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.

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 .

Biholomorphic Mappings on Banach Spaces

2019 ◽  
Vol 62 (4) ◽  
pp. 913-924
Author(s):  
H. Carrión ◽  
P. Galindo ◽  
M. L. Lourenço
Keyword(s):  
Banach Space ◽  
Hilbert Space ◽  
Open Subset ◽  
Separable Hilbert Space ◽  
Derivative Operator ◽  
Infinite Dimensional ◽  
Dimensional Version ◽  
Dual Banach Space ◽  
Bounded Convex ◽  
Power Bounded

AbstractWe present an infinite-dimensional version of Cartan's theorem concerning the existence of a holomorphic inverse of a given holomorphic self-map of a bounded convex open subset of a dual Banach space. No separability is assumed, contrary to previous analogous results. The main assumption is that the derivative operator is power bounded, and which we, in turn, show to be diagonalizable in some cases, like the separable Hilbert space.

scholarly journals ON THE ANNIHILATOR OF A DOLBEAULT GROUP

2010 ◽  
Vol 21 (03) ◽  
pp. 317-331
Author(s):  
IMRE PATYI
Keyword(s):  
Holomorphic Function ◽  
Open Subset ◽  
Cohomology Group ◽  
Holomorphic Functions ◽  
Dolbeault Cohomology ◽  
Infinite Dimensional ◽  
Continuous Character

We show that any Dolbeault cohomology group Hp,q(D), p ≥ 0, q ≥ 1, of an open subset D of a closed finite codimensional complex Hilbert submanifold of ℓ2 is either zero or infinite dimensional. We also show that any continuous character of the algebra of holomorphic functions of a closed complex Hilbert submanifold M of ℓ2 is induced by its evaluation at a point of M. Lastly, we prove that any closed split infinite dimensional complex Banach submanifold of ℓ2 admits a nowhere critical holomorphic function.

scholarly journals Application of the functional calculus to solving of infinite dimensional heat equation

2016 ◽  
Vol 8 (2) ◽  
pp. 313-322
Author(s):  
S.V. Sharyn
Keyword(s):  
Cauchy Problem ◽  
Heat Equation ◽  
Functional Calculus ◽  
Ultradifferentiable Functions ◽  
Infinite Dimensional ◽  
Semigroup Approach

In this paper we study infinite dimensional heat equation associated with the Gross Laplacian. Using the functional calculus method, we obtain the solution of appropriate Cauchy problem in the space of polynomial ultradifferentiable functions. The semigroup approach is considered as well.

Robust Hyperbolic Chaos in Froude Pendulum with Delayed Feedback and Periodic Braking

2019 ◽  
Vol 29 (12) ◽  
pp. 1930035
Author(s):  
Sergey P. Kuznetsov ◽  
Yuliya V. Sedova
Keyword(s):  
Parameter Space ◽  
Invariant Subspaces ◽  
Constant Angular Velocity ◽  
Diagnostic Tools ◽  
Small Perturbations ◽  
Infinite Dimensional ◽  
Dimensional State Space ◽  
Froude Pendulum ◽  
Numerical Research ◽  
Hyperbolic Chaos

We indicate a possibility of implementing hyperbolic chaos using a Froude pendulum that is able to produce self-oscillations due to the suspension on a shaft rotating at constant angular velocity, in the presence of time-delay feedback and of periodic braking by the application of additional frictional force. We formulate a mathematical model and carry out its numerical research. In the parameter space we reveal areas of chaotic and regular dynamics using the analysis of Lyapunov exponents and some other diagnostic tools. It is shown that there are regions in the parameter space where the Poincaré stroboscopic map has an attractor, which is a kind of Smale–Williams solenoid embedded in the infinite-dimensional state space. We confirm the hyperbolicity of the attractor by numerical calculations including the analysis of angles of intersections of stable and unstable invariant subspaces of vectors of small perturbations for trajectories on the attractor and verify the absence of tangencies between these subspaces.

Vector Spaces

2021 ◽  
pp. 47-102
Author(s):  
Adel N. Boules
Keyword(s):  
Invariant Subspaces ◽  
Extensive Study ◽  
Vector Spaces ◽  
Inner Product ◽  
Product Spaces ◽  
Direct Sums ◽  
Infinite Dimensional ◽  
Quotient Spaces ◽  
Inner Product Spaces ◽  
Finite Dimensional

The first three sections of this chapter provide a thorough presentation of the concepts of basis and dimension. The approach is unified in the sense that it does not treat finite and infinite-dimensional spaces separately. Important concepts such as algebraic complements, quotient spaces, direct sums, projections, linear functionals, and invariant subspaces make their first debut in section 3.4. Section 3.5 is a brief summary of matrix representations and diagonalization. Then the chapter introduces normed linear spaces followed by an extensive study of inner product spaces. The presentation of inner product spaces in this section and in section 4.10 is not limited to finite-dimensional spaces but rather to the properties of inner products that do not require completeness. The chapter concludes with the finite-dimensional spectral theory.

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