On quasinilpotent operators

Authors:
Il Bong Jung, Eungil Ko and Carl Pearcy

Journal:
Proc. Amer. Math. Soc. **131** (2003), 2121-2127

MSC (2000):
Primary 47A15

Published electronically:
February 5, 2003

MathSciNet review:
1963758

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Abstract | References | Similar Articles | Additional Information

Abstract: In this note we modify a new technique of Enflo for producing hyperinvariant subspaces to obtain a much improved version of his ``two sequences'' theorem with a somewhat simpler proof. As a corollary we get a proof of the ``best'' theorem (due to V. Lomonosov) known about hyperinvariant subspaces for quasinilpotent operators that uses neither the Schauder-Tychonoff fixed point theorem nor the more recent techniques of Lomonosov.

**[1]**William B. Arveson and Jacob Feldman,*A note on invariant subspaces*, Michigan Math. J.**15**(1968), 61–64. MR**0223922****[2]**Shamim Ansari and Per Enflo,*Extremal vectors and invariant subspaces*, Trans. Amer. Math. Soc.**350**(1998), no. 2, 539–558. MR**1407476**, 10.1090/S0002-9947-98-01865-0**[3]**N. Aronszajn and K. T. Smith,*Invariant subspaces of completely continuous operators*, Ann. of Math. (2)**60**(1954), 345–350. MR**0065807****[4]**Allen R. Bernstein and Abraham Robinson,*Solution of an invariant subspace problem of K. T. Smith and P. R. Halmos*, Pacific J. Math.**16**(1966), 421–431. MR**0193504****[5]**Scott W. Brown,*Hyponormal operators with thick spectra have invariant subspaces*, Ann. of Math. (2)**125**(1987), no. 1, 93–103. MR**873378**, 10.2307/1971289**[6]**Bernard Chevreau, Wing Suet Li, and Carl Pearcy,*A new Lomonosov lemma*, J. Operator Theory**40**(1998), no. 2, 409–417. MR**1660394****[7]**Don Deckard, R. G. Douglas, and Carl Pearcy,*On invariant subspaces of quasitriangular operators*, Amer. J. Math.**91**(1969), 637–647. MR**0256202****[8]**P. Enflo and V. Lomonosov,*Some aspects of the invariant subspace problem*, preprint.**[9]**P. R. Halmos,*Invariant subspaces of polynomially compact operators*, Pacific J. Math.**16**(1966), 433–437. MR**0193505****[10]**P. R. Halmos,*Quasitriangular operators*, Acta Sci. Math. (Szeged)**29**(1968), 283–293. MR**0234310****[11]**V. I. Lomonosov,*Invariant subspaces of the family of operators that commute with a completely continuous operator*, Funkcional. Anal. i Priložen.**7**(1973), no. 3, 55–56 (Russian). MR**0420305****[12]**V. Lomonosov,*An extension of Burnside’s theorem to infinite-dimensional spaces*, Israel J. Math.**75**(1991), no. 2-3, 329–339. MR**1164597**, 10.1007/BF02776031**[13]**Victor I. Lomonosov,*On real invariant subspaces of bounded operators with compact imaginary part*, Proc. Amer. Math. Soc.**115**(1992), no. 3, 775–777. MR**1086334**, 10.1090/S0002-9939-1992-1086334-4**[14]**Carl Pearcy and Norberto Salinas,*An invariant-subspace theorem*, Michigan Math. J.**20**(1973), 21–31. MR**0317075****[15]**Carl Pearcy and Allen L. Shields,*A survey of the Lomonosov technique in the theory of invariant subspaces*, Topics in operator theory, Amer. Math. Soc., Providence, R.I., 1974, pp. 219–229. Math. Surveys, No. 13. MR**0355639****[16]**Aleksander Simonič,*An extension of Lomonosov’s techniques to non-compact operators*, Trans. Amer. Math. Soc.**348**(1996), no. 3, 975–995. MR**1348869**, 10.1090/S0002-9947-96-01612-1

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Additional Information

**Il Bong Jung**

Affiliation:
Department of Mathematics, Kyungpook National University, Taegu 702-701, Korea

Email:
ibjung@kyungpook.ac.kr

**Eungil Ko**

Affiliation:
Department of Mathematics, Ewha Women’s University, Seoul 120-750, Korea

Email:
eiko@mm.ewha.ac.kr

**Carl Pearcy**

Affiliation:
Department of Mathematics, Texas A&M University, College Station, Texas 77843

Email:
pearcy@math.tamu.edu

DOI:
https://doi.org/10.1090/S0002-9939-03-06895-3

Keywords:
Invariant subspaces,
Enflo technique,
extremal vectors

Received by editor(s):
February 6, 2002

Published electronically:
February 5, 2003

Communicated by:
David R. Larson

Article copyright:
© Copyright 2003
American Mathematical Society