Von Neumann Operators in

1984 ◽  
Vol 27 (2) ◽  
pp. 146-156
Author(s):  
Karim Seddighi
Keyword(s):  
Operator Theory ◽  
Functional Calculus ◽  
Complex Geometry ◽  
Main Concern ◽  
Spectral Mapping ◽  
Special Situation ◽  
Von Neumann ◽  
Connected Open Subset ◽  
Spectral Set ◽  
Simply Connected

AbstractFor a connected open subset Ω of the plane and n a positive integer, let be the space introduced by Cowen and Douglas in their paper, “Complex geometry and operator theory”. Our main concern is the case n = 1, in which case we show the existence of a functional calculus for von Neumann operators in for which a spectral mapping theorem holds. In particular we prove that if the spectrum of , is a spectral set for T, and if , then σ(f(T)) = f(Ω)- for every bounded analytic function f on the interior of L, where L is compact, σ(T) ⊂ L, the interior of L is simply connected and L is minimal with respect to these properties. This functional calculus turns out to be nice in the sense that the general study of von Neumann operators in is reduced to the special situation where Ω is an open connected subset of the unit disc with .

scholarly journals Perturbation of the generator of a quaternionic evolution operator

2015 ◽  
Vol 13 (04) ◽  
pp. 347-370 ◽  
Author(s):  
Daniel Alpay ◽  
Fabrizio Colombo ◽  
Irene Sabadini
Keyword(s):  
Operator Theory ◽  
Functional Calculus ◽  
Evolution Operator ◽  
Classical Case ◽  
Closed Operator ◽  
Point Of View ◽  
Linear Operators ◽  
Von Neumann ◽  
Research Areas ◽  
Hyperholomorphic Functions

The theory of slice hyperholomorphic functions, introduced in recent years, has important applications in operator theory. The quaternionic version of this function theory and its Cauchy formula yield to a definition of the quaternionic version of the Riesz–Dunford functional calculus which is based on the notion of S-spectrum. This quaternionic functional calculus allows to define the quaternionic evolution operator which appears in the quaternionic version of quantum mechanics proposed by J. von Neumann and later developed by S. L. Adler. Generation results such as the Hille–Phillips–Yosida theorem have been recently proved. In this paper, we study the perturbation of the generator. The motivation of this study is that, as it happens in the classical case of closed complex linear operators, to verify the generation conditions of the Hille–Phillips–Yosida theorem, in the concrete cases, is often difficult. Thus in this paper we study the generation problem from the perturbation point of view. Precisely, given a quaternionic closed operator T that generates the evolution operator [Formula: see text] we study under which condition a closed operator P is such that T + P generates the evolution operator [Formula: see text]. This paper is addressed to people working in different research areas such as hypercomplex analysis and operator theory.

scholarly journals Certain invariant subspaces for operators with rich eigenvalues

1991 ◽  
Vol 14 (1) ◽  
pp. 69-76
Author(s):  
Karim Seddighi
Keyword(s):  
Positive Integer ◽  
Open Subset ◽  
Functional Calculus ◽  
Invariant Subspaces ◽  
Infinite Dimensional ◽  
Connected Open Subset ◽  
Spectral Set

For a connected open subsetΩof the plane andna positive integer, letBn(Ω)be the space introduced by Cowen and Douglas. In this article we study the spectrum of restrictions ofTin order to obtain more information about the invariant subspaces ofT. Whenn=1andT ϵ B1(Ω)such thatσ(T)=Ω¯is a spectral set forTwe use the functional calculus we have developed for such operators to give some infinite dimensional cyclic invariant subspaces forT.

scholarly journals Lefschetz Operators, Hodge–Riemann Forms, and Representations

2020 ◽  
Author(s):  
Peter Fiebig
Keyword(s):  
Root System ◽  
Bilinear Form ◽  
Chevalley Group ◽  
Simple Root ◽  
Complex Geometry ◽  
Linear Operators ◽  
Vector Spaces ◽  
Tilting Module ◽  
Weight Lattice ◽  
Simply Connected

Abstract For a field of characteristic $\ne 2$, we study vector spaces that are graded by the weight lattice of a root system and are endowed with linear operators in each simple root direction. We show that these data extend to a weight lattice graded semisimple representation of the corresponding Lie algebra, if and only if there exists a bilinear form that satisfies properties (roughly) analogous to those of the Hodge–Riemann forms in complex geometry. In the 2nd part of the article, we replace the field by the $p$-adic integers (with $p\ne 2$) and show that in this case the existence of a certain bilinear form is equivalent to the existence of a structure of a tilting module for the associated simply connected $p$-adic Chevalley group.

CESÀRO–VOLTERRA PATH INTEGRAL FORMULA ON A SURFACE

2009 ◽  
Vol 19 (03) ◽  
pp. 419-441 ◽  
Author(s):  
PHILIPPE G. CIARLET ◽  
LILIANA GRATIE ◽  
MICHELE SERPILLI
Keyword(s):  
Path Integral ◽  
Open Subset ◽  
Displacement Field ◽  
Tensor Field ◽  
Displacement Vector ◽  
Integral Formula ◽  
Symmetric Matrix ◽  
Explicit Function ◽  
Connected Open Subset ◽  
Simply Connected

If a symmetric matrix field e = (eij) of order three satisfies the Saint-Venant compatibility relations in a simply-connected open subset Ω of ℝ3, then e is the linearized strain tensor field of a displacement field v of Ω, i.e. [Formula: see text] in Ω. A classical result, due to Cesàro and Volterra, asserts that, if the field e is smooth, the unknown displacement field v(x) at any point x ∈ Ω can be explicitly written as a path integral inside Ω with endpoint x, and whose integrand is an explicit function of the functions eij and their derivatives. Now let ω be a simply-connected open subset in ℝ2 and let θ : ω → ℝ3 be a smooth immersion. If two symmetric matrix fields (γαβ) and (ραβ) of order two satisfy appropriate compatibility relations in ω, then (γαβ) and (ραβ) are the linearized change of metric and change of curvature tensor field corresponding to a displacement vector field η of the surface θ(ω). We show here that a "Cesàro–Volterra path integral formula on a surface" likewise holds when the fields (γαβ) and (ραβ) are smooth. This means that the displacement vector η(y) at any point θ(y), y ∈ ω, of the surface θ(ω) can be explicitly computed as a path integral inside ω with endpoint y, and whose integrand is an explicit function of the functions γαβ and ραβ and their covariant derivatives. Such a formula has potential applications to the mathematical analysis and numerical simulation of linear "intrinsic" shell models.

scholarly journals FUNCTIONAL CALCULI FOR SECTORIAL OPERATORS AND RELATED FUNCTION THEORY

2021 ◽  
pp. 1-81
Author(s):  
Charles Batty ◽  
Alexander Gomilko ◽  
Yuri Tomilov
Keyword(s):  
Banach Spaces ◽  
Functional Calculus ◽  
Function Theory ◽  
Operator Norm ◽  
Spectral Mapping ◽  
Related Function ◽  
Norm Estimates ◽  
Sectorial Operators ◽  
Associated Function ◽  
Class Of Functions

Abstract We construct two bounded functional calculi for sectorial operators on Banach spaces, which enhance the functional calculus for analytic Besov functions, by extending the class of functions, generalising and sharpening estimates and adapting the calculus to the angle of sectoriality. The calculi are based on appropriate reproducing formulas, they are compatible with standard functional calculi and they admit appropriate convergence lemmas and spectral mapping theorems. To achieve this, we develop the theory of associated function spaces in ways that are interesting and significant. As consequences of our calculi, we derive several well-known operator norm estimates and provide generalisations of some of them.

scholarly journals Simultaneous Upper Triangular Forms for Commuting Operators in a Finite von Neumann Algebra

2019 ◽  
Vol 72 (5) ◽  
pp. 1188-1245
Author(s):  
Ian Charlesworth ◽  
Ken Dykema ◽  
Fedor Sukochev ◽  
Dmitriy Zanin
Keyword(s):  
Functional Calculus ◽  
Von Neumann Algebra ◽  
Von Neumann ◽  
Holomorphic Functional Calculus ◽  
Joint Spectrum ◽  
Natural Properties ◽  
Tracial State ◽  
Commuting Operators ◽  
Finite Von Neumann Algebra

AbstractThe joint Brown measure and joint Haagerup–Schultz projections for tuples of commuting operators in a von Neumann algebra equipped with a faithful tracial state are investigated, and several natural properties are proved for these. It is shown that the support of the joint Brown measure is contained in the Taylor joint spectrum of the tuple, and also in the ostensibly smaller left Harte spectrum. A simultaneous upper triangularization result for finite commuting tuples is proved, and the joint Brown measure and joint Haagerup–Schultz projections are shown to behave well under the Arens multivariate holomorphic functional calculus of such a commuting tuple.

scholarly journals Complex geometry and operator theory

1978 ◽  
Vol 141 (0) ◽  
pp. 187-261 ◽  
Author(s):  
M. J. Cowen ◽  
R. G. Douglas
Keyword(s):  
Operator Theory ◽  
Complex Geometry
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