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A Banach space operator $Tin B(X)$ is left polaroid if for each $lambdainhbox{iso}sigma_a(T)$ there is an integer $d(lambda)$ such that asc $(T-lambda)=d(lambda)<infty$ and $(T-lambda)^{d(lambda)+1}X$ is closed; $T$ is finitely left polaroid if asc $(T-lambda)<infty$, $(T-lambda)X$ is closed and $dim(T-lambda)^{-1}(0)<infty$ at each $lambdainhbox{iso }sigma_a(T)$. The left polaroid property transfers from $A$ and $B$ to their tensor product $Aotimes B$, hence also from $A$ and $B^*$ to the left-right multiplication operator $tau_{AB}$, 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 $Aotimes B$ if and only if $0 otinhbox{iso}sigma_a(Aotimes B)$; a similar result holds for $tau_{AB}$ for finitely left polaroid $A$ and $B^*$.
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Given Banach spaces $X$ and $Y$ and operators $Ain B(X)$ and $Bin B(Y)$, property $(gw)$ does not in general transfer from $A$ and $B$ to the tensor product operator $Aotimes Bin B(Xoverline{otimes} Y)$ or to the elementary operator defined by $A$ and $B$, $tau_{AB}=L_AR_Bin B(B(Y,X))$. In this article necessary and sufficient conditions ensuring that property $(gw)$ transfers from $A$ and $B$ to $Aotimes B$ and to $tau_{AB}$ will be given.
A Banach space operator $Ain B({cal{X}})$ is polaroid, $Ain {cal{P}}$, if the isolated points of the spectrum $sigma(A)$ are poles of the operator; $A$ is hereditarily polaroid, $Ain{cal{HP}}$, if every restriction of $A$ to a closed invariant subspace is polaroid. Operators $Ain{cal{HP}}$ have SVEP on $Phi_{sf}(A)={lambda: A-lambda$ 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 $Phi_{sf}^+(A)=emptyset$. A sufficient condition for $Ain B({cal{X}})$ to have SVEP on $Phi_{sf}(A)$ is that its component $Omega_a(A)={lambdainPhi_{sf}(A): rm{ind}(A-lambda)leq 0}$ is connected. We prove: If $Ain B({cal{H}})$ is a Hilbert space operator, then a necessary and sufficient condition for there to exist a compact operator $K$ such that $A+Kin{cal{HP}}$ is that $Omega_a(A)$ is connected.
The transfer property for the generalized Browders 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.
Let $sigma(A)$, $rho(A)$ and $r(A)$ denote the spectrum, spectral radius and numerical radius of a bounded linear operator $A$ on a Hilbert space $H$, respectively. We show that a linear operator $A$ satisfying $$rho(AB)le r(A)r(B) quadtext{ for all bounded linear operators } B$$ if and only if there is a unique $mu in sigma (A)$ satisfying $|mu| = rho(A)$ and $A = frac{mu(I + L)}{2}$ for a contraction $L$ with $1insigma(L)$. One can get the same conclusion on $A$ if $rho(AB) le r(A)r(B)$ for all rank one operators $B$. If $H$ is of finite dimension, we can further decompose $L$ as a direct sum of $C oplus 0$ under a suitable choice of orthonormal basis so that $Re(C^{-1}x,x) ge 1$ for all unit vector $x$.
It is known that the spatial product of two product systems is intrinsic. Here we extend this result by analyzing subsystems of the tensor product of product systems. A relation with cluster systems is established. In a special case, we show that the amalgamated product of product systems through strictly contractive units is independent of the choices of the units. The amalgamated product in this case is isomorphic to the tensor product of the spatial product of the two and the type I product system of index one.
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