A perturbed elementary operator and range-kernel orthogonality

Duggal, B. P.

doi:10.1090/s0002-9939-05-08337-1

Proc. Amer. Math. Soc.

2005

DOI: 10.1090/s0002-9939-05-08337-1

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A perturbed elementary operator and range-kernel orthogonality

B. P. Duggal

Abstract: Abstract. Let B(H) denote the algebra of operators on a Hilbert H. If A j and B j ∈ B(H) are commuting normal operators, and C j and D j ∈ B(H) are commuting quasi-nilpotents such thatwhere k ≥ 1 is some scalar and H 0 (E) is the quasi-nilpotent part of the operator E.

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Cited by 3 publications

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Size of the peripheral point spectrum under power or resolvent growth conditions

Badea¹,

Grivaux²

2007

Journal of Functional Analysis

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We characterize Jamison sequences, that is sequences (n k ) of positive integers with the following property: every bounded linear operator T acting on a separable Banach space with sup k T n k < +∞ has a countable set of peripheral eigenvalues. We also discuss partially power-bounded operators acting on Banach or Hilbert spaces having peripheral point spectra with large Hausdorff dimension. For a Lavrentiev domain Ω in the complex plane, we show the uniform minimality of some families of eigenvectors associated with peripheral eigenvalues of operators satisfying the Kreiss resolvent condition with respect to Ω. We introduce and study the notion of Ω-Jamison sequence, which is defined by replacing the partial power-boundedness condition sup k T n k < +∞ by sup k F Ω n k (T ) < +∞, where F Ω n is the nth Faber polynomial of Ω. A characterization of Ω-Jamison sequences is obtained for domains with sufficiently smooth boundary.

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Size of the peripheral point spectrum under power or resolvent growth conditions

Badea¹,

Grivaux²

2007

Journal of Functional Analysis

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Remarks on the range and the kernel of generalized derivation

et al. 2022

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Let $L(H)$ denote the algebra of operators on a complexinfinite dimensional Hilbert space $H$ and let $\;\mathcal{J}$denote a two-sided ideal in $L(H)$. Given $A,B\in L(H)$, definethe generalized derivation $\delta_{A,B}$ as an operator on$L(H)$ by \centerline{$\delta_{A,B}(X)=AX-XB.$} \smallskip\noi We say that the pair ofoperators $(A,B)$ has the Fuglede-Putnam property$(PF)_{\mathcal{J}}$ if $AT=TB$ and $T\in \mathcal{J}$ implies$A^{\ast}T=TB^{\ast}$. In this paper, we give operators $A,B$ forwhich the pair $(A,B)$ has the property $(PF)_{\mathcal{J}}$. Weestablish the orthogonality of the range and the kernel of ageneralized derivation $\delta_{A,B}$ for non-normal operators $A,B\in L(H)$. We also obtain new results concerning the intersectionof the closure of the range and the kernel of $\delta_{A,B}$.

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On Self-commutator Approximants

Duggal¹

2009

Kyungpook mathematical journal

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Let B(X) denote the algebra of operators on a complex Banach space X , H(X) = {h ∈ B(X) : h is hermitian}, and J(X) = {x ∈ B(X) : x = x1 + ix2, x1 and x2 ∈ H(X)}. Let δa ∈ B(B(X)) denote the derivation δa(x) = ax − xa. If J(X) is an algebra and δa −1 (0) ⊆ δ −1 a * (0) for some a ∈ J(X), then ||a|| ≤ ||a−(x * x−xx *)|| for all x ∈ J(X) ∩ δa −1 (0). The cases J(X) = B(H), the algebra of operators on a complex Hilbert space, and J(X) = Cp, the von Neumann-Schatten p-class, are considered.

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A perturbed elementary operator and range-kernel orthogonality

Abstract: Abstract. Let B(H) denote the algebra of operators on a Hilbert H. If A j and B j ∈ B(H) are commuting normal operators, and C j and D j ∈ B(H) are commuting quasi-nilpotents such thatwhere k ≥ 1 is some scalar and H 0 (E) is the quasi-nilpotent part of the operator E.

Cited by 3 publications

References 20 publications

Size of the peripheral point spectrum under power or resolvent growth conditions

Size of the peripheral point spectrum under power or resolvent growth conditions

Remarks on the range and the kernel of generalized derivation

On Self-commutator Approximants

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