ACTA UNIVERSITATIS SZEGEDIENSIS ACTA SCIENTIARUM MATHEMATICARUM ADIUVANTIB U S B. CSÁKÁNY L. LEINDLER J. NÉMETH S. CSÖRGŐ L. LOVÁSZ L. PINTÉR E. DURSZT L. MEGYESI G. POLLÁK F. GÉCSEG F. MÓRICZ I. SZALAY L. HATVANI P. T. NAGY Á. SZÉNDREI A. HUHN K. TÁNDORI REDIGIT B. SZ.-NAGY TOMUS 43 FASC. 1—2 I: . vV; i.. ií Sr"' SZEGED, 1981 INSTITUTUM BOLYAIANUM UNIVERSITATIS SZEGEDIENSIS » .. 4 A J<Q)Z§Jgg ATTILA TUDOMÁNYEGYETEM KÖZLEMÉNYEI ACTA &QENTHAKUM MATMEMATICAMUM CSÁKÁNY BÉLA LEINDLER LÁSZLÓ NÉMETH JÓZSEF CSÖRGŐ SÁNDOR LOVÁSZ LÁSZLÓ PINTÉR LAJOS DURSZT ENDRE MEGYESI LÁSZLÓ POLLÁK GYÖRGY GÉCSEG FERENC MÓRICZ FERENC SZALAY ISTVÁN HATVANI LÁSZLÓ NAGY PÉTER SZENDREI ÁGNES HUHN ANDRÁS TANDORI KÁROLY KÖZREMŰKÖDÉSÉVEL SZERKESZTI SZŐKEFALVI-NAGY BÉLA 43. KÖTET FASC. 1—2 SZEGED, 1981 JÓZSEF ATTILA TUDOMÁNYEGYETEM BOLYAI INTÉZETE ACTA UNIVERSITATIS SZEGEDIENSIS ACTA SCIENTIARUM MATHEMATICARUM ADIUVANTIBUS B. CSÁKÁNY L. LEINDLER J. NÉMETH S. CSÖRGŐ L. LOVÁSZ L. PINTÉR E. DURSZT L. MEGYESI G. POLLÁK F. GÉCSEG F. MÓRICZ I. SZALAY L. HATVANI P. T. NAGY Á. SZENDREI A. HUHN K. TANDORI REDIGIT B. SZ.-NAGY TOMUS 43 SZEGED, 1981 INSTITUTUM BOLYAIANUM UNIVERSITATIS SZEGEDIENSIS A JÓZSEF ATTILA TUDOMÁNYEGYETEM KÖZLEMÉNYEI ACTA SCIENTIARUM MATHEMATICARUM CSÁKÁNY BÉLA LEINDLER LÁSZLÓ NÉMETH JÓZSEF CSÖRGŐ SÁNDOR LOVÁSZ LÁSZLÓ PINTÉR LAJOS DURSZT ENDRE MEGYESI LÁSZLÓ POLLÁK GYÖRGY GÉCSEGFERENC MÓRICZ FERENC SZALAY ISTVÁN HATVANI LÁSZLÓ NAGY PÉTER SZENDREI ÁGNES HUHN ANDRÁS TANDORI KÁROLY KÖZREMŰKÖDÉSÉVEL SZERKESZTI SZŐKEFALVI-NAGY BÉLA 43. KÖTET SZEGED, 1981 JÓZSEF ATTILA TUDOMÁNYEGYETEM BOLYAI INTÉZETE [LÁSZLÓ RÉDEI (1900—1980) On November 21, 1980, six days after completing his eightieth year, passed away László Rédei, member of the editorial board of our Acta since its eleventh volume, professor emeritus and doctor honoris causa of our University, Nestor of the Hungarian algebraists. He was born near Budapest and went to secondary school in the capital. His mathematical abilities presented themselves these years already; in 1918 he was a prize-winner of the Eötvös Competition. After having graduated from the Uni- versity of Budapest, he became a secondary school teacher of mathematics and physics first in the town Miskolc, and then in Mezőtúr. In 1940, he was appointed professor in the University of Szeged, where he spent 27 years: the most fruitful years of his life. In 1967 he moved to Budapest, where he lead the Algebra Depart- ment of the Mathematical Institute of the Hungarian Academy of Sciences until his retirement in 1971. The creative activity of László Rédei started in the twenties with algebraic number theory. He achieved deep results in the study of class groups of quadratic number fields, he determined among others the number of those invariants which are divisible by 2k (the case k = 1 was already settled by Gauss). Since 1940 his interests were more and more shifted to abstract algebra, and his enthusiastic works and lectures attracted many of his pupils to this subject also. Almost all algebraists working now in Hungary are his direct or indirect disciples. Particularly productive was his collaboration with the untimely deceased young mathematician Tibor Szele; among others, they were the first to recognize the functional completeness of finite fields. Rédei's further work on finite fields has also a growing interest in view of its connections with combinatorial theory. In abstract algebra, group theory was Rédei's most favourite topic; even in his last year of life he studied the subdirect irreducibles in the variety of nilpotent groups of class 2. Today, when the old problem of finite simple groups is completely solved, we have to remind of Rédei's pioneering paper "Ein Satz über die endlichen einfachen Gruppen" (Acta Math., 84 (1950), 129—153), which was a forerunner for the development of this area in the last three decades. Besides about one and a half hundred papers, László Rédei was also the author of the books "Algebra", "The theory of finitely generated commutative semigroups", "Foundation of the Euclidean and non-Euclidean geometries", and "Lacunary polynomials over finite fields". The first and the third of these are advanced text- books with rich contents while the other two books summarize Rédei's own results in monographical form. The merits of Professor Rédei were widely acknowledged by the competent communities. He was elected a corresponding member of the Hungarian Academy of Sciences in 1949, and an ordinary member in 1955. He was awarded the Kossuth prize two times and also several further high decorations. László Rédei was one of the great Masters of Hungarian mathematics. He was a teacher and advisor for generations, and also an interesting personality, hero of numerous anecdotes and legends. He was an attractive lecturer with a fine sense of humour and self-irony. Last but not least, he had the rare endowment of being fully devoted to his beloved profession. His decease is a heavy loss for this journal also. We shall cherish his memory. The Editors Acta Sci. Math., 43 (1981), 5—13 Reflexive and hyper-reflexive operators of class C0 H. BERCOVICI, C. FOIA§, B. SZ.-NAGY Dedicated to P. R. Ha I mo s on his 65 th birthday The Jordan model of a finite matrix was used for the first time in the study of reflexive operators (on finite dimensional spaces) by DEDDENS and FILLMORE [5]. Their result was extended in [1] to the class of algebraic operators on Hilbert space, using the quasi-similar Jordan model (in fact in [1] the notion of para-reflexivity is studied, but one can easily see that reflexivity and para-reflexivity are equivalent for algebraic operators). The possibility of extending these results to the entire class C0 was then indicated in [6] for the separable case and [2] (where a sketch of proof is done) for the nonseparable case. It appeared that the reflexivity of an operator of class C0 is equivalent, to the reflexivity of a single "Jordan block" S(m) (cf. § 1 below for the precise statement). In this note we give a simplified version of the proofs of [6] and [2]. We further study the related notion of hyper-reflexivity (stronger than reflexivity for the class C„) and prove an analogous characterization of hyper-reflexive operators of class C„. 1. Notations and results We shall denote by § a complex Hilbert space and by the algebra of linear and bounded operators acting on For an algebra ¿/cz &(§>), Lat si will stand for the set of closed linear subspaces 9Jlc:£j invariant with respect to all elements of st\ XaJlcaR, X^stf. For a family if of closed linear subspaces of §>, Alg £f will denote the algebra of operators Xd & (§) for which X9Jlc9ft whenever 9Jl£ ¿¡P. The algebra is called reflexive if js/=AlgLat stf. An operator 36 ($) is reflexiveii the weakly closed algebra s>2T generated by Tand /g is a reflexive algebra. An operator T£ id (§) will be called hyper-reflexive if its commutant {T}'=(stfT)' is a reflexive algebra. Received September 26, 1980. 6 H. Bercovici, C. Foia$, B. Sz.-Nagy Recall that a completely nonunitary contraction T(i3S(Sj) is an operator of m class C0 if u(T) = 0 for some u£H , u^O (cf. [10], ch. V). The simplest operators of class C0 are the "Jordan blocks" S(m), with m£H°° an inner function, defined by 2 2 (1.1) S(m)u = P^m)(zu(z)),u^(m) = H QmH . By the results of [11], [4] and [3], every operator T of class C0 is quasi-similar to a unique Jordan operator, that is to an operator of the form (1.2) S=®S(0 a where the values of a are ordinal numbers and the inner functions ma are subject to the conditions (1.3) mx = 1 for some a ^ 0; (1.4) m3 divides inwhenever a = (1.5) ma = mp whenever card (a) = card (/?). Let us note that m0 coincides with the minimal function mT of T. The operators quasi-similar to some S(m) are precisely the cyclic operators of class C0 (multiplicity- free operators). For multiplicity-free T it follows from [12] that Lat r=Lat {T}' and so for such operators reflexivity and hyper-reflexivity are equiv- alent. We are now able to state the main results of this note. Theorem A. An operator T of class C0 with Jordan model 5 = ® S(ma) is a reflexive if and only if S(mQ/m^) is reflexive. Theorem B. Let T and S be as in Theorem A.. Then T is hyper-reflexive if and only if S(m0) is reflexive. Recently P. Y. Wu [15] published a proof of Theorem A for the particular case of operators of class C0 with finite defect indices. 2. Preliminary results The following theorem plays an important role in the study of reflexive opera- tors of class C0 (cf. [13] and [14] for the proof). Theorem 2.1. For every operator T of class C0 we have (2.1) = {T}" = {T}' n Alg Lat T. Reflexive and hyper-reflexive operators of class C0 7 Corollary 2.2. An operator T of class C0 is reflexive if and only if Alg Lat Tcz c {J}'. — Obvious from relation (2.1). Corollary 2.3. Let be an operator of class C0 and let Sit, £ Lat T (j£J) be such that rlSOl.- is reflexive for each j. If §= V then T is reflexive. J€J Proof. It follows from Corollary 2.2 that it is enough to show that every X£ Alg Lat T commutes with T. But it is obvious that for XgAlg Lat T we have Alg Lat (r|9Jl7) so that {rp,}' by the hypothesis. Therefore, ktr(XT-TX) 3 V = that is Xe{T}'.