AMD-Numbers, Compactness, Strict Singularity and the Essential Spectrum of Operators
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Abstract
For an operator 𝑇 acting from an infinite-dimensional Hilbert space 𝐻 to a normed space 𝑌 we define the upper AMD-number 
and the lower AMD-number 
as the upper and the lower limit of the net (δ(𝑇|𝐸))𝐸∈𝐹𝐷(𝐻), with respect to the family 𝐹𝐷(𝐻) of all finite-dimensional subspaces of 𝐻. When these numbers are equal, the operator is called AMD-regular.
It is shown that if an operator 𝑇 is compact, then 
and, conversely, this property implies the compactness of 𝑇 provided 𝑌 is of cotype 2, but without this requirement may not imply this. Moreover, it is shown that an operator 𝑇 has the property if and only if it is superstrictly singular. As a consequence, it is established that any superstrictly singular operator from a Hilbert space to a cotype 2 Banach space is compact.
For an operator 𝑇, acting between Hilbert spaces, it is shown that and are respectively the maximal and the minimal elements of the essential spectrum of 
, and that 𝑇 is AMD-regular if and only if the essential spectrum of |𝑇| consists of a single point.
© Heldermann Verlag
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- A Problem of Linear Conjugation for Analytic Functions with Boundary Values from the Zygmund Class
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Articles in the same Issue
- Hyperconvex Spaces and Fixed Points
- Asymptotic Behaviour and Hopf Bifurcation of a Three-Dimensional Nonlinear Autonomous System
- AMD-Numbers, Compactness, Strict Singularity and the Essential Spectrum of Operators
- On an Optimal Decomposition in Zygmund Spaces
- Ordinary Differential Equations with Nonlinear Boundary Conditions
- Geometry of Modulus Spaces
- A Note on Convexly Independent Subsets of an Infinite Set of Points
- A Problem of Linear Conjugation for Analytic Functions with Boundary Values from the Zygmund Class
- On Common Fixed Points
- Characteristic Functions and s-Orthogonality Properties of Chebyshev Polynomials of Third and Fourth Kind
- Periodic Points and Chaotic-Like Dynamics of Planar Maps Associated to Nonlinear Hill's Equations with Indefinite Weight
- Conformal and Quasiconformal Mappings of Close Multiply-Connected Domains
- Strong Innovation and its Applications to Information Diffusion Modelling in Finance
- Boundary Integral Equations of Plane Elasticity in Domains with Peaks. Addendum
