Explore chapters and articles related to this topic
Compact operators on Hilbert space
Published in Orr Moshe Shalit, A First Course in Functional Analysis, 2017
Recall that the nonzero points of the spectrum of a compact operator consist of eigenvalues. If we define Nλ = ker(λ − A), we can reformulate the above theorem by saying that for every compact self-adjoint operator, H decomposes as the direct sum () H=⊕λ∈σ(A)Nλ=⊕λ∈σp(A)Nλ,
Diffraction by a bounded planar screen
Published in A.S. Ilyinsky, Yu.G. Smirnov, Yu.V. Shestopalov, Electromagnetic Wave Diffraction by Conducting Screens, 2022
A.S. Ilyinsky, Yu.G. Smirnov, Yu.V. Shestopalov
The proof that T and L are Fredholm operators with a zero index (in the corresponding spaces) constitutes a central part of this section. We recall that a bounded operator F with a closed domain of values is called the Fredholm operator, if dim ker F<∞ and dim coker F<∞ the difference ind F = dim ker F —dim coker F is the index of operator F. In order to prove that F is a Fredholm operator with a zero index, it is sufficient to show that F = S + K, where S is a continuously invertible operator and K is a compact operator [25, 26]). We will prove that T and L are Fredholm operators using an important property of form (2.34), which can be directly verified: t(u, v) = 0 when u≠Wi and v≠Wj for i≠j,
Measures of Noncompactness
Published in Eberhard Malkowsky, Vladimir Rakočević, Advanced Functional Analysis, 2019
Eberhard Malkowsky, Vladimir Rakočević
Measures of noncompactness are very useful tools in functional analysis, for instance in metric fixed point theory and the theory of operator equations in Banach spaces. They are also used in the studies of functional equations, ordinary and partial differential equations, fractional partial differential equations, integral and integro–differential equations, optimal control theory, and in the characterizations of compact operators between Banach spaces.
A computer-assisted proof of dynamo growth in the stretch-fold-shear map
Published in Dynamical Systems, 2023
F. A. Pramy, B. D. Mestel, A. D. Gilbert
We note that this operator is well defined for functions on , since for all , both and provided r>1. Below we review some properties of this operator, but here we note, in passing, that on the Banach space , S is a compact operator and therefore its spectrum consists of discrete eigenvalues together with 0. We refer to an eigenvector corresponding to an eigenvalue λ of S as an eigenfunction and the pair as an eigenvalue–eigenfunction pair.
Schwarz problem in a ring domain*
Published in Applicable Analysis, 2022
A. Okay Çelebi, Pelin Ayşe Gökgöz
Since is compact, the operator is the sum of the invertible operator and compact operator which implies that is a Fredholm operator with index zero. The Fredholm alternative applies to singular integral Equation (14). Thus is the solution of (10) subject to the homogeneous conditions.