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

MathSciNet review:
857432

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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.''

**[1]**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**0487559****[2]**C. Apostol, L. A. Fialkow, D. A. Herrero and D. Voiculescu,*Approximation of Hilbert space operators*. II, Pitman, Boston, Mass., 1984.**[3]**William Arveson,*Interpolation problems in nest algebras*, J. Functional Analysis**20**(1975), no. 3, 208–233. MR**0383098****[4]**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) Springer, Berlin, 1973, pp. 1–6, Lecture Notes in Math., Vol. 345. MR**0361885****[5]**T. Crimmins and P. Rosenthal,*On the decomposition of invariant subspaces*, Bull. Amer. Math. Soc.**73**(1967), 97–99. MR**0203463**, 10.1090/S0002-9904-1967-11659-8**[6]**Kenneth R. Davidson,*Similarity and compact perturbations of nest algebras*, J. Reine Angew. Math.**348**(1984), 72–87. MR**733923**, 10.1515/crll.1984.348.72**[7]**Theodore W. Gamelin,*Uniform algebras*, Prentice-Hall, Inc., Englewood Cliffs, N. J., 1969. MR**0410387****[8]**P. R. Halmos,*Quasitriangular operators*, Acta Sci. Math. (Szeged)**29**(1968), 283–293. MR**0234310****[9]**P. R. Halmos,*Ten problems in Hilbert space*, Bull. Amer. Math. Soc.**76**(1970), 887–933. MR**0270173**, 10.1090/S0002-9904-1970-12502-2**[10]**Domingo A. Herrero,*On the spectra of the restrictions of an operator*, Trans. Amer. Math. Soc.**233**(1977), 45–58. MR**0473870**, 10.1090/S0002-9947-1977-0473870-0**[11]**-,*Approximation of Hilbert space operators*. I. Pitman, Boston, Mass., 1982.**[12]**Domingo A. Herrero,*Economical compact perturbations. I. Erasing normal eigenvalues*, J. Operator Theory**10**(1983), no. 2, 289–306. MR**728910****[13]**Domingo A. Herrero,*Compact perturbations of continuous nest algebras*, J. London Math. Soc. (2)**27**(1983), no. 2, 339–344. MR**692539**, 10.1112/jlms/s2-27.2.339**[14]**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**, 10.1016/0022-1236(84)90020-X**[15]**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**, 10.1112/plms/s3-48.2.247**[16]**Domingo A. Herrero,*A trace obstruction to approximation by block-diagonal nilpotents*, Amer. J. Math.**108**(1986), no. 2, 451–484. MR**833364**, 10.2307/2374680**[17]**Domingo Antonio Herrero and Norberto Salinas,*Analytically invariant and bi-invariant subspaces*, Trans. Amer. Math. Soc.**173**(1972), 117–136. MR**0312294**, 10.1090/S0002-9947-1972-0312294-9**[18]**Richard V. Kadison and I. M. Singer,*Triangular operator algebras. Fundamentals and hyperreducible theory.*, Amer. J. Math.**82**(1960), 227–259. MR**0121675****[19]**Tosio Kato,*Perturbation theory for linear operators*, Die Grundlehren der mathematischen Wissenschaften, Band 132, Springer-Verlag New York, Inc., New York, 1966. MR**0203473****[20]**S. C. Power,*The distance to upper triangular operators*, Math. Proc. Cambridge Philos. Soc.**88**(1980), no. 2, 327–329. MR**578277**, 10.1017/S0305004100057637**[21]**J. R. Ringrose,*Compact non-self-adjoint operators*, Van Nostrand-Reinhold, London, 1971.**[22]**Joseph G. Stampfli,*Compact perturbations, normal eigenvalues and a problem of Salinas*, J. London Math. Soc. (2)**9**(1974/75), 165–175. MR**0365196****[23]**Dan Voiculescu,*A non-commutative Weyl-von Neumann theorem*, Rev. Roumaine Math. Pures Appl.**21**(1976), no. 1, 97–113. MR**0415338****[24]**D. A. Herrero and D. R. Larson,*Ideals of nest algebras and models for operators*, (in preparation).**[25]**David R. Larson,*Nest algebras and similarity transformations*, Ann. of Math. (2)**121**(1985), no. 3, 409–427. MR**794368**, 10.2307/1971180

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC:
47A55,
47A66

Retrieve articles in all journals with MSC: 47A55, 47A66

Additional Information

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