Quasidiagonality of direct sums of weighted shifts

Narayan, Sivaram K.

doi:10.1090/S0002-9947-1992-1012511-9

Quasidiagonality of direct sums of weighted shifts
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Trans. Amer. Math. Soc. 332 (1992), 757-774 Request permission

Abstract:

Let $\mathcal {H}$ be a separable Hilbert space. A bounded linear operator $A$ defined on $\mathcal {H}$ is said to be quasidiagonal if there exists a sequence $\{ {P_n}\}$ of projections of finite rank such that ${P_n} \to I$ strongly and $\left \| A{P_n} - {P_n}A\right \| \to 0$ as $n \to \infty$. We give a necessary and sufficient condition for a finite direct sum of weighted shifts to be quasidiagonal. The condition is stated using a marked graph (a graph with a $(0)$, $( + )$ or $( - )$ attached to its vertices) that can be associated with the direct sum.

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