## The diagonal entries in the formula “quasitriangular $-$ compact $=$ triangular” and restrictions of quasitriangularity

HTML 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
- PDF | Request permission

## 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.”

## References

- Constantin Apostol,
*The correction by compact perturbation of the singular behavior of operators*, Rev. Roumaine Math. Pures Appl.**21**(1976), no. 2, 155–175. MR**487559**
C. Apostol, L. A. Fialkow, D. A. Herrero and D. Voiculescu, - William Arveson,
*Interpolation problems in nest algebras*, J. Functional Analysis**20**(1975), no. 3, 208–233. MR**0383098**, DOI 10.1016/0022-1236(75)90041-5 - C. A. Berger and B. I. Shaw,
*Intertwining, analytic structure, and the trace norm estimate*, Proceedings of a Conference on Operator Theory (Dalhousie Univ., Halifax, N.S., 1973) Lecture Notes in Math., Vol. 345, Springer, Berlin, 1973, pp. 1–6. MR**0361885** - T. Crimmins and P. Rosenthal,
*On the decomposition of invariant subspaces*, Bull. Amer. Math. Soc.**73**(1967), 97–99. MR**203463**, DOI 10.1090/S0002-9904-1967-11659-8 - Kenneth R. Davidson,
*Similarity and compact perturbations of nest algebras*, J. Reine Angew. Math.**348**(1984), 72–87. MR**733923**, DOI 10.1515/crll.1984.348.72 - Theodore W. Gamelin,
*Uniform algebras*, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1969. MR**0410387** - P. R. Halmos,
*Quasitriangular operators*, Acta Sci. Math. (Szeged)**29**(1968), 283–293. MR**234310** - P. R. Halmos,
*Ten problems in Hilbert space*, Bull. Amer. Math. Soc.**76**(1970), 887–933. MR**270173**, DOI 10.1090/S0002-9904-1970-12502-2 - Domingo A. Herrero,
*On the spectra of the restrictions of an operator*, Trans. Amer. Math. Soc.**233**(1977), 45–58. MR**473870**, DOI 10.1090/S0002-9947-1977-0473870-0
—, - Domingo A. Herrero,
*Economical compact perturbations. I. Erasing normal eigenvalues*, J. Operator Theory**10**(1983), no. 2, 289–306. MR**728910** - Domingo A. Herrero,
*Compact perturbations of continuous nest algebras*, J. London Math. Soc. (2)**27**(1983), no. 2, 339–344. MR**692539**, DOI 10.1112/jlms/s2-27.2.339 - Domingo A. Herrero,
*Compact perturbations of nest algebras, index obstructions, and a problem of Arveson*, J. Funct. Anal.**55**(1984), no. 1, 78–109. MR**733035**, DOI 10.1016/0022-1236(84)90020-X - Domingo A. Herrero,
*On multicyclic operators. II. Two extensions of the notion of quasitriangularity*, Proc. London Math. Soc. (3)**48**(1984), no. 2, 247–282. MR**729070**, DOI 10.1112/plms/s3-48.2.247 - Domingo A. Herrero,
*A trace obstruction to approximation by block-diagonal nilpotents*, Amer. J. Math.**108**(1986), no. 2, 451–484. MR**833364**, DOI 10.2307/2374680 - Domingo Antonio Herrero and Norberto Salinas,
*Analytically invariant and bi-invariant subspaces*, Trans. Amer. Math. Soc.**173**(1972), 117–136. MR**312294**, DOI 10.1090/S0002-9947-1972-0312294-9 - Richard V. Kadison and I. M. Singer,
*Triangular operator algebras. Fundamentals and hyperreducible theory*, Amer. J. Math.**82**(1960), 227–259. MR**121675**, DOI 10.2307/2372733 - Tosio Kato,
*Perturbation theory for linear operators*, Die Grundlehren der mathematischen Wissenschaften, Band 132, Springer-Verlag New York, Inc., New York, 1966. MR**0203473** - S. C. Power,
*The distance to upper triangular operators*, Math. Proc. Cambridge Philos. Soc.**88**(1980), no. 2, 327–329. MR**578277**, DOI 10.1017/S0305004100057637
J. R. Ringrose, - Joseph G. Stampfli,
*Compact perturbations, normal eigenvalues and a problem of Salinas*, J. London Math. Soc. (2)**9**(1974/75), 165–175. MR**365196**, DOI 10.1112/jlms/s2-9.1.165 - Dan Voiculescu,
*A non-commutative Weyl-von Neumann theorem*, Rev. Roumaine Math. Pures Appl.**21**(1976), no. 1, 97–113. MR**415338**
D. A. Herrero and D. R. Larson, - David R. Larson,
*Nest algebras and similarity transformations*, Ann. of Math. (2)**121**(1985), no. 3, 409–427. MR**794368**, DOI 10.2307/1971180

*Approximation of Hilbert space operators*. II, Pitman, Boston, Mass., 1984.

*Approximation of Hilbert space operators*. I. Pitman, Boston, Mass., 1982.

*Compact non-self-adjoint operators*, Van Nostrand-Reinhold, London, 1971.

*Ideals of nest algebras and models for operators*, (in preparation).

## Bibliographic Information

- © Copyright 1986 American Mathematical Society
- 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