Wadhwa, Bhushan L.

English

Wadhwa Bhushan L.

University of Szeged

Acta Set. Math., 38 (1976), 187—148 A note on reductive operators B H U S H A N L . W A D H W A A bounded linear operator T on a Hilbert space H is called a reductive operator if every invariant subspace of T reduces T. In this note, we shall study a local spectral theoretic condition which is satisfied by certain types of reductive operators. Con-sequently we shall obtain a set of conditions which are sufficient for a reductive operator to be equal to the sum of a normal operator and a commuting quasinil-potent operator. This will provide alternative proofs of some of the results of JAFARIAN [2], NORDGREN—RADJAVI—ROSENTHAL [3] a n d RADJABALIPOUR [4]. We shall be using the notation and terminology of DUNFORD and SCHWARTZ [ 1 ] . If an operator T has the single valued extension property, called property (A), then it satisfies Dunford's condition (B) if a(T, x)f\a(T,y) = q> implies that S ^ L L X + ^ H , where Kis independent of x and y. In [5], STAMPFLI introduced an ortho-gonality version of condition (B), that is, a(T, x )P lo(T ,y ) = cp implies that (x, y) = 0, for all vectors x and y in the Hilbert space H. An operator T with property (A) satisfies Dunford's condition (C) if for each closed set d, H(S) = {x£H: a(T, x)cz<5}is closed subspace. A basic theorem of DUNFORD [1, page 2147] asserts that an operator, T on a Hilbert space H is spectral (i.e., T~S + 0 where S is similar to a normal operator, Q is quasinilpotent and SQ = QS) if and only if T satisfies conditions (A), (B), (C), and (D). It is easy to prove that if in this result the condition (B) is replaced by the orthogonality version of condition (B) then S will be a normal operator and conversely [5, Lemma 7]. P r o p o s i t i o n 1. If T is a reductive operator and if T satisfies (A) and (C) then T satisfies the orthogonality version of condition (B). P r o o f . For any closed-set <5, since T is reductive, TP6=PST where P6 denotes the projection of H onto H(<5). Thus o(T, Pdx)ca(T,x) for all x£H. If <7(7", x)f l f]a(T,y)=(p then a(T, P}y)cz<j(T, y)r\8 where S = o(T, x). Thus a(T,Psy)=0, and hence Psy=0. Received February 1 , revised April 11, 1975. 188 Bhushan L. Wadhwa C o r o l l a r y 1. If T is a reductive spectral operator then T=N+Q where N is normal, Q is quasinilpotent and NQ — QN. C o r o l l a r y 2. [3] If T is a reductive operator which is similar to a normal oper-ator then T is normal. C o r o l l a r y 3. If T is reductive and if a(T) (the spectrum of T) is totally dis-connected then T=N+Q. P r o o f . If G{T) is totally disconnected then T is spectral if and only if T satisfies the Dunford condition (B) [1, page 2149]. Since T is reductive, the result follows f rom the proposition. If T is a reductive operator and if T satisfies conditions (A) and (C), then for each closed set S, a(T, Psx)(Za(T, x) for all x£H, where Ps is the projection of H onto H{ß). This is the condition (I), introduced by the author in [6], where in it was shown that a decomposable operator T which satisfies condition (I), (in particular if T i s reductive decomposable operatoi), then T is the sum of a normal operator and a commuting quasinilpotent operator. The next theorem is a generalization of this result. T h e o r e m 1. If T satisfies conditions (A), (C), (I), and if for each closed set S, H=H(ö)+H(ö') [5' denotes the closure of the complement of <5] then T=N+Q where N is normal, Q is quasinilpotent, and NQ = QN. P r o o f . By Dunford's theorem, we only need to show that Tsatisfies Dunford's condition (D). In order to show this, it is enough to show that for each closed set 6, H=H(Ö)®H(Ö'). Since T satisfies condition (I), it satisfies the orthogonality version of condition (B) and hence H(Ö')C:H±(Ö) where HX(Ö) is the orthogonal complement of H(S) in H. Now for any x£Hx(5), let x=u+v, where u£H(5) and V£H(5'). Thus 0 = Pdx = u+P3v and a(T, u) = o(T, Pdv)cz5f\a{T, v). Hence £H(S'). Thus for any closed set <5, H(¥)(ZH±(S)(ZH(S'). N o w for any open set G which contains Ö, H±(G)^H(Ö') and hence HL(B')(ZC\H{G)=H(5), where the intersection is taken over all open sets G which contain S. C o r o l l a r y 4. If T is a reductive operator and if T satisfies conditions (A) and (C) and if for each closed set 5, H=H(ö) + H(5') then T=N+Q where N is a normal operator, Q is quasinilpotent and NQ = QN. This result appears to be a generalization of Theorem 1 . 1 of J A F A R I A N [ 2 ] . Let g: s1 = {z: |z| = 1}—/ be an arc length parametrization of a rectifiable Jordan curve. Since J is rectifiable, exists almost everywhere (with respect to Lebesgue measure) on the unit circle j1. An operator T satisfies the growth condition (Gm) if A note on reductive operators 189 ||(A — T)~11| ^M[d i s t (A, <r(r))]-m for all A£Q(T) and |A| ^ | | r | | 4-1, where Q(T) denotes the resolvent set of T. T h e o r e m 2. Let T be an operator such that <J(T) is contained in a rectifiable Jordan curve J. If T satisfies the growth condition (Gm) and the condition (I) then T=N+Q where N is normal operator, Q is quasinilpotent and NQ = QN. P r o o f . Since Tsatisfies the growth condition (Gm) and <r(T) lies on J, it follows from [7, Theorem 11] that H(8) is closed subspace of H for every closed set 8. Also from the proof of [7, Theorem 10], it follows that for any wx, w2dJ, w;=g(.?,•) are such that g'(si) exists, / / = //[wj^, w 2 ]+/ / [w 2 , w j where [wx, w2] = {g(.s): By Theorem 1, we only need to show that for each closed set 8, H=H(S)+H(B'), i.e., Hx (<5)cH(5'). Now suppose that there exist x£HL (8) such that a(T, x)fl8°?±<p (<5° denotes the interior of 8). Then we can find w l5 w2fJ such that wt = g(st) and g'(si) exists, a(T, x)Pi[u^, w2]^(p and [wx, w2]c<5. Let x=xx+x2 where xidH[w1, w2] and x 2 £ H [ w 2 , w j , then 0 = P s x = x 1 + Pix2. Thus a(T, xx) = o(T, Psx2)<^8C\o(T, x2), i.e., <r(T, X)C8. References [1] N . DLTNFORD and J. T . SCHWARTZ, Linear operators. Ill, Spectral operators, Wiley-Interscience (New York, 1971 ) . [2] A. A. JAFARIAN, On reductive operators, Indiana University Math. J., 23 (1974), 607—613. [3] E . NORDGREN, H . RADJAVI and P . ROSENTHAL, On operators with reducing invariant subspace, to appear. [4] M. RADJABALIPOUR, Growth conditions, spectral operators and reductive operators, Indiana University Math. J., 2 3 ( 1 9 7 4 ) , 9 8 1 — 9 9 0 . [5] J. G . STAMPFLI, Analytic extensions and spectral localization, J. Math. Mech., 16 ( 1 9 6 6 ) , 287—296. [6] B . L . W A D H W A , Decomposable and spectral operators on a Hilbert space, Proc. Amer. Math. Soc., 4 0 ( 1 9 7 3 ) , 1 1 2 — 1 1 4 . [7] B. L . W A D H W A , Operators with spectrum in a C'-Jordan curve, Duke Math. J., 41 ( 1 9 7 4 ) , 645—654. T H E CLEVELAND STATE UNIVERSITY D E P A R T M E N T O F M A T H E M A T I C S CLEVELAND, OHIO 44115, USA 

A note on reductive operators

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A note on reductive operators

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