scholarly journals Perturbation theory for the approximation of stability spectra by QR methods for sequences of linear operators on a Hilbert space

2012 ◽  
Vol 437 (1) ◽  
pp. 37-59 ◽  
Author(s):  
M. Badawy ◽  
E. Van Vleck
Keyword(s):  
Hilbert Space ◽  
Perturbation Theory ◽  
Linear Operators ◽  
Qr Methods

Global Perturbation of Nonlinear Eigenvalues

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Julián López-Gómez ◽  
Juan Carlos Sampedro
Keyword(s):  
Perturbation Theory ◽  
Spectral Parameter ◽  
Classical Theory ◽  
General Setting ◽  
Perturbation Parameter ◽  
Linear Operators ◽  
Fredholm Operators ◽  
Index Zero ◽  
Finite Dimensional ◽  
Nonlinear Eigenvalues

Abstract This paper generalizes the classical theory of perturbation of eigenvalues up to cover the most general setting where the operator surface 𝔏 : [ a , b ] × [ c , d ] → Φ 0 ⁢ ( U , V ) {\mathfrak{L}:[a,b]\times[c,d]\to\Phi_{0}(U,V)} , ( λ , μ ) ↦ 𝔏 ⁢ ( λ , μ ) {(\lambda,\mu)\mapsto\mathfrak{L}(\lambda,\mu)} , depends continuously on the perturbation parameter, μ, and holomorphically, as well as nonlinearly, on the spectral parameter, λ, where Φ 0 ⁢ ( U , V ) {\Phi_{0}(U,V)} stands for the set of Fredholm operators of index zero between U and V. The main result is a substantial extension of a classical finite-dimensional theorem of T. Kato (see [T. Kato, Perturbation Theory for Linear Operators, 2nd ed., Class. Math., Springer, Berlin, 1995, Chapter 2, Section 5]).

Some properties of D-weak operator topology

2020 ◽  
Vol 70 (3) ◽  
pp. 753-758
Author(s):  
Marcel Polakovič
Keyword(s):  
Hilbert Space ◽  
Effect Algebra ◽  
Linear Subspace ◽  
Linear Operators ◽  
Positive Linear Operators ◽  
Weak Operator Topology ◽  
Generalized Effect Algebra ◽  
Weak Operator ◽  
Dense Linear Subspace

AbstractLet 𝓖D(𝓗) denote the generalized effect algebra consisting of all positive linear operators defined on a dense linear subspace D of a Hilbert space 𝓗. The D-weak operator topology (introduced by other authors) on 𝓖D(𝓗) is investigated. The corresponding closure of the set of bounded elements of 𝓖D(𝓗) is the whole 𝓖D(𝓗). The closure of the set of all unbounded elements of 𝓖D(𝓗) is also the set 𝓖D(𝓗). If Q is arbitrary unbounded element of 𝓖D(𝓗), it determines an interval in 𝓖D(𝓗), consisting of all operators between 0 and Q (with the usual ordering of operators). If we take the set of all bounded elements of this interval, the closure of this set (in the D-weak operator topology) is just the original interval. Similarly, the corresponding closure of the set of all unbounded elements of the interval will again be the considered interval.

scholarly journals Duality and the existence of weakly completely continuous elements in aB*-algebra

1972 ◽  
Vol 13 (1) ◽  
pp. 56-60 ◽  
Author(s):  
B. J. Tomiuk
Keyword(s):  
Hilbert Space ◽  
Left Ideal ◽  
Linear Operators ◽  
Completely Continuous ◽  
Identity Modulo

Ogasawara and Yoshinaga [9] have shown that aB*-algebra is weakly completely continuous (w.c.c.) if and only if it is*-isomorphic to theB*(∞)-sum of algebrasLC(HX), where eachLC(HX)is the algebra of all compact linear operators on the Hilbert spaceHx.As Kaplansky [5] has shown that aB*-algebra isB*-isomorphic to theB*(∞)-sum of algebrasLC(HX)if and only if it is dual, it follows that a5*-algebraAis w.c.c. if and only if it is dual. We have observed that, if only certain key elements of aB*-algebraAare w.c.c, thenAis already dual. This observation constitutes our main theorem which goes as follows.A B*-algebraAis dual if and only if for every maximal modular left idealMthere exists aright identity modulo M that isw.c.c.

One method for the boundary value problem eigenvalues calcuating for a second-order differential equation with a fractional derivative

2019 ◽  
Vol 10 (01) ◽  
pp. 1941004
Author(s):  
Temirkhan Aleroev ◽  
Hedi Aleroeva ◽  
Lyudmila Kirianova
Keyword(s):  
Differential Equation ◽  
Perturbation Theory ◽  
Dirichlet Problem ◽  
Boundary Value Problem ◽  
Fractional Derivative ◽  
Fractional Derivatives ◽  
Order Differential Equation ◽  
Boundary Value ◽  
Second Order ◽  
Linear Operators

In this paper, we give a formula for computing the eigenvalues of the Dirichlet problem for a differential equation of second-order with fractional derivatives in the lower terms. We obtained this formula using the perturbation theory for linear operators. Using this formula we can write out the system of eigenvalues for the problem under consideration.

scholarly journals Periodicity of functionals and representations of normed algebras on reflexive spaces

1976 ◽  
Vol 20 (2) ◽  
pp. 99-120 ◽  
Author(s):  
N. J. Young
Keyword(s):  
Hilbert Space ◽  
Continuous Homomorphism ◽  
Linear Operators ◽  
The Other ◽  
Continuous Linear ◽  
Normed Algebra ◽  
Reflexive Space ◽  
To Come ◽  
Algebra Of Operators ◽  
Continuous Linear Operators

It is a well-known fact that any normed algebra can be represented isometrically as an algebra of operators with the operator norm. As might be expected from the very universality of this property, it is little used in the study of the structure of an algebra. Far more helpful are representations on Hilbert space, though these are correspondingly hard to come by: isometric representations on Hilbert space are not to be expected in general, and even continuous nontrivial representations may fail to exist. The purpose of this paper is to examine a class of representations intermediate in both availability and utility to those already mentioned—namely, representations on reflexive spaces. There certainly are normed algebras which admit isometric representations of the latter type but have not even faithful representations on Hilbert space: the most natural example is the algebra of all continuous linear operators on E where E = lp with 1 < p ≠ 2 < ∞, for Berkson and Porta proved in (2) that if E, F are taken from the spaces lp with 1 < p < ∞ and E ≠ F then the only continuous homomorphism from into is the zero mapping. On the other hand there are also algebras which have no continuous nontrivial representation on any reflexive space—for example the algebra of finite-rank operators on an irreflexive Banach space (see Berkson and Porta (2) or Barnes (1) or Theorem 3, Corollary 1 below).

The Sobolev space of half-differentiable functions and quasisymmetric homeomorphisms

2016 ◽  
Vol 23 (4) ◽  
pp. 615-622 ◽  
Author(s):  
Armen Sergeev
Keyword(s):  
Hilbert Space ◽  
Sobolev Space ◽  
Noncommutative Geometry ◽  
Function Theory ◽  
Banach Algebras ◽  
Linear Operators ◽  
Boundary Values ◽  
Differentiable Functions ◽  
Quasisymmetric Homeomorphisms

AbstractIn this paper, we give an interpretation of some classical objects of function theory in terms of Banach algebras of linear operators in a Hilbert space. We are especially interested in quasisymmetric homeomorphisms of the circle. They are boundary values of quasiconformal homeomorphisms of the disk and form a group ${\operatorname{QS}(S^{1})}$ with respect to composition. This group acts on the Sobolev space ${H^{1/2}_{0}(S^{1},\mathbb{R})}$ of half-differentiable functions on the circle by reparameterization. We give an interpretation of the group ${\operatorname{QS}(S^{1})}$ and the space ${H^{1/2}_{0}(S^{1},\mathbb{R})}$ in terms of noncommutative geometry.

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