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B. P. Duggal

Publications and source records attributed to B. P. Duggal.

25 records · Page 2Linked to original sources
arXiv2018

Isolated eigenvalues, poles and compact perturbations of Banach space operators

Given a Banach space operator $A$, the isolated eigenvalues $E(A)$ and the poles $Π(A)$ (resp., eigenvalues $E^a(A)$ which are isolated points of the approximate point spectrum and the left ploles $Π^a(A)$) of the spectrum of $A$ satisfy $Π(A)\subseteq E(A)$ (resp., $Π^a(A)\subseteq E^a(A)$), and the reverse inclusion holds if and only if $E(A)$ (resp., $E^a(A)$) has empty intersection with the B-Weyl spectrum (resp., upper B-Weyl spectrum) of $A$. Evidently $Π(A)\subseteq E^a(A)$, but no such inclusion exists for $E(A)$ and $Π^a(A)$. The study of identities $E(A)=Π^a(A)$ and $E^a(A)=Π(A)$, and their stability under perturbation by commuting Riesz operators, has been of some interest in the recent past. This paper studies the stability of these identities under perturbation by (non-commuting) compact operators. Examples of analytic Toeplitz operators and operators satisfying the abstract shift condition are considered.

math.FA↗
arXiv2015

Compact perturbations and consequent hereditarily polaroid operators

A Banach space operator $A\in B({\cal{X}})$ is polaroid, $A\in {\cal{P}}$, if the isolated points of the spectrum $σ(A)$ are poles of the operator; $A$ is hereditarily polaroid, $A\in{\cal{HP}}$, if every restriction of $A$ to a closed invariant subspace is polaroid. Operators $A\in{\cal{HP}}$ have SVEP on $Φ_{sf}(A)=\{λ: A-λ$ is semi Fredholm $\}$: This, in answer to a question posed by Li and Zhou (Studia Math. 221(2014), 175-192), proves the necessity of the condition $Φ_{sf}^+(A)=\emptyset$. A sufficient condition for $A\in B({\cal{X}})$ to have SVEP on $Φ_{sf}(A)$ is that its component $Ω_a(A)=\{λ\inΦ_{sf}(A): \rm{ind}(A-λ)\leq 0\}$ is connected. We prove: If $A\in B({\cal{H}})$ is a Hilbert space operator, then a necessary and sufficient condition for there to exist a compact operator $K$ such that $A+K\in{\cal{HP}}$ is that $Ω_a(A)$ is connected.

math.FA↗
arXiv2014

Tensor product of left polaroid operators

A Banach space operator $T\in B(X)$ is left polaroid if for each $λ\in\hbox{iso}σ_a(T)$ there is an integer $d(λ)$ such that asc $(T-λ)=d(λ)<\infty$ and $(T-λ)^{d(λ)+1}X$ is closed; $T$ is finitely left polaroid if asc $(T-λ)<\infty$, $(T-λ)X$ is closed and $\dim(T-λ)^{-1}(0)<\infty$ at each $λ\in\hbox{iso }σ_a(T)$. The left polaroid property transfers from $A$ and $B$ to their tensor product $A\otimes B$, hence also from $A$ and $B^*$ to the left-right multiplication operator $τ_{AB}$, for Hilbert space operators; an additional condition is required for Banach space operators. The finitely left polaroid property transfers from $A$ and $B$ to their tensor product $A\otimes B$ if and only if $0\not\in\hbox{iso}σ_a(A\otimes B)$; a similar result holds for $τ_{AB}$ for finitely left polaroid $A$ and $B^*$.

math.FA↗
arXiv2014

Property (gw) for tensor product and left-right multiplication operators

Given Banach spaces $X$ and $Y$ and operators $A\in B(X)$ and $B\in B(Y)$, property $(gw)$ does not in general transfer from $A$ and $B$ to the tensor product operator $A\otimes B\in B(X\overline{\otimes} Y)$ or to the elementary operator defined by $A$ and $B$, $τ_{AB}=L_AR_B\in B(B(Y,X))$. In this article necessary and sufficient conditions ensuring that property $(gw)$ transfers from $A$ and $B$ to $A\otimes B$ and to $τ_{AB}$ will be given.

math.FA↗
arXiv2013

Generalized Browder's theorem for tensor product and elementary operators

The transfer property for the generalized Browder's theorem both of the tensor product and of the left-right multiplication operator will be characterized in terms of the $B$-Weyl spectrum inclusion. In addition, the isolated points of these two classes of operators will be fully characterized.

math.FA↗
arXiv2008

Upper Triangular Operator Matrices, SVEP and Browder. Weyl Theorems

A Banach space operator $T\in B({\cal X})$ is polaroid if points $λ\in\isoσσ(T)$ are poles of the resolvent of $T$. Let $σ_a(T)$, $σ_w(T)$, $σ_{aw}(T)$, $σ_{SF_+}(T)$ and $σ_{SF_-}(T)$ denote, respectively, the approximate point, the Weyl, the Weyl essential approximate, the upper semi--Fredholm and lower semi--Fredholm spectrum of $T$. For $A$, $B$ and $C\in B({\cal X})$, let $M_C$ denote the operator matrix $(A & C 0 & B)$. If $A$ is polaroid on $π_0(M_C)=\{λ\in\isoσ(M_C) 0<\dim(M_C-λ)^{-1}(0)<\infty\}$, $M_0$ satisfies Weyl's theorem, and $A$ and $B$ satisfy either of the hypotheses (i) $A$ has SVEP at points $λ\inσ_w(M_0)\setminusσ_{SF_+}(A)$ and $B$ has SVEP at points $μ\inσ_w(M_0)\setminusσ_{SF_-}(B)$, or, (ii) both $A$ and $A^*$ have SVEP at points $λ\inσ_w(M_0)\setminusσ_{SF_+}(A)$, or, (iii) $A^*$ has SVEP at points $λ\inσ_w(M_0)\setminusσ_{SF_+}(A)$ and $B^*$ has SVEP at points $μ\inσ_w(M_0)\setminusσ_{SF_-}(B)$, then $σ(M_C)\setminusσ_w(M_C)=π_0(M_C)$. Here the hypothesis that $λ\inπ_0(M_C)$ are poles of the resolvent of $A$ can not be replaced by the hypothesis $λ\inπ_0(A)$ are poles of the resolvent of $A$. For an operator $T\in B(\X)$, let $π_0^a(T)=\{λ:λ\in\isoσ_a(T), 0<\dim(T-λ)^{-1}(0)<\infty\}$. We prove that if $A^*$ and $B^*$ have SVEP, $A$ is polaroid on $π_0^a(\M)$ and $B$ is polaroid on $π_0^a(B)$, then $σ_a(\M)\setminusσ_{aw}(\M)=π_0^a(\M)$.

math.FA↗