Van Casteren, Jan A.

English

Van Casteren Jan A.

University of Szeged

Acta Sci. Math., 42 (1980), 189—164 A problem of Sz.-Nagy JAN A. V A N CASTEREN 1. Introduction Let § be a complex Hilbert space. Relatively simple proofs of the following results arc given. (a) A power bounded operator T on § is similar to a unitary operator if and only if T is surjective and if there exists a constant M such that (i-IA|)ll*ll ^Af||7,*-;jf||> Ml < i, (b) Let iA be the generator of a strongly continuous group {P,: /€R} in Suppose that sup {HP-,!!: /=0} is finite. Then A is similar to a selfadjoint operator if and only if there is a constant M such that ReA||x|| ^ M^Xx—iAx\, Re A > 0, x<LD(A). By spectral theory the "only i f " parts are obvious. For a contraction T, sta-tement (a) is due to GOHBERG and KREIN [3], who deduced it from a theorem of SZ.-NAGY and FOIA§ [10]. In the latter theorem the authors provide a sufficient condition for an invertible contraction T to be similar to a unitary operator, in terms of the characteristic operator function 0 r (A ) of T. This condition is that a constant N exists for which 11*11 ^iV||er(A)*||, |A| < i, For the concept of characteristic operator function and its connection with the theory of unitary dilations we refer to SZ.-NAGY and FOIA§ [11, ChapitreVI, pp. 228—230, and Chapitre IX, p. 334]. The problem of finding a simpler proof of statement (a), avoiding characteristic functions and dilation theory, was pointed out by SZ.-NAGY in [2]. In the present paper we shall give a solution. We shall even do it for non-contractive, but power Received February 22, and in revised form, May 25, 1979. 190 Jan A. Van Casteren bounded operators. Indeed, the proof of (a) shall be reduced to the comparatively simpler theorem of SZ.-NAGY [9] which asserts that an invertible operator S is sim-ilar to a unitary operator if (and only i f ) sup {[|S"||: /i£Z} is finite. Statement (b), the continuous counterpart of (a), is entirely new. 2. Main results We shall need a few definitions. A linear operator T on § is said to be power bounded if sup {HT"!!: n€N} is finite. Let A and B be linear operators with domain and range in §>. Then A is said to be similar to B if there exists a bounded linear operator V with bounded everywhere defined inverse such that AV=VB. Theorem 1. A power bounded operator T on § is similar to a unitary operator if and only if it satisfies one of the following conditions (in (ii)' T is supposed to be a contraction): (i) T has power bounded inverse S. (ii) The operators (T— A/)-1, |A|<1, exist and SUP{(1— |A|)||(T— A/)-1||: |A| < (ii)' The operators 6R(A)-1, |A|<1, exist and sup {ll©,.^)-1»: (iii) T has an inverse S for which the operators (/—AS)-1, |A|<1, exist and for which liminf sup{(1 —r2)||(/—AS)-1||: |A|=r} (iv) T is surjective and there is a constant M such that (1—|A|)||*|| S M\\Tx—kx\\, |A| -<= 1, Proof. SZ.-NAGY [9] proves the sufficiency of (i) by means of an invariant mean on Z. The necessity of (i) is trivial. The implications (i)=>(ii), (ii)=>(iii) and (iii)=>(iv) are more or less trivial. The implication (iv)=>(ii) follows from the fact that boundary points of the spectrum of a closed linear operator are approximate eigenvalues; e.g. HALMOS [4, Problem 63, p. 39]. In [10] SZ.-NAGY and FOIA§ use unitary dilation theory to prove the sufficiency of (ii)'. By establishing certain mutual inequalities between ||0T(A)-1|| and ||(R-A/)-1||, |A|<1, GOHBERG and KREIN [3] prove the equivalency of (ii) and (ii)'. See also KREIN [5, 6] and SZ.-NAGY and FOIA§ [11, Chapitre IX, p. 334]. A simple proof of the implication (iii)=>(i) runs as follows. Since it neither uses unitary dilation theory nor characteristic functions it solves a problem posed by SZ.-NAGY in [2, p. 585]. A problem of Sz.-Nagy 191 Fix x in § and r in [0,1). Denote M(r) = sup {(1 -rOlKZ-AS)-1 ! !: W = r} for 0 < r < l and put M0=sup {||rn||: From (iii) it follows that the spectral radius Q(S) of S satisfies Since ||rn||^M0, n€N, it also follows that e ( T ) s 1. Hence, for |/.|<1, we have norm convergence in both expansions (I-J.S)-1 = 2 (i-ZT)-1 = 2 ^"T". n=0 n=0 So, since ST=I, we have with l=re", 0 S r < l , co oo oo 2 rMeintSn= 2rne~in'T" = n= — OO R — 0 /1 = 1 = (/— re"5)_1 + re~" T(J— re~u T)~l = (1 - r2) (/ - rj'S)-1 (I-re~ u T)~K Thus, by (iii), it follows that 2 /-2'n|||S',x||2 = f 2 rMeimSnx dt 1 = ^ f \\(l—r2)(I—rei'S)~1(I—re~"T)~1x\\2dt S —n 1 + n sM(r)2-— f I\(I-re~i,T)-1x\\2dt = —it +it 2 rne-in,T"x n—0 dt = M{rf 2r*n\\T"x\\2. n = 0 = M(r)2 • — J —n Consequently, 2 r ' lS"* ! ! 2 ^ ( M ( r ) 2 - i ) 2 ^"lli""*!!2. n=l (1=0 Next, fix m in N, m£ l . Then, r2m||5m*||2 = (1-r2) 2 r2n\\Tn-mS"x\\2S n = m 3=( l - r 2 ) -M 0 2 2 »•2n||5",x||2s ( l - r^Mo 2 2 ||S"jc||8 n=m w=l and, by what is proved above, r2m||Smx||2=s ( l - r 2 )M 0 2 (M ( r ) 2 - l ) 2^"\\Tnx\\2S n=0 Si (1 -r2) A f 0 2 (M ( r ) 2 -1 ) Ml(1 - r2 ) -1 IIJC||2 = M%(M(r)2-1)|| xIP-192 Jan A. Van Casteren Since 0 < r < l is arbitrary, we conclude that ||Sm|| ^ liminf M02(M(r)2-1)1 '2, m S 1. >r»l Hence (i) follows. Remark 1. The operator (1—r2 ) (/-rS) - 1 (/—rS - 1 ) - 1 can be considered as ikind of an operator valued Poisson kernel. Remark 2. In [7] SHIELDS discusses a number of boundedness properties of powers of an operator in relation to the boundedness properties of its resolvent family. See also VAN CASTEREN [12] where similar questions are considered. * Next we describe the continuous analogue of Theorem 1. For a proof the reader will need Stone's theorem and some other standard facts on strongly continuous semigroups. For all this we refer to YOSIDA [13]. Theorem 2. Let iA be the generator of a strongly continuous group {Pt: /£R}. Assume that sup {||P_,||: i s 0} is finite. Then A is similar to a selfadjoint operator if and only if it satisfies one of the following conditions: (i) sup{||7>J: s s O H « . (ii) The inverses (XI — iA)-1, Re A>0, exist and sup {Re XKXI-iAyW: Re X > 0} < =°. '(iii) The inverses (XI— iA)~x, Re A>0, exist and lim inf sup {to||A7—i/4)_1||: Re A = <w} < <o40 (iv) There is a constant M such that Re A||x|| == M\\Xx-iAx\\, Re A > 0, x£D(A). Proof. We only prove the implication (iii)=>-(i). Here we use Plancherel's theorem in L 2 ( R , § ) ; e.g. see EDWARDS and GAUDRY [1, § 3.4, p. 53] or STEIN [8, Chapter II, § 5, pp. 45—47]. Fix x in § and to >0. Put M(oo) = sup {2(o\$XI-iA)~1\\: R e A = to}. From standard semigroup considerations it follows by (iii) that the integral f e-a^-*sPsxds A problem of Sz.-Nagy 193 exists and that OO CO oo f e-°'ul~iisPsxds= f e~t0S~'SsPsxds+ f e~ms+i^P.sxds = 0 0 = = 2(a({co + i^)I-iA)-%(o-ii)IJriA)-1x. So by Plancherel's theorem it follows that CO | » » f e-°-<a^\\Psx\\2ds = — f | f e-°>M-'SsPsxds\fdQ = = ¿ 7 WMin+iOI-i^doi-iQI+iA^xWtdlis I OO CO oo = M(co)2 — f || f e-as^sP^xds\\2d^M((of- f e-^P.^'ds. 2 j r -CO 0 0 Put M0=sup {||P_t||: issO} and fix 0. Then oo e-*"s\\PsxP = 2co f e-*™\\P_is^Psx\\*ds^ s S2a)M2 7e-2rasH^xll2rfs ^2ojM02 7 S 0 and by what is proved above, oo e-2<oS\\Psx\\2 2coM$(M((o)2-l) f e-2as\\ P-Sx\\2ds^ o 2cuM04(M(co)2-l) 7e~2asds-\\x\\2 = M^(M((o)2-$\\x\\2. o Consequently, we conclude that |jPJ s M02 lim inf (M(w)2 -1)1/2, s fe 0. Acknowledgement. The author is indebted to R. A. Hirschfeld (Antwerp) and to C. Foia§ (Orsay) for several comments and suggestions. Special thanks are due to B. Sz.-Nagy (Szeged) for several references and for suggesting a number of improvements in an earlier version of the present paper. The author is also obliged to the National Fund for Scientific Research of Belgium (NFWO) and to the Uni-versity of Antwerp (UIA) for their material support. 13 194 Jan A. Van Casteren: A problem of Sz.-Nagy Bibliography [1] R. E. EDWARDS and G. I. GAUDRY, Littlewood—Paley and multiplier theory, Springer-Verlag (Berlin, 1977). [2] H. G. GARNIR, K. R. UNNI and J. H. WILLIAMS (editors), Functional analysis and its applica-tions, Lecture Notes in Math. 399, Springer-Verlag (Berlin, 1974). M R 49 # 11187. [3] I. C. GOHBERG and M. G. KREIN, On a description of contraction operators similar to unit-ary ones, Funkcionál. Anal, i Prilozen., 1:1 (1967), 38—60. (Russian), M R 35 #4761. [4] P. R. HALMOS, A Hilbert space problem book, American Book Company (New York, 1967). M R 34 #8178. [5] M. G. KREIN, Analytic problems and results in the theory of linear operators in Hilbert space (abstract), Amer. Math. Soc. Transi., (2) 70 (1968), 68—72. M R 37 #1213. [6] M. G. KREIN, Analytic problems and results in the theory of linear operators in Hilbert space, Proc. Internat. Congr. Math. Moscow, 1966, 189—216, Izdat. „ M i r " (Moscow, 1968). (Russian), MR 38 #3733. [7] A . L. SHIELDS, On Möbius-bounded operators, Acta Sci. Math., 40 (1978), 371—374. [8] E. M. STEIN, Singular integral operators and differentiability properties of functions, Princeton Math. Series 30 (Princeton, 1970). M R 44 #7280. [9] B. SZ.-NAGY, On uniformly bounded linear transformations in Hilbert space, Acta Sci. Math., 11 (1947), 152—157, M R 9 — p. 191. [10] B. SZ.-NAGY and C. FOIAS, Sur les contractions de 1' espace de Hilbert. X . Contractions similaires à des transformations unitaires, Acta Sci. Math., 26 (1965), 79—91. M R 34 #1856. [11] B. SZ.-NAGY and C. FOIAÇ, Analyse harmonique des opérateurs de l'espace de Hilbert, Aka-démiai Kiadó (Budapest, 1967). M R 37 #778. [12] J. A . VAN CASTEREN, Boundedness properties of resolvents and semigroups, U.I.A. preprint 78—17. [13] K. YOSJDA, Functional analysis, third edition, Springer-Verlag (Berlin, 1971). M R 50#2851. DEPARTMENT OF MATHEMATICS UNIVERSITAIRE INSTELLING ANTWERPEN UNIVERSITEITSPLEIN I 2610 WILRIJK, BELGIUM 

A problem of Sz.-Nagy

http://acta.bibl.u-szeged.hu/14767/1/math_042_fasc_001_002_189-194.pdf

A problem of Sz.-Nagy

Abstract

Similar works

Full text

Available Versions

University of Szeged