Small Compact Perturbations Research Articles

Compact perturbations of Moore-Penrose invertible operators

Firstly, we study the stability of the Moore-Penrose invertibility of Hilbert space operators, establishing necessary and sufficient conditions for the invariance of the Moore-Penrose invertibility under small perturbations, compact perturbations and small compact perturbations. Secondly, we study the lifting problem of Moore-Penrose invertible elements of the Calkin algebra, showing that essentially Moore-Penrose invertible operators are small compact perturbations of Moore-Penrose invertible operators. Finally, we discuss the compactness of Moore-Penrose spectra, showing that each Hilbert space operator can be perturbed into an operator with nonempty compact Moore-Penrose spectrum by an arbitrarily small compact perturbation.

Complex symmetric operators with closed numerical ranges

A bounded linear operator T:H→H is said to be complex symmetric if there exists a conjugation C on H such that CT⁎C=T. In this paper, we study the numerical ranges of complex symmetric operators. We show that every complex symmetric operator T on H has a small compact perturbation being complex symmetric and having a closed numerical range. We also show that this holds for skew symmetric operators but fails to hold for unitary operators.

Perturbation of the spectra of complex symmetric operators

An operator T on a complex Hilbert space H is called complex symmetric if T has a symmetric matrix representation relative to some orthonormal basis for H. This paper focuses on the perturbation theory for the spectra of complex symmetric operators. We prove that each complex symmetric operator on a complex separable Hilbert space has a small compact perturbation being complex symmetric and having the single-valued extension property. Also it is proved that each complex symmetric operator on a complex separable Hilbert space has a small compact perturbation being complex symmetric and satisfying generalized Weyl?s theorem.

Perturbation of minimum attaining operators

We prove that the minimum attaining property of a bounded linear operator on a Hilbert space H whose minimum modulus lies in the discrete spectrum, is stable under small compact perturbations. We also observe that given a bounded operator with strictly positive essential minimum modulus, the set of compact perturbations which fail to produce a minimum attaining operator is smaller than a nowhere dense set. In fact, it is a porous set in the ideal of all compact operators on H. Further, we try to extend these stability results to perturbations by all bounded linear operators with small norm and obtain subsequent results.

Orthogonality Property $$\mathcal {O}$$ O and Compact Perturbations

A bounded linear operator T acting on a Hilbert space is said to have orthogonality property \(\mathcal {O}\) if the subspaces \(\ker (T-\alpha )\) and \(\ker (T-\beta )\) are orthogonal for all \(\alpha , \beta \in \sigma _p(T)\) with \(\alpha \ne \beta \). In this paper, the authors investigate the compact perturbations of operators with orthogonality property \(\mathcal {O}\). We give a sufficient and necessary condition to determine when an operator T has the following property: for each \(\varepsilon >0\), there exists \(K\in \mathcal {K(H)}\) with \(\Vert K\Vert <\varepsilon \) such that \(T+K\) has orthogonality property \(\mathcal {O}\). Also, we study the stability of orthogonality property \(\mathcal {O}\) under small compact perturbations and analytic functional calculus.

Two classes of operators with irreducibility and the small and compact perturbations of them

This paper gives the concepts of finite dimensional irreducible operators ((FDI) operators) and infinite dimensional irreducible operators ((IDI) operators). Discusses the relationships of (FDI) operators, (IDI) operators and strongly irreducible operators ((SI) operators) and illustrates some properties of the three classes of operators. Some sufficient conditions for the finite-dimensional irreducibility of operators which have the forms of upper triangular operator matrices are given. This paper proves that every operator with a singleton spectrum is a small compact perturbation of an (FDI) operator on separable Banach spaces and shows that every bounded linear operator T can be approximated by operators in (ΣFDI)(X) with respect to the strong-operator topology and every compact operator K can be approximated by operators in (ΣFDI)(X) with respect to the norm topology on a Banach space X with a Schauder basis, where (ΣFDI)(X):= {T ∈ B(X): T = Σi=1k ⊕Ti, Ti ∈ (FDI), k ∈ ℕ}.

Enforced stability of a simple eigenvalue in the continuous spectrum of a waveguide

It is shown in the paper that, under several orthogonality and normalization conditions and a proper choice of accessory parameters, a simple eigenvalue lying between thresholds of the continuous spectrum of the Dirichlet problem in a domain with a cylindrical outlet to infinity is not taken out from the spectrum by a small compact perturbation of the Helmholtz operator. The result is obtained by means of an asymptotic analysis of the augmented scattering matrix.

Existence of the best n-term approximants for structured dictionaries

This paper investigates the existence of elements of the best n-term approximation in infinite dimensional Hilbert spaces. The notion of uniform linear independence (ULI) for a dictionary is introduced. It is shown that if the dictionary used for approximation satisfies the Bessel inequality and has the ULI property, then for every element of the Hilbert space there exists an element of the best n-term approximation. It is also shown that if a dictionary does not satisfy the ULI property, then there exists an arbitrarily small compact perturbation of this dictionary for which the elements of the best n-term approximation need not exist. The obtained results are applied to frames.

WEYL TYPE THEOREMS FOR FUNCTIONS OF OPERATORS

AbstractA-Weyl's theorem and property (ω), as two variations of Weyl's theorem, were introduced by Rakočević. In this paper, we study a-Weyl's theorem and property (ω) for functions of bounded linear operators. A necessary and sufficient condition is given for an operator T to satisfy that f(T) obeys a-Weyl's theorem (property (ω)) for all f ∈ Hol(σ(T)). Also we investigate the small-compact perturbations of operators satisfying a-Weyl's theorem (property (ω)) in the setting of separable Hilbert spaces.

The approximate Jordan forms of operators on Σ e 1 type Banach spaces

This paper studies the structure of operators on Σ e 1 type Banach spaces. It solves the problem of the small compact perturbations of operators with connected spectra. Namely, it shows that every operator with a connected spectrum on separable Σ e 1 type Banach spaces is a small compact perturbation of a strongly irreducible operator. Based on this result, this paper establishes the approximate Jordan forms of operators on Σ e 1 type Banach spaces with Schauder bases.

Some dynamical properties for linear operators

In this article, some dynamical properties for continuous linear operators are studied. We investigate that neither normal operators nor compact operators can be Li–Yorke chaotic. In addition, we show that a small compact perturbation of the unit operator could be distributionally chaotic.

Small compact perturbation of strongly irreducible operators

An operatorT onH is called strongly irreducible ifT is not similar to any reducible operators. In this paper, we shall say ‘yes’ to answer the following question raised by D. A. Herrero.

Preservation of the Inndex of p-Adic Linear Operators Under Compact Perturbations

In this paper it is proved that the index of a Fredholm operator between $p$-adic Banach spaces is preserved under compact perturbations. A case of special interest is provided when the ground field is nonspherically complete. In this case the classical techniques are no longer valid and the relation between the kernels of a Fredholm operator and that of a small compact perturbation turn out to be in general much richer than in the complex context.

Incompatibility of compact perturbations with the Sz. Nagy–Foias functional calculus

For every absolutely continuous contraction T with spectrum on the unit circle, we exhibit an H ∞ {H^\infty } function h and a sequence of operators T n {T_n} which are unitarily equivalent to T and differ from T by a sequence of compact operators converging to 0 in norm such that h ( T n ) h({T_n}) is never a compact perturbation of h ( T ) h(T) . When T is diagonal, it can also be arranged that T − T n T - {T_n} is trace class, and T n {T_n} commutes with T. Pour toute contraction absolument continue T dont le spectre rencontre le cercle unité, il existe une fonction h de H ∞ {H^\infty } et une suite T n {T_n} d’opérateurs unitairement equivalents à T telle que T − T n T - {T_n} soit compact et convergent en norme vers 0, mais h ( T n ) − h ( T ) h({T_n}) - h(T) soit non compact pour tout n. Dans le cas où T est diagonal, la suite T n {T_n} vérifie en plus T − T n T - {T_n} est un opérateur à trace, et T n {T_n} commute avec T.

Approximate equivalence for representations of C∗-algebras and C∗-dynamical systems

Approximate equivalence, unitary equivalence up to arbitrarily small compact perturbations, is studied for representations of C ∗-algebras and covariant representations of C ∗-dynamical systems. Approximate equivalence for representations of C ∗-algebras is shown to be determined by a trace condition, analogous to the trace condition for unitary equivalence of representations of finite groups. For C ∗-dynamical systems (with a compact, abelian group acting) approximate equivalence is shown to be determined by certain associated representations of the fixed point algebra of the action.

On a class of selfadjoint operators in Krein space and their compact perturbations

For a class of selfadjoint operators in a Krein space containing the definitizable selfadjoint operators a funetional calculus and the spectral function are studied. Stability properties of the spectral function with respect to small compact perturbations of the resolvent are proved.

Nest Algebras and Similarity Transformations

In recent years the theory of algebras of operators on Hilbert space has been stimulated by developments in the theory of quasitriangularity. Andersen [1] has shown that up to unitary equivalence there is only one quasitrianguilar algebra. We use this to provide the following answer to a question posed by J. R. Ringrose approximately 20 years ago: Similar continuous nests on separable Hilbert space can fail to be unitarily equivalent (Theorem 2.2). A consequence is the existence of a nonhyperintransitive compact operator (Corollary 2.3), which answers a question of Kadison and Singer [12] and of Gohberg and Krein [11]. We extend our initial theorem to show that arbitrary continuous nests are similar (Theorem 2.10), and that every maximal nest is similar to a multiplicity-one nest (Theorem 2.11). A consequence is that every compact operator is similar to a hyperintransitive compact operator (Corollary 2.12). The similarity transformation can be induced by an arbitrarily small compact perturbation of a unitary operator. The methods of Section 2 apply only to the continuous parts of nests. For general results the atomic core part must be dealt with. In Section 3 different methods are developed for this purpose, again utilizing Andersen's results. These are used in Section 4 to prove that a complete nest X admits an Arveson factorization for every positive invertible operator if and only if X is countable as a family of subspaces (Theorem 4.7). A consequence is that if X is an uncountable nest with atomic core then some similarity transformation of X has a continuous part. This could not be deduced from Section 2. These methods also yield a weak factorization result which concludes the paper. It should be noted that a negative resolution to the Ringrose question was conjectured in recent years by several mathematicians, including W. Arveson and J. Ringrose. Also, many of the results presented in this paper were announced in an A.M.S. Bulletin article [16]. Finally, we wish to thank the referee for suggesting that the original manuscript could be condensed and improved.