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The diagonal entries in the formula ``quasitriangular $ -$ compact $ =$ triangular'' and restrictions of quasitriangularity


Author: Domingo A. Herrero
Journal: Trans. Amer. Math. Soc. 298 (1986), 1-42
MSC: Primary 47A55; Secondary 47A66
DOI: https://doi.org/10.1090/S0002-9947-1986-0857432-4
MathSciNet review: 857432
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Abstract: A (bounded linear) Hilbert space operator $ T$ is called quasitriangular if there exists an increasing sequence $ \{ {P_n}\} _{n = 0}^\infty $ of finite-rank orthogonal projections, converging strongly to 1, such that $ \left\Vert {(1 - {P_n})T{P_n}} \right\Vert \to 0\,(n \to \infty )$. This definition, due to P. R. Halmos, plays a very important role in operator theory. The core of this article is a concrete answer to the following problem: Suppose $ T$ is a quasitriangular operator and $ \Gamma = \{ {\lambda _j}\} _{j = 1}^\infty $ is a sequence of complex numbers. Find necessary and sufficient conditions for the existence of a compact operator $ K$ (of arbitrarily small norm) so that $ T - K$ is triangular with respect to some orthonormal basis, and the sequence of diagonal entries of $ T - K$ coincides with $ \Gamma $. For instance, if no restrictions are put on the norm of $ K$, then $ T$ and $ \Gamma $ must be related as follows: (a) if $ {\lambda _0}$ is a limit point of $ \Gamma $ and $ {\lambda _0} - T$ is semi-Fredholm, then $ {\operatorname{ind}}({\lambda _0} - T) > 0$; and (b) if $ \Omega $ is an open set intersecting the Weyl spectrum of $ T$, whose boundary does not intersect this set, then $ \{ j:{\lambda _j} \in \Omega \} $ is a denumerable set of indices.

Particularly important is the case when $ \Gamma = \{ 0,0,0, \ldots \} $. The following are equivalent for an operator $ T$: (1) there is an integral sequence $ \{ {P_n}\} _{n = 0}^\infty $ of orthogonal projections, with rank $ {P_n} = n$ for all $ n$, converging strongly to 1, such that $ \left\Vert {(1 - {P_n})T{P_{n + 1}}} \right\Vert \to 0\,(n \to \infty )$; (2) from some compact $ K,\,T - K$ is triangular, with diagonal entries equal to 0; (3) $ T$ is quasitriangular, and the Weyl spectrum of $ T$ is connected and contains the origin. The family $ {({\text{StrQT}})_{ - 1}}$ of all operators satisfying (1) (and hence (2) and (3)) is a (norm) closed subset of the algebra of all operators; moreover, $ {({\text{StrQT}})_{ - 1}}$ is invariant under similarity and compact perturbations and behaves in many senses as an analog of Halmos's class of quasitriangular operators, or an analog of the class of extended quasitriangular operators $ {({\text{StrQT}})_{ - 1}}$, introduced by the author in a previous article.

If $ \{ {P_n}\} _{n = 0}^\infty $ is as in (1), but condition $ \left\Vert {(1 - {P_n})T{P_{n + 1}}} \right\Vert \to 0\,(n \to \infty )$ is replaced by (1') $ \left\Vert {(1 - {P_{{n_k}}})T{P_{{n_k} + 1}}} \right\Vert \to 0\,(k \to \infty )$ for some subsequence $ \{ {n_k}\} _{k = 1}^\infty $, then (1') is equivalent to (3'), $ T$ is quasitriangular, and its Weyl spectrum contains the origin. The family $ {({\text{QT}})_{ - 1}}$ of all operators satisfying (1') (and hence (3')) is also a closed subset, invariant under similarity and compact perturbations, and provides a different analog to Halmos's class of quasitriangular operators. Both classes have ``$ m$-versions'' ( $ {({\text{StrQT}})_{ - m}}$ and, respectively, $ {({\text{QT}})_{ - m}}$, $ m = 1,2,3, \ldots $) with similar properties. ( $ {({\text{StrQT}})_{ - m}}$ is the class naturally associated with triangular operators $ A$ such that the main diagonal and the first $ (m - 1)$ superdiagonals are identically zero, etc.)

The article also includes some applications of the main result to certain nest algebras ``generated by orthonormal bases.''


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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1986-0857432-4
Keywords: Quasitriangular operator, compact perturbation, triangular operator, sequence of diagonal entries, $ ( \pm m)$-quasitriangular operator, strictly $ ( \pm m)$-quasitriangular operator, spectral characterization, Weyl spectrum, nest algebra generated by an orthonormal basis
Article copyright: © Copyright 1986 American Mathematical Society

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