# Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48 .

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## The diagonal entries in the formula “quasitriangular $-$ compact $=$ triangular” and restrictions of quasitriangularityHTML articles powered by AMS MathViewer

by Domingo A. Herrero
Trans. Amer. Math. Soc. 298 (1986), 1-42
DOI: https://doi.org/10.1090/S0002-9947-1986-0857432-4

## 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 $\left \| {(1 - {P_n})T{P_n}} \right \| \to 0 (n \to \infty )$. 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 $\left \| {(1 - {P_n})T{P_{n + 1}}} \right \| \to 0 (n \to \infty )$; (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 $\left \| {(1 - {P_n})T{P_{n + 1}}} \right \| \to 0 (n \to \infty )$ is replaced by (1’) $\left \| {(1 - {P_{{n_k}}})T{P_{{n_k} + 1}}} \right \| \to 0 (k \to \infty )$ 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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