Publication: Consequences of universality among Toeplitz operators
Loading...
Full text at PDC
Publication Date
2015
Advisors (or tutors)
Editors
Journal Title
Journal ISSN
Volume Title
Publisher
Elsevier
Citations
Exportar
Abstract
The Invariant Subspace Problem for Hilbert spaces is a long-standing question and the use of universal operators in the sense of Rota has been one tool for studying the problem. The best known universal operators have been adjoints of analytic Toeplitz operators or unitarily equivalent to them. We present many examples of Toeplitz operators whose adjoints are universal operators and exhibit some of their common properties. Some ways in which the invariant subspaces of these universal operators interact with operators in their commutants are given. Special attention is given to the closed subalgebra, not always the zero algebra, of compact operators in their commutants. Finally, three questions connecting shift invariant subspaces and invariant subspaces of analytic Toeplitz operators are raised. Positive answers for both of the first two imply the existence of non-trivial invariant subspaces for every bounded operator on separable Hilbert spaces of dimension two or more.
Description
UCM subjects
Unesco subjects
Keywords
Citation
[1]P.R. Ahern, D.N. Clark, Functions orthogonal to invariant subspaces, Acta Math. 124 (1970) 191–204.
[2] S.R. Caradus, Universal operators and invariant subspaces, Proc. Amer. Math. Soc. 23 (1969) 526–527.
[3] I. Chalendar, J.R. Partington, Modern Approaches to the Invariant Subspace Problem, Cambridge University Press,2011.
[4] C.C. Cowen, The commutant of an analytic Toeplitz operator, Trans. Amer. Math. Soc. 239 (1978) 1–31.
[5] C.C. Cowen, The commutant of an analytic Toeplitz operator, II, Indiana Math. J. 29 (1980) 1–12.
[6] C.C. Cowen, An analytic Toeplitz operator that commutes with a compact operator and a related class of Toeplitz operators, J. Funct. Anal. 36 (1980) 169–184.
[7] C.C. Cowen, E.A. Gallardo-Gutiérrez, Unitary equivalence of one-parameter groups of Toeplitz and composition operators,J. Funct. Anal. 261 (2011) 2641–2655.
[8] C.C. Cowen, E.A. Gallardo-Gutiérrez, Rota’s universal operators and invariant subspaces in Hilbert spaces,submitted for publication.
[9] C.C. Cowen, B.D. MacCluer, Composition Operators on Spaces of Analytic Functions, CRC Press, 1995.
[10] R.G. Douglas, Banach Algebra Techniques in Operator Theory, Academic Press, New York, 1972.
[11] R.G. Douglas, H.S. Shapiro, A.L. Shields, Cyclic vectors and invariant subspaces for the backward shift operator, Ann. Inst. Fourier (Grenoble) 20 (1970) 37–76.
[12] E.A. Gallardo-Gutiérrez, P. Gorkin, Minimal invariant subspaces for composition operators, J. Math. Pures Appl. 95 (2011) 245–259.
[13] G. Gunatillake, Compact weighted composition operators on the Hardy space, Proc. Amer. Math. Soc. 136 (2008)2895–2899.
[14] D.A. Herrero, On analytically invariant subspaces and spectra, Trans. Amer. Math. Soc. 233 (1977) 37–44.
[15] D.A. Herrero, M.J. Sherman, Non-cyclic vectors for S∗, Proc. Amer. Math. Soc. 48 (1975) 193–196.
[16] J.E. Littlewood, On inequalities in the theory of functions, Proc. Lond. Math. Soc. 23 (1925) 481–519.
[17] V. Lomonosov, On invariant subspaces of families of operators commuting with a completely continuous operator, Funktsional. Anal. i Prilozhen. 7 (1973) 55–56 (in Russian).
[18] N.K. Nikolski, Operators, Functions, and Systems: An Easy Reading, vol. 1: Hardy, Hankel and Toeplitz, Math.Surveys Monogr., vol. 92, AMS, 2002.
[19] E.A. Nordgren, P. Rosenthal, F.S. Wintrobe,Composition operators and the invariant subspace problem, C. R. Math.Rep. Acad. Sci. Can. 6 (1984) 279–282.
[20] E.A. Nordgren, P. Rosenthal, F.S. Wintrobe, Invertible composition operators on Hp, J. Funct. Anal. 73 (1987) 324–344.
[21] H. Radjavi, P. Rosenthal, Invariant Subspaces,pringer-Verlag, New York, 1973.
[22] G.-C. Rota, On models for linear operators, Comm. Pure Appl. Math. 13 (1960) 469–472.