Weyl’s Theorem for Algebraically Quasi-class A Operators

If T or T* is an algebraically quasi-class A operator acting on an infinite dimensional separable Hilbert space then we prove that Weyl’s theorem holds for f(T) for every f ∈ H(σ(T)), where H(σ(T)) denotes the set of all analytic functions in an open neighborhood of σ(T). Moreover, if T* is algebraically quasi-class A then a-Weyl’s theorem holds for f(T). Also, if T or T* is an algebraically quasi-class A operator then we establish that the spectral mapping theorems for the Weyl spectrum and the essential approximate point spectrum of T for every f ∈ H(σ(T)), respectively. This research was supported by the Kyung Hee University Research Fund in 2007 (KHU- 20071605).     

On Weyl’s Theorem for Functions of Operators

Let H be a complex separable infinite dimensional Hilbert space. In this paper, a variant of the Weyl spectrum is discussed. Using the new spectrum, we characterize the necessary and sufficient conditions for both T and f(T) satisfying Weyl's theorem, where f ∈ Hol(σ(T)) and Hol(σ(T)) is defined by the set of all functions f which are analytic on a neighbourhood of σ(T) and are not constant on any component of σ(T). Also we consider the perturbations of Weyl's theorem for f(T).     

Weyl’s Theorem for Functions of Operators and Approximation

Let H{\mathcal{H}} be a complex separable infinite dimensional Hilbert space. In this paper, we characterize those operators T on H{\mathcal{H}} satisfying that Weyl’s theorem holds for f(T) for each function f analytic on some neighborhood of σ(T). Also, it is proved that, given an operator T on H{\mathcal{H}} and ε > 0, there exists a compact operator K with ||K|| < e{\|K\| < \varepsilon} such that Weyl’s theorem holds for T + K.     

Riesz Idempotent and Weyl’s Theorem for w-hyponormal Operators

Let T be a w-hyponormal operator on a Hilbert space H, its Aluthge transform, λ an isolated point of the spectrum of T, and E_λ and the Riesz idempotents, with respect to λ, of T and respectively. It is shown that Consequently, E_λ is self-adjoint, and if λ ≠ 0. Moreover, it is shown that Weyl’s theorem holds for f(T), where f ∈ H(σ (T)).     

Hörmander’s Hypoelliptic Theorem for Nonlocal Operators

Journal of Theoretical Probability - In this paper we show the Hörmander hypoelliptic theorem for nonlocal operators by a purely probabilistic method: the Malliavin calculus. Roughly speaking,...     

Property (aw) and Weyl’s Theorem

In this note we study the property (aw), a variant of Weyl’s theorem introduced by Berkani and Zariouh, by means of the localized single valued extension property (SVEP). We establish for a bounded linear operator defined on a Banach space several sufficient and necessary conditions for which property (aw) holds. We also relate this property with Weyl’s theorem, a-Weyl’s theorem and property (w). Finally, we show that if T is a-polaroid and either T or T* has SVEP then f(T) satisfies property (aw) for each ${f \in H_1(\sigma(T))}$ .     

Operator Matrices: SVEP and Weyl’s Theorem

It is known that if and are Banach space operators with the single-valued extension property, SVEP, then the matrix operator has SVEP for every operator and hence obeys Browder’s theorem. This paper considers conditions on operators A, B, and M₀ ensuring Weyls theorem for operators M_C.     

The Schur–Weyl Graph and Thoma’s Theorem

Functional Analysis and Its Applications - We define a graded graph, called the Schur–Weyl graph, which arises naturally when one considers simultaneously the RSK algorithm and the classical...     

Semi-Fredholm Spectrum and Weyl＇s Theorem for Operator Matrices

When A ∈ B（H） and B ∈ B（K） are given, we denote by Mc an operator acting on the Hilbert space HΘ K of the form Me = （ A0 CB）. In this paper, first we give the necessary and sufficient condition for Mc to be an upper semi-Fredholm （lower semi-Fredholm, or Fredholm） operator for some C ∈B（K,H）. In addition, let σSF＋（A） = {λ ∈ C ： A-λI is not an upper semi-Fredholm operator} bc the upper semi-Fredholm spectrum of A ∈ B（H） and let σrsF- （A） = {λ∈ C ： A-λI is not a lower semi-Fredholm operator} be the lower semi Fredholm spectrum of A. We show that the passage from σSF±（A） U σSF±（B） to σSF±（Mc） is accomplished by removing certain open subsets of σSF-（A） ∩σSF＋ （B） from the former, that is, there is an equality σSF±（A） ∪σSF± （B） = σSF± （Mc） ∪＆ where L is the union of certain of the holes in σSF±（Mc） which ilappen to be subsets of σSF- （A） A σSF＋ （B）. Weyl＇s theorem and Browder＇s theorem are liable to fail for 2 × 2 operator matrices. In this paper, we also explore how Weyl＇s theorem, Browder＇s theorem, a-Weyl＇s theorem and a-Browder＇s theorem survive for 2 × 2 upper triangular operator matrices on the Hilbert space.     

On Pitt’s Theorem for Operators between Scalar and Vector-Valued Quasi-Banach Sequence Spaces

 We find natural conditions under which all continuous linear operators between two scalar or vector-valued quasi-Banach sequence spaces are compact. In the case of scalar-valued Banach sequence spaces we show that all such operators essentially factorize through diagonal operators between suitable -spaces.     

Noether’s Theorem for Constrained Hamiltonian System Under Generalized Operators北大核心CSCD

Noether’s symmetry and conserved quantity of singular systems under generalized operators were studied. Firstly, the Lagrangian equation of singular systems under generalized operators was established, and the primary constraints on the system were derived. Then the Lagrangian multiplier was introduced to establish the constrained Hamilton equation and the compatibility condition under generalized operators. Secondly, based on the invariance of the Hamilton action under the infinitesimal transformation, Noether’s theorem for constrained Hamiltonian systems under generalized operators was established, and the symmetry and corresponding conserved quantity of the system were given. Under certain conditions, Noether’s conservation of constrained Hamiltonian systems under generalized operators can be reduced to Noether’s conservation of integer-order constrained Hamiltonian systems. Finally, an example illustrates the application of the results. © 2022 Editorial Office of Applied Mathematics and Mechanics. All rights reserved.     

On Pitt’s Theorem for Operators between Scalar and Vector-Valued Quasi-Banach Sequence Spaces

 We find natural conditions under which all continuous linear operators between two scalar or vector-valued quasi-Banach sequence spaces are compact. In the case of scalar-valued Banach sequence spaces we show that all such operators essentially factorize through diagonal operators between suitable -spaces. (Received 21 June 1999; in revised form 27 September 1999)     

Putnam’s Inequality for log-Hyponormal Operators

Let T be a bounded linear operator on a complex Hilbert space H. T $/in$ B(H) is called a log-hyponormal operator if T is invertible and log (TT ^*) log (T ^* T). Since a function log : (0,) (-,) is operator monotone, every invertible p-hyponormal operator T, i.e., (TT ^*) ^p (T ^* T ^p is log-hyponormal for 0 < p 1. Putnams inequality for p-hyponormal operator T is the following:$ \| (T^*T)^p-(TT^*)^p \|\leq\frac{p}{\pi}\int\int_{\sigma(T)}r^{2p-1}drd\theta $.In this paper, we prove that if T is log-hyponormal, then$ \| log(T^*T)-log(TT^*) \|\leq\frac{1}{\pi}\int\int_{\sigma(T)}r^{-1}drd\theta $.     

Cartan’s Second Main Theorem and Mason’s Theorem for Jackson Difference Operator

Let f : C → Pⁿ be a holomorphic curve of order zero. The authors establish a Jackson difference analogue of Cartan’s second main theorem for the Jackson q-Casorati determinant and introduce a truncated second main theorem of Jackson difference operator for holomorphic curves. In addition, a Jackson difference Mason’s theorem is proved by using a Jackson difference radical of a polynomial. Furthermore, they extend the Mason’s theorem for m + 1 polynomials. Some examples are constructed to show that their results are accurate.     

The Equivalence between Property (ω) and Weyl’s Theorem

We call T ∈ B(H) consistent in Fredholm and index (briefly a CFI operator) if for each B ∈ B(H),T B and BT are Fredholm together and the same index of B,or not Fredholm together.Using a new spectrum defined in view of the CFI operator,we give the equivalence of Weyl’s theorem and property (ω) for T and its conjugate operator T* .In addition,the property (ω) for operator matrices is considered.     

The Weyl’s Theorem and Its Perturbations in Semisimple Banach Algebra

Denote a semisimple Banach algebra with an identity e by A.This paper studies the Fredholm,Weyl and Browder spectral theories in a semisimple Banach algebra,and meanwhile considers the properties of the Fredholm element,the Weyl element and the Browder element.Further,for a∈A,we give the Weyl's theorem and the Browder's theorem for a,and characterize necessary and sufficient conditions that both a and f(a) satisfy the Weyl's theorem or the Browder's theorem,where f is a complex-valued function analytic on a neighborhood of σ(a).In addition,the perturbations of the Weyl's theorem and the Browder's theorem are investigated.     

On the Equivalence and Generalized of Weyl Theorem Weyl Theorem

We know that an operator T acting on a Banach space satisfying generalized Weyl＇s theorem also satisfies Weyl＇s theorem. Conversely we show that if all isolated eigenvalues of T are poles of its resolvent and if T satisfies Weyl＇s theorem, then it also satisfies generalized Weyl＇s theorem. We give also a sinlilar result for the equivalence of a-Weyl＇s theorem and generalized a-Weyl＇s theorem. Using these results, we study the case of polaroid operators, and in particular paranormal operators.     

A Refined Luecking’s Theorem and Finite-Rank Products of Toeplitz Operators

For a bounded measurable function f on the open unit disk \({\mathbb{D}}\) , let T _f denote the corresponding Toeplitz operator on the Bergman space \({A^2(\mathbb{D})}\) . A recent result of Luecking shows that if T _f has finite rank, then f must be the zero function. Using a refined version of this result, we show that if all, except possibly one, of the functions f ₁,..., f _m are radial and \({T_{f_1}\cdots T_{f_m}}\) has finite rank, then one of these functions must be zero.