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 is called *quasitriangular* if there exists an increasing sequence of finite-rank orthogonal projections, converging strongly to 1, such that . 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 is a quasitriangular operator and is a sequence of complex numbers. Find necessary and sufficient conditions for the existence of a compact operator (of arbitrarily small norm) so that is triangular with respect to some orthonormal basis, and the sequence of diagonal entries of coincides with . For instance, if no restrictions are put on the norm of , then and must be related as follows: (a) if is a limit point of and is semi-Fredholm, then ; and (b) if is an open set intersecting the Weyl spectrum of , whose boundary does not intersect this set, then is a denumerable set of indices.

Particularly important is the case when . The following are equivalent for an operator : (1) there is an integral sequence of orthogonal projections, with rank for all , converging strongly to 1, such that ; (2) from some compact is triangular, with diagonal entries equal to 0; (3) is quasitriangular, and the Weyl spectrum of is connected and contains the origin. The family of all operators satisfying (1) (and hence (2) and (3)) is a (norm) closed subset of the algebra of all operators; moreover, 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 , introduced by the author in a previous article.

If is as in (1), but condition is replaced by (1') for some subsequence , then (1') is equivalent to (3'), is quasitriangular, and its Weyl spectrum contains the origin. The family 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 ``-versions'' ( and, respectively, , ) with similar properties. ( is the class naturally associated with triangular operators such that the main diagonal and the first 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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DOI:
https://doi.org/10.1090/S0002-9947-1986-0857432-4

Keywords:
Quasitriangular operator,
compact perturbation,
triangular operator,
sequence of diagonal entries,
-quasitriangular operator,
strictly -quasitriangular operator,
spectral characterization,
Weyl spectrum,
nest algebra generated by an orthonormal basis

Article copyright:
© Copyright 1986
American Mathematical Society