Hostname: page-component-8448b6f56d-gtxcr Total loading time: 0 Render date: 2024-04-24T17:40:41.669Z Has data issue: false hasContentIssue false
  • English
  • Français

On Universal Operators and Universal Pairs

Part of: Special classes of linear operators

Published online by Cambridge University Press:  08 June 2018

Riikka Schroderus  and
Hans-Olav Tylli
Riikka Schroderus
Affiliation:
Department of Mathematics and Statistics, University of Helsinki, Box 68, FI-00014 Helsinki, Finland (riikka.schroderus@helsinki.fi; hans-olav.tylli@helsinki.fi)
Hans-Olav Tylli*
Affiliation:
Department of Mathematics and Statistics, University of Helsinki, Box 68, FI-00014 Helsinki, Finland (riikka.schroderus@helsinki.fi; hans-olav.tylli@helsinki.fi)
*
*Corresponding author.
Get access Rights & Permissions [Opens in a new window]

Abstract

We study some basic properties of the class of universal operators on Hilbert spaces, and provide new examples of universal operators and universal pairs.

Keywords

Hilbert spaceuniversal operatoruniversal commuting paircomposition operator

MSC classification

Primary: 47B37: Operators on special spaces (weighted shifts, operators on sequence spaces, etc.)
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 61 , Issue 3 , August 2018 , pp. 891 - 908
Copyright
Copyright © Edinburgh Mathematical Society 2018 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1.Caradus, S. R., Universal operators and invariant subspaces, Proc. Amer. Math. Soc. 23 (1969), 526–527.CrossRefGoogle Scholar
2.Caradus, S. R., Pfaffenberger, W. E. and Yood, Bertram, Calkin algebras and algebras of operators on Banach spaces, Lecture Notes in Pure and Applied Mathematics, Volume 9 (Marcel Dekker, New York, 1974).Google Scholar
3.Chalendar, I. and Partington, J. R., Modern approaches to the invariant-subspace problem, Cambridge Tracts in Mathematics, Volume 188 (Cambridge University Press, Cambridge, 2011).CrossRefGoogle Scholar
4.Cowen, C. C. and Gallardo-Gutiérrez, E. A., Unitary equivalence of one-parameter groups of Toeplitz and composition operators, J. Funct. Anal. 261(9) (2011), 2641–2655.Google Scholar
5.Cowen, C. C. and Gallardo-Gutiérrez, E. A., Consequences of universality among Toeplitz operators, J. Math. Anal. Appl. 432(1) (2015), 484–503.Google Scholar
6.Cowen, C. C. and Gallardo-Gutiérrez, E. A., An introduction to Rota's universal operators: properties, old and new examples and future issues, Concr. Oper. 3 (2016), 43–51.Google Scholar
7.Cowen, C. C. and Gallardo-Gutiérrez, E. A., Rota's universal operators and invariant subspaces in Hilbert spaces, J. Funct. Anal. 271(5) (2016), 1130–1149.Google Scholar
8.Cowen, C. C. and Gallardo-Gutiérrez, E. A., A new proof of a Nordgren, Rosenthal and Wintrobe theorem on universal operators, In Problems and recent methods in operator theory (eds Bothelho, F., King, R. and Rao, T.S.S.R.K.), pp. 97–102, Contemporary Mathematics, Volume 687 (American Mathematical Society, Memphis, TN, 2017).Google Scholar
9.Cowen, C. C. and MacCluer, B. D., Composition operators on spaces of analytic functions, Studies in Advanced Mathematics (CRC Press, Boca Raton, FL, 1995).Google Scholar
10.Elliott, S. and Jury, M. T., Composition operators on Hardy spaces of a half-plane, Bull. Lond. Math. Soc. 44(3) (2012), 489–495.Google Scholar
11.Elliott, S. J. and Wynn, A., Composition operators on weighted Bergman spaces of a half-plane, Proc. Edinb. Math. Soc. (2) 54(2) (2011), 373–379.Google Scholar
12.Gallardo-Gutiérrez, E. A. and Gorkin, P., Minimal invariant subspaces for composition operators, J. Math. Pures Appl. (9) 95(3) (2011), 245–259.Google Scholar
13.Gallardo-Gutiérrez, E. A. and Schroderus, R., The spectra of linear fractional composition operators on weighted Dirichlet spaces, J. Funct. Anal. 271(3) (2016), 720–745.Google Scholar
14.Heller, K., Adjoints of linear fractional composition operators on S2(𝔻), J. Math. Anal. Appl. 394(2) (2012), 724–737.CrossRefGoogle Scholar
15.Higdon, W. M., The spectra of composition operators from linear fractional maps acting upon the Dirichlet space, J. Funct. Anal. 220(1) (2005), 55–75.Google Scholar
16.Hurst, P. R., Relating composition operators on different weighted Hardy spaces, Arch. Math. (Basel) 68(6) (1997), 503–513.Google Scholar
17.Lindenstrauss, J. and Tzafriri, L., Classical Banach spaces I: sequence spaces, Ergebnisse der Mathematik und ihrer Grenzgebiete, Volume 92 (Springer-Verlag, Berlin–New York, 1977).Google Scholar
18.Matache, V., On the minimal invariant subspaces of the hyperbolic composition operator, Proc. Amer. Math. Soc. 119(3) (1993), 837–841.Google Scholar
19.Matache, V., Composition operators on Hardy spaces of a half-plane Proc. Amer. Math. Soc. 127(5) (1999), 1483–1491.CrossRefGoogle Scholar
20.Matache, V., Invertible and normal composition operators on the Hilbert Hardy space of a half–plane, Concr. Oper. 3 (2016), 77–84.Google Scholar
21.Müller, V., Spectral theory of linear operators and spectral systems in Banach algebras, Operator Theory: Advances and Applications, Volume 139 (Birkhäuser Verlag, Basel, 2003).Google Scholar
22.Müller, V., Universal n-tuples of operators, Math. Proc. R. Ir. Acad. 113A(2) (2013), 143–150.Google Scholar
23Murphy, G. J., C*-Algebras and operator theory (Academic Press, Boston, MA, 1990).Google Scholar
24.Nordgren, E., Rosenthal, P. and Wintrobe, F. S., Invertible composition operators on H p, J. Funct. Anal. 73(2) (1987), 324–344.Google Scholar
25.Partington, J. R. and Pozzi, E., Universal shifts and composition operators, Oper. Matrices 5(3) (2011), 455–467.Google Scholar
26.Pozzi, E., Universality of weighted composition operators on L 2([0, 1]) and Sobolev spaces, Acta Sci. Math. (Szeged) 78(3–4) (2012), 609–642.Google Scholar
27.Rosenblum, M. and Rovnyak, J., Hardy classes and operator theory. Corrected reprint of the 1985 original (Dover Publications, Mineola, NY, 1997).Google Scholar
28.Rota, G. C., Note on the invariant subspaces of linear operators, Rend. Circ. Mat. Palermo (2) 8 (1959), 182–184.CrossRefGoogle Scholar
29.Rudin, W., Real and complex analysis, 3rd edn (McGraw-Hill, New York, 1987).Google Scholar
30.Schroderus, R., Spectra of linear fractional composition operators on the Hardy and weighted Bergman spaces of the half-plane, J. Math. Anal. Appl. 447(2) (2017), 817–833.Google Scholar
31.Shapiro, J. H., Composition operators and classical function theory. Universitext: Tracts in Mathematics (Springer-Verlag, New York, 1993).Google Scholar
32.Zorboska, N., Composition operators on S a spaces, Indiana Univ. Math. J. 39(3) (1990), 847–857.Google Scholar