Landweber scheme for compact operator equation in Hilbert space and its applications

2009 ◽  
Vol 25 (6) ◽  
pp. 771-786 ◽  
Author(s):  
Gangrong Qu ◽  
Ming Jiang
Keyword(s):  
Hilbert Space ◽  
Compact Operator ◽  
Operator Equation

On an operator equation in Hilbert space

1974 ◽  
Vol 25 (6) ◽  
pp. 660-661
Author(s):  
A. T. Zaplitnaya
Keyword(s):  
Hilbert Space ◽  
Operator Equation

Weyl's theorem for the square of operator and perturbations

2015 ◽  
Vol 17 (05) ◽  
pp. 1450042
Author(s):  
Weijuan Shi ◽  
Xiaohong Cao
Keyword(s):  
Hilbert Space ◽  
Compact Operator ◽  
Complex Hilbert Space ◽  
Linear Operators ◽  
Weyl’S Theorem ◽  
Bounded Linear Operators ◽  
Bounded Linear ◽  
Infinite Dimensional ◽  
Weyl Spectrum ◽  
The Stability

Let H be an infinite-dimensional separable complex Hilbert space and B(H) the algebra of all bounded linear operators on H. T ∈ B(H) satisfies Weyl's theorem if σ(T)\σw(T) = π00(T), where σ(T) and σw(T) denote the spectrum and the Weyl spectrum of T, respectively, π00(T) = {λ ∈ iso σ(T) : 0 < dim N(T - λI) < ∞}. T ∈ B(H) is said to have the stability of Weyl's theorem if T + K satisfies Weyl's theorem for all compact operator K ∈ B(H). In this paper, we characterize the operator T on H satisfying the stability of Weyl's theorem holds for T2.

A contribution to the solution of the compact correction problem for operators on a Banach space

1989 ◽  
Vol 31 (2) ◽  
pp. 219-229
Author(s):  
Mícheál Ó Searcóid
Keyword(s):  
Banach Space ◽  
Hilbert Space ◽  
Compact Operator ◽  
Complex Number ◽  
Space Problem ◽  
Riesz Operators ◽  
Correction Problem ◽  
Bounded Below ◽  
General Banach Space

We consider the hypothesis that an operator T on a given Banach space can always be perturbed by a compact operator K in such a way that, whenever a complex number A is in the semi-Fredholm region of T + K, then T + K – λ is either bounded below or surjective. The hypothesis has its origin in the work of West [11], who proved it for Riesz operators on Hilbert space. In this paper, we reduce the general Banach space problem to one of considering only operators of a special type, operators which are, in a spectral sense, natural generalizations of the Riesz operators studied by West.

On Compact Perturbations of Operators

1974 ◽  
Vol 26 (1) ◽  
pp. 247-250 ◽  
Author(s):  
Joel Anderson
Keyword(s):  
Hilbert Space ◽  
Compact Operator ◽  
Unitary Operator ◽  
General Theorem ◽  
Infinite Dimensional ◽  
Infinite Dimensional Hilbert Space ◽  
Dimensional Hilbert Space ◽  
Compact Perturbations ◽  
Small Norm

Recently R. G. Douglas showed [4] that if V is a nonunitary isometry and U is a unitary operator (both acting on a complex, separable, infinite dimensional Hilbert space ), then V — K is unitarily equivalent to V ⊕ U (acting on ⊕ ) where K is a compact operator of arbitrarily small norm. In this note we shall prove a much more general theorem which seems to indicate "why" Douglas' theorem holds (and which yields Douglas' theorem as a corollary).

scholarly journals An algorithm for solving generalized algebraic Lyapunov equations in Hilbert space, applications to boundary value problems

1988 ◽  
Vol 31 (1) ◽  
pp. 99-105 ◽  
Author(s):  
Lucas Jódar
Keyword(s):  
Hilbert Space ◽  
Boundary Value Problem ◽  
Boundary Value Problems ◽  
Operator Equation ◽  
Boundary Value ◽  
Linear Operators ◽  
Bounded Linear Operators ◽  
Space Applications ◽  
Bounded Linear ◽  
Image Position

Let L(H) be the algebra of all bounded linear operators on a separable complex Hubert space H. In a recent paper [7], explicit expressions for solutions of a boundary value problem in the Hubert space H, of the typeare given in terms of solutions of an algebraic operator equation

On the Hyperinvariant Subspace Problem. IV

2008 ◽  
Vol 60 (4) ◽  
pp. 758-789 ◽  
Author(s):  
H. Bercovici ◽  
C. Foias ◽  
C. Pearcy
Keyword(s):  
Hilbert Space ◽  
Compact Operator ◽  
Essential Spectrum ◽  
Complete Lattice ◽  
Unit Disc ◽  
Diagonal Operator ◽  
Hyperinvariant Subspace ◽  
Infinite Dimensional ◽  
Infinite Dimensional Hilbert Space ◽  
Subspace Problem

AbstractThis paper is a continuation of three recent articles concerning the structure of hyperinvariant subspace lattices of operators on a (separable, infinite dimensional) Hilbert space . We show herein, in particular, that there exists a “universal” fixed block-diagonal operator B on such that if ε > 0 is given and T is an arbitrary nonalgebraic operator on , then there exists a compact operator K of norm less than ε such that (i) Hlat(T) is isomorphic as a complete lattice to Hlat(B + K) and (ii) B + K is a quasidiagonal, C00, (BCP)-operator with spectrum and left essential spectrum the unit disc. In the last four sections of the paper, we investigate the possible structures of the hyperlattice of an arbitrary algebraic operator. Contrary to existing conjectures, Hlat(T) need not be generated by the ranges and kernels of the powers of T in the nilpotent case. In fact, this lattice can be infinite.

10.—An Abstract Relation for Multiparameter Eigenvalue Problems

1976 ◽  
Vol 74 ◽  
pp. 135-143 ◽  
Author(s):  
A. Källström ◽  
B. D. Sleeman
Keyword(s):  
Hilbert Space ◽  
Operator Equation ◽  
Eigenvalue Problems ◽  
Separable Hilbert Space ◽  
Bounded Operator ◽  
Image Position ◽  
Symmetric Operators ◽  
Densely Defined

SynopsisConsider the multiparameter systemwhere ut is an element of a separable Hilbert space Hi, i = 1, …, n. The operators Sij are assumed to be bounded symmetric operators in Hi and Ai is assumed self-adjoint. In addition consider the operator equationwhere B is densely defined and closed in a separable Hilbert space H and Tj, j = 1, …, n is a bounded operator in H. The problem treated in this paper is to seek an expression for a solution v of (**) in terms of the eigenfunctions of the system (*).

Sums and products of quasi-nilpotent operators

1984 ◽  
Vol 99 (1-2) ◽  
pp. 193-200 ◽  
Author(s):  
C. K. Fong ◽  
A. R. Sourour
Keyword(s):  
Hilbert Space ◽  
Compact Operator ◽  
Necessary Conditions ◽  
Bounded Operator ◽  
Sufficient Conditions ◽  
Nilpotent Operators

SynopsisIt is proved that a bounded operator on Hilbert space is the sum of two quasi-nilpotent operators if and only if it is not a non-zero scalar plus a compact operator. Necessary conditions and sufficient conditions for an operator to be the product of two quasi-nilpotent operators are given.

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