scholarly journals Finite dimensional invariant subspaces for algebras of linear operators and amenable Banach algebras

2016 ◽  
Vol 510 ◽  
pp. 329-345
Author(s):  
Rasoul Nasr-Isfahani ◽  
Mehdi Nemati ◽  
Somayeh Shahmoradi
Keyword(s):  
Banach Algebras ◽  
Invariant Subspaces ◽  
Linear Operators ◽  
Finite Dimensional

scholarly journals 2-локальные дифференцирования на алгебрах матрично-значных функций на компакте

2018 ◽  
Author(s):  
S.A. Ayupov ◽  
F.N. Arzikulov
Keyword(s):  
Hilbert Space ◽  
Banach Algebras ◽  
Algebraic Approach ◽  
Linear Operators ◽  
Bounded Linear ◽  
Infinite Dimensional ◽  
Matrix Algebras ◽  
Dimensional Matrix ◽  
Finite Dimensional ◽  
Local Derivation

The present paper is devoted to 2-local derivations. In 1997, P. Semrl introduced the notion of 2-local derivations and described 2-local derivations on the algebra B(H) of all bounded linear operators on the infinite-dimensional separable Hilbert space H. After this, a number of paper were devoted to 2-local maps on different types of rings, algebras, Banach algebras and Banach spaces. A similar description for the finite-dimensional case appeared later in the paper of S. O. Kim and J. S. Kim. Y. Lin and T. Wong described 2-local derivations on matrix algebras over a finite-dimensional division ring. Sh. A. Ayupov and K. K. Kudaybergenov suggested a new technique and have generalized the above mentioned results for arbitrary Hilbert spaces. Namely they considered 2-local derivations on the algebra B(H) of all linear bounded operators on an arbitrary Hilbert space H and proved that every 2-local derivation on B(H) is a derivation. Then there appeared several papers dealing with 2-local derivations on associative algebras. In the present paper 2-lo\-cal derivations on various algebras of infinite dimensional matrix-valued functions on a compactum are described. We develop an algebraic approach to investigation of derivations and \mbox{2-local} derivations on algebras of infinite dimensional matrix-valued functions on a compactum and prove that every such 2-local derivation is a derivation. As the main result of the paper it is established that every \mbox{2-local} derivation on a ∗-algebra C(Q,Mn(F)) or C(Q,Nn(F)), where Q is a compactum, Mn(F) is the ∗-algebra of infinite dimensional matrices over complex numbers (real numbers or quaternoins) defined in section 1, Nn(F) is the ∗-subalgebra of Mn(F) defined in section 2, is a derivation. Also we explain that the method developed in the paper can be applied to Jordan and Lie algebras of infinite dimensional matrix-valued functions on a compactum.

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]).

scholarly journals Compact linear operators from an algebraic standpoint

1967 ◽  
Vol 8 (1) ◽  
pp. 41-49 ◽  
Author(s):  
F. F. Bonsall
Keyword(s):  
Banach Algebra ◽  
Banach Algebras ◽  
Linear Operators ◽  
Identity Operator ◽  
Algebra Structure ◽  
Natural Setting ◽  
Norm Topology ◽  
Bounded Linear ◽  
Underlying Space ◽  
Closed Subalgebra

Let B(X) denote the Banach algebra of all bounded linear operators on a Banach space X. Let t be an element of B(X), and let edenote the identity operator on X. Since the earliest days of the theory of Banach algebras, ithas been understood that the natural setting within which to study spectral properties of t is the Banach algebra B(X), or perhaps a closed subalgebra of B(X) containing t and e. The effective application of this method to a given class of operators depends upon first translating the data into terms involving only the Banach algebra structure of B(X) without reference to the underlying space X. In particular, the appropriate topology is the norm topology in B(X) given by the usual operator norm. Theorem 1 carries out this translation for the class of compact operators t. It is proved that if t is compact, then multiplication by t is a compact linear operator on the closed subalgebra of B(X) consisting of operators that commute with t.

scholarly journals Conditional Lie-Bäcklund Symmetries and Reductions of the Nonlinear Diffusion Equations with Source

2014 ◽  
Vol 2014 ◽  
pp. 1-17
Author(s):  
Junquan Song ◽  
Yujian Ye ◽  
Danda Zhang ◽  
Jun Zhang
Keyword(s):  
Invariant Subspace ◽  
Dynamic Systems ◽  
Nonlinear Diffusion ◽  
Invariant Subspaces ◽  
Higher Order ◽  
Diffusion Equations ◽  
Canonical Forms ◽  
Nonlinear Diffusion Equations ◽  
Symmetry Approach ◽  
Finite Dimensional

Conditional Lie-Bäcklund symmetry approach is used to study the invariant subspace of the nonlinear diffusion equations with sourceut=e−qx(epxP(u)uxm)x+Q(x,u),m≠1. We obtain a complete list of canonical forms for such equations admit multidimensional invariant subspaces determined by higher order conditional Lie-Bäcklund symmetries. The resulting equations are either solved exactly or reduced to some finite-dimensional dynamic systems.

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.

scholarly journals Boundary problems for Riccati and Lyapunov equations

1986 ◽  
Vol 29 (1) ◽  
pp. 15-21 ◽  
Author(s):  
Lucas Jódar
Keyword(s):  
Boundary Condition ◽  
Boundary Problem ◽  
Transport Theory ◽  
Adjoint Operator ◽  
Linear Operators ◽  
Dimensional Case ◽  
Bounded Linear Operators ◽  
Bounded Linear ◽  
Finite Dimensional ◽  
Image Position

The resolution problem of the systemwhere U(t), A, B, D and Uo are bounded linear operators on H and B* denotes the adjoint operator of B, arises in control theory, [9], transport theory, [12], and filtering problems, [3]. The finite-dimensional case has been introduced in [6,7], and several authors have studied the infinite-dimensional case, [4], [13], [18]. A recent paper, [17],studies the finite dimensional boundary problemwhere t ∈[0,b].In this paper we consider the more general boundary problemwhere all operators which appear in (1.2) are bounded linear operators on a separable Hilbert space H. Note that we do not suppose C = −B* and the boundary condition in (1.2) is more general than the boundary condition in (1.1).

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