Approximate Point Spectrum Research Articles

Jacobson’s Lemma in the ring of quaternionic linear operators

Abstract In the present paper, we study the Jacobson’s Lemma in the unital ring of all bounded right linear operators ℬ R (X) acting on a two-sided quaternionic Banach space X. In particular, let A, B ∈ ℬ R (X) and let q ∈ ℍ \ {0}, we prove that w(AB) \ {0} = w(BA) \ {0} where w belongs to the spherical spectrum, the spherical approximate point spectrum, the right spherical spectrum, the left spherical spectrum, the spherical point spectrum, the spherical residual spectrum and the spherical continuous spectrum. We also prove that the range of (AB)2 − 2Re(q)AB + |q|2 I is closed if and only if (BA)2 − 2Re(q)BA + |q|2 I has closed range. Finally, we show that (AB)2 − 2Re(q)AB + |q|2 I is Drazin invertible if and only if (BA)2 − 2Re(q)BA + |q|2 I is.

On the spectral mapping theorem for Weyl spectra of Toeplitz operators

In this paper we study the approximate point spectrum and the surjectivity spectrum of a Toeplitz operators $$T_\phi$$ , defined on Hardy spaces $$H^2(\mathbf{T })$$ of the unit circle $$\mathbf{T }$$ of $${\mathbb {C}}$$ , where the symbol $$\phi \in L^\infty (\mathbf{T })$$ , or $$\phi$$ is a continuous symbol. We also extend some results obtained by Farenick and Lee (Trans Am Math Soc 348(10):4153–4174, 1996). In particular, we study Weyl type theorems and the spectral mapping theorem for the Weyl spectra.

Berberian Extension and its S-spectra in a Quaternionic Hilbert Space

For a bounded right linear operators A, in a right quaternionic Hilbert space $$V_\mathbb {H}^R$$, following the complex formalism, we study the Berberian extension $$A^\circ $$, which is an extension of A in a right quaternionic Hilbert space obtained from $$V_\mathbb {H}^R$$. In the complex setting, the important feature of the Berberian extension is that it converts approximate point spectrum of A into point spectrum of $$A^\circ $$. We show that the same is true for the quaternionic S-spectrum. As in the complex case, we use the Berberian extension to study some properties of the commutator of two quaternionic bounded right linear operators.

Partition of the spectra for the lower triangular double band matrix as generalized difference operator Δv over the sequence spaces c and lp (1 < p < ∞)

Let the sequence (vk) is assumed to be either constant or strictly decreasing sequence of positive real numbers satisfying limk?? vk = L > 0 and supk vk ? 2L. Then the generalized difference operator ?v is ?v x = ?v (xn) = (vnxn ? vn?1xn?1)? n=0 with x?1 = v?1 = 0. The aim of this paper is to obtain the approximate point spectrum, the defect spectrum and the compression spectrum of the operator ?v and modified of the operator ?v on the sequence spaces c and ????p (1 < p < ?).

Spectrum and fine spectrum of generalized lower triangular triple band matrices over the sequence space lp

ABSTRACT The spectrum of triangular band matrices defined on the sequence spaces where the entries of each band are a constant or convergent sequence is well studied. In this article, the spectrum and fine spectrum of a new generalized difference operator defined by a lower triangular triple band matrix on the sequence space are obtained where the bands are considered as periodic sequences. The approximate point spectrum, defect spectrum, compression spectrum and the Goldberg classification of the spectrum are also discussed. Suitable examples are given in order to supplement the results. Several special cases of our findings are discussed which confirm that our study is more general and extensive.

Subdivision of Spectra for Some Lower Triangular Double-Band Matrices as Operators on c0

The generalized difference operator ∆a,b was defined by El-Shabrawy: $$ {\varDelta}_{a,b}x={\varDelta}_{a,b}\left({x}_n\right)={\left({a}_n{x}_n+{b}_{n-1}\right)}_{n=0}^{\infty}\;\mathrm{with}\;{x}_{-1}={b}_{-1}=0, $$ where (ak) and (bk) are convergent sequences of nonzero real numbers satisfying certain conditions. We completely determine the approximate point spectrum, the defect spectrum, and the compression spectrum of the operator ∆a,b in a sequence space c0.

On new strong versions of Browder type theorems

Abstract An operator T acting on a Banach space X satisfies the property (UWΠ ) if σa (T)∖ σ S F + − $\begin{array}{} \sigma_{SF_{+}^{-}} \end{array} $ (T) = Π(T), where σa (T) is the approximate point spectrum of T, σ S F + − $\begin{array}{} \sigma_{SF_{+}^{-}} \end{array} $ (T) is the upper semi-Weyl spectrum of T and Π(T) the set of all poles of T. In this paper we introduce and study two new spectral properties, namely (VΠ ) and (VΠa ), in connection with Browder type theorems introduced in [1], [2], [3] and [4]. Among other results, we have that T satisfies property (VΠ ) if and only if T satisfies property (UWΠ ) and σ(T) = σa (T).

Genelleştirilmiş Fark Operatörü B(r,s) nin cs Dizi Uzayı Üzerinde Spektral Ayrışımı

The goal of this study gives the approximate point spectrum, the defect spectrum and the compression spectrum of generalized difference operator B (r, s) over the class of convergent series.

Generalized Drazin-type spectra of Operator matrices

In this paper, we investigate the limit points set of surjective and approximate point spectra of upper triangular operator matrices . We prove that σ*(MC) ∪ W=σ*(A)∪ σ*(B) where W is the union of certain holes in σ*(MC), which happen to be subsets of σ lgD(B) ∩ σrgD(A), σ* ∈ {σlgD, σrgD} are the limit points set of surjective and approximate point spectra. Furthermore, several sufficient conditions for σ* (MC) = σ* (A)∪σ* (B) holds for every C ∈ ℬ(Y,X) are given.

Spectral inclusion for unbounded diagonally dominant $n\times n$ operator matrices

In this paper, we establish an analytic enclosure for the spectrum of unbounded linear operators~$\mathcal{A} $ admitting an $n \times n$ matrix representation in a Hilbert space $\mathcal{H} =\mathcal{H} _1\oplus \cdots \oplus \mathcal{H} _n$. For diagonally dominant operator matrices of order 0, we show that this new enclosing set, the block numerical range $W^n(\mathcal{A} )$, contains the eigenvalues of $\mathcal{A} $ and that the approximate point spectrum of $\mathcal{A} $ is contained in its closure $\overline {W^n(\mathcal{A} )}$. Since the block numerical range turns out to be a subset of the usual numerical range, $W^n(\mathcal{A} )\subset W(\mathcal{A} )$, it may give a tighter enclosure of the spectrum. Moreover, we prove Gershgorin theorems for diagonally dominant $n \times n$ operator matrices and compare our results to both Gershgorin bounds and classical perturbation theory. Our results are illustrated by deriving new lower bounds for $3\times 3$ self-adjoint operator matrices and applying the latter to three-channel Hamiltonians in quantum~mechanics.

Lower spectral radius and spectral mapping theorem for suprema preserving mappings

We study Lipschitz, positively homogeneous and finite suprema preserving mappings defined on a max-cone of positive elements in a normed vector lattice. We prove that the lower spectral radius of such a mapping is always a minimum value of its approximate point spectrum. We apply this result to show that the spectral mapping theorem holds for the approximate point spectrum of such a mapping. By applying this spectral mapping theorem we obtain new inequalites for the Bonsall cone spectral radius of max-type kernel operators.

On properties of the operator equation TT*=T+T*

In this paper, we study properties of the operator equation TT*=T+T* which T.T. West observed in [12]. We first investigate the structure of solutions T 2 B(H) of such equation. Moreover, we prove that if T is a polynomial root of solutions of that operator equation, then the spectral mapping theorem holds for Weyl and essential approximate point spectra of T and f(T) satisfies a-Weyl?s theorem for f?H(?(T)), where H(?(T)) is the space of functions analytic in an open neighborhood of ?(T).

Circles in the spectrum and the geometry of orbits: A numerical ranges approach

We prove that a bounded linear Hilbert space operator has the unit circle in its essential approximate point spectrum if and only if it admits an orbit satisfying certain orthogonality and almost-orthogonality relations. This result is obtained via the study of numerical ranges of operator tuples where several new results are also obtained. As consequences of our numerical ranges approach, we derive in particular wide generalizations of Arveson's theorem as well as show that the weak convergence of operator powers implies the uniform convergence of their compressions on an infinite-dimensional subspace.

Spectral Properties of \(k\)-Quasi \(^*\)Parahyponormal Operators

In this paper, we prove some basic properties of \(k\)-quasi-\(^*\)parahyponormal operators and spectrum of class of \(k\)-quasi-\(^*\)parahyponormal operators is continuous. Also, we proved the non zero points of its approximate point spectrum and joint approximate point spectrum are identical.

Tensor product and property (b)

A Banach space operator satisfies property (b) if the complement of its essential Weyl approximate point spectrum in its approximate point spectrum is the set of all poles of the resolvent of finite rank. Property (b) does not transfer from operators A and B to their tensor product $$A\otimes B;$$ we give necessary and/or sufficient conditions ensuring the passage of property (b) from A and B to $$A\otimes B.$$ Perturbations by Riesz operators are considered.

Subdivision of the Spectra for the Generalized Difference Operator Δ_(a,b) on the Sequence Space l_p " " (1&lt;p&lt;∞)

El-Shabrawy has introduced the generalized difference operator denoted by . Let nonzero real numbers sequences and be convergent sequences such that and The generalized difference operator is with . In this paper we study the approximate point spectrum, the defect spectrum and the compression spectrum of the operator on the p-absolutely summable sequences space .

Spectrum and fine spectrum of the Zweier matrix over the sequence space cs

In this article we have determined the spectrum and fine spectrum of the Zweier matrix Z_s on the sequence space cs. In a further development, we have also determined the approximate point spectrum, the defect spectrum and the compression spectrum of the operator Z_s on the sequence space cs.

Perturbation of semi-Browder linear relations by commuting Riesz operators

This paper is devoted to the investigation of the stability of upper semi-Browder, lower semi-Browder and Browder linear relations which satisfy the stabilization property, under commuting Riesz operator perturbations. As applications, we infer the invariance of the corresponding Browder’s essential approximate point spectrum, Browder’s essential defect spectrum and Browder’s essential spectrum under commuting Riesz operator perturbations.

Hypercyclicity, supercyclicity and generalized rakočević's property

A Banach space operator satisfies generalized Rakocevic's property (gw) if the complement of its upper semi B-Weyl spectrum in its approximate point spectrum is the set of eigenvalues of T which are isolated in the spectrum of T. In this note, we characterize hypecyclic and supercyclic operators satisfying the property (gw).

On the Bonsall cone spectral radius and the approximate point spectrum

We study the Bonsall cone spectral radius and the approximate point spectrum of (in general non-linear) positively homogeneous, bounded and supremum preserving maps, defined on a max-cone in a given normed vector lattice. We prove that the Bonsall cone spectral radius of such maps is always included in its approximate point spectrum. Moreover, the approximate point spectrum always contains a (possibly trivial) interval. Our results apply to a large class of (nonlinear) max-type operators. We also generalize a known result that the spectral radius of a positive (linear) operator on a Banach lattice is contained in the approximate point spectrum. Under additional generalized compactness type assumptions our results imply Krein-Rutman type results.