Geodesic
On reflexivity and hyperreflexivity of some spaces of intertwining operators
Mathematica Bohemica, Tome 133 (2008) no. 1, pp. 75-83

Voir la notice de l'article provenant de la source Czech Digital Mathematics Library

MR Zbl
Let $T,T^{\prime }$ be weak contractions (in the sense of Sz.-Nagy and Foiaş), $m,m^{\prime }$ the minimal functions of their $C_0$ parts and let $d$ be the greatest common inner divisor of $m,m^{\prime }$. It is proved that the space $I(T,T^{\prime })$ of all operators intertwining $T,T^{\prime }$ is reflexive if and only if the model operator $S(d)$ is reflexive. Here $S(d)$ means the compression of the unilateral shift onto the space $H^2\ominus dH^2$. In particular, in finite-dimensional spaces the space $I(T,T^{\prime })$ is reflexive if and only if all roots of the greatest common divisor of minimal polynomials of $T,T^{\prime }$ are simple. The paper is concluded by an example showing that quasisimilarity does not preserve hyperreflexivity of $I(T,T^{\prime })$.
Let $T,T^{\prime }$ be weak contractions (in the sense of Sz.-Nagy and Foiaş), $m,m^{\prime }$ the minimal functions of their $C_0$ parts and let $d$ be the greatest common inner divisor of $m,m^{\prime }$. It is proved that the space $I(T,T^{\prime })$ of all operators intertwining $T,T^{\prime }$ is reflexive if and only if the model operator $S(d)$ is reflexive. Here $S(d)$ means the compression of the unilateral shift onto the space $H^2\ominus dH^2$. In particular, in finite-dimensional spaces the space $I(T,T^{\prime })$ is reflexive if and only if all roots of the greatest common divisor of minimal polynomials of $T,T^{\prime }$ are simple. The paper is concluded by an example showing that quasisimilarity does not preserve hyperreflexivity of $I(T,T^{\prime })$.
DOI : 10.21136/MB.2008.133939
Classification : 47A10, 47A15, 47A45
Keywords: intertwining operator; reflexivity; $C_0$ contraction; weak contraction; hyperreflexivity 
Zajac, Michal. On reflexivity and hyperreflexivity of some spaces of intertwining operators. Mathematica Bohemica, Tome 133 (2008) no. 1, pp. 75-83. doi: 10.21136/MB.2008.133939
@article{10_21136_MB_2008_133939,
     author = {Zajac, Michal},
     title = {On reflexivity and hyperreflexivity of some spaces of intertwining operators},
     journal = {Mathematica Bohemica},
     pages = {75--83},
     year = {2008},
     volume = {133},
     number = {1},
     doi = {10.21136/MB.2008.133939},
     mrnumber = {2400152},
     zbl = {1199.47024},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.21136/MB.2008.133939/}
}
TY  - JOUR
AU  - Zajac, Michal
TI  - On reflexivity and hyperreflexivity of some spaces of intertwining operators
JO  - Mathematica Bohemica
PY  - 2008
SP  - 75
EP  - 83
VL  - 133
IS  - 1
UR  - http://geodesic.mathdoc.fr/articles/10.21136/MB.2008.133939/
DO  - 10.21136/MB.2008.133939
LA  - en
ID  - 10_21136_MB_2008_133939
ER  - 
%0 Journal Article
%A Zajac, Michal
%T On reflexivity and hyperreflexivity of some spaces of intertwining operators
%J Mathematica Bohemica
%D 2008
%P 75-83
%V 133
%N 1
%U http://geodesic.mathdoc.fr/articles/10.21136/MB.2008.133939/
%R 10.21136/MB.2008.133939
%G en
%F 10_21136_MB_2008_133939

[1] W. B. Arveson: Ten Lectures in Operator Algebras. C.B.M.S. Regional Conf. Ser. in Math, Amer. Math. Soc., Providence, 1984.

[2] H. Bercovici: Operator Theory and Arithmetic in $H^{\infty }$. Mathematical surveys and monographs 26, A.M.S. Providence, Rhode Island, 1988. | MR

[3] H. Bercovici, C. Foiaş, B. Sz.-Nagy: Reflexive and hyper-reflexive operators of class $C_0$. Acta Sci. Math. (Szeged) 43 (1981), 5–13. | MR

[4] J. B. Conway: A Course in Operator Theory. American Mathematical Society, Providence RI, 2000. | MR | Zbl

[5] Š. Drahovský, M. Zajac: Hyperreflexive operators on finite dimensional Hilbert spaces. Math. Bohem. 118 (1993), 249–254. | MR

[6] Š. Drahovský, M. Zajac: Hyperinvariant subspaces of operators on Hilbert spaces. Functional Analysis and Operator Theory, Banach Center Publications, vol 30, Institute of Mathematics, Warszawa, 1994, pp. 117–126. | MR

[7] V. V. Kapustin: Reflexivity of operators: general methods and a criterion for almost isometric contractions. Algebra i Analiz 4 (1992), 141–160. (Russian) | MR | Zbl

[8] K. Kliś, M. Ptak: $k$-hyperreflexive subspaces. Houston J. Math. l32 (2006), 299–313. | MR

[9] J. Kraus, D. Larson: Some applications of technique for constructing reflexive operator algebras. J. Operator Theory 13 (1985), 227–236. | MR

[10] J. Kraus, D. Larson: Reflexivity and distance formulae. Proc. London Math. Soc. 53 (1986), 340–356. | MR

[11] V. Müller, M. Ptak: Hyperreflexivity of finite-dimensional subspaces. J. Funct. Anal. 218 (2005), 395–408. | DOI | MR

[12] D. Sarason: Invariant subspaces and unstarred operator algebras. Pacific J. Math. 17 (1966), 511–517. | DOI | MR | Zbl

[13] V. S. Shul’man: The Fuglede-Putnam theorem and reflexivity. Dokl. Akad. Nauk SSSR 210 (1973), 543–544. (Russian) | MR

[14] B. Sz.-Nagy, C. Foiaş: Harmonic Analysis of Operators on Hilbert Space. North-Holland, Akadémiai kiadó, Budapest, 1970. | MR

[15] P. Y. Wu: Hyperinvariant subspaces of weak contractions. Acta Sci. Math. (Szeged) 41 (1979), 259–266. | MR | Zbl

[16] M. Zajac: Hyperinvariant subspaces of weak contractions. Operator Theory: Advances and Applications vol. 14, Birkhäuser, Basel, 1984, pp. 291–299. | MR | Zbl

[17] M. Zajac: Hyperinvariant subspaces of weak contractions. II. Operator Theory: Advances and Applications vol. 28, Birkhäuser, Basel, 1988, pp. 317–322. | MR | Zbl

[18] M. Zajac: On the singular unitary part of a contraction. Revue Roum. Math. Pures Appl. 35 (1990), 379–384. | MR | Zbl

[19] M. Zajac: Hyper-reflexivity of isometries and weak contractions. J. Operator Theory 25 (1991), 43–51. | MR | Zbl

[20] M. Zajac: Reflexivity of intertwining operators in finite dimensional spaces. Proc. 2nd Workshop Functional Analysis and its Applications, Sept. 16–18, 1999, Nemecká, Slovakia, 75–78.

Cité par Sources :