p\geq |{\widetilde{T}}^{*}|^p$, де $\widetilde{T}$ --- перетворення Алутге. В цій роботі досліджені основні властивості таких операторів. Показано також, що якщо $P$ --- ідемпотент Рісса, який відповідає ненульовій ізольованій точці $\lambda$ спектру $T$, то оператор $P$ самоспряжений. Доведено, що ці оператори мають скінченний підйом і що для ненульових $p$-$w$-гіпонормальних $T$ і $S$, $T\otimes S$ є $p$-$w$-гіпонормальним тоді й тільки тоді, коли $T$ і $S$ $p$-$w$-гіпонормальні. Крім того, доведено, що властивість $(gt)$ має місце для $f(T)$, де $f\in H_{nc}(\sigma(T)).$