A Theory of Projections in Complex Banach Spaces Citation Woolson, John Robert (1941) A Theory of Projections in Complex Banach Spaces. Master's thesis, California Institute of Technology. doi:10.7907/4ss0-3b22. https://resolver.caltech.edu/CaltechTHESIS:02242025-225545467 Abstract No abstract. Item Type: Thesis (Master's thesis) Subject Keywords: (Mathematics) Degree Grantor: California Institute of Technology Division: Physics, Mathematics and Astronomy Major Option: Mathematics Thesis Availability: Public (worldwide access) Research Advisor(s): Michal, Aristotle D. Thesis Committee: Unknown, Unknown Defense Date: 1 January 1941 Record Number: CaltechTHESIS:02242025-225545467 Persistent URL: https://resolver.caltech.edu/CaltechTHESIS:02242025-225545467 DOI: 10.7907/4ss0-3b22 Default Usage Policy: No commercial reproduction, distribution, display or performance rights in this work are provided. ID Code: 17022 Collection: CaltechTHESIS Deposited By: Benjamin Perez Deposited On: 25 Feb 2025 00:15 Last Modified: 25 Feb 2025 00:24 Thesis Files PDF - Final Version See Usage Policy. 5MB Repository Staff Only: item control page

A Theory of Projections in Complex Banach Spaces Thesis by J. R. Woolson In Partial Fulfilment of the Requirements for the Degree of Master of Science California Institute of Technology Pasadena, California 1941 The author is indebted to Professor A. D. Michal for suggestions and help in the formulation of this thesis. A Theory of Projections in Complex Banach Spaces Introduction. P.:z = P. A Projection is defined as a linear operator P, such that A theory of projections has been developed in complex Banach Spaces, which are reflexive. This was done in 1939 by E. R. Lorch~ A complex Banach Space is a complex linear vector space in which a norm, with the u sual properties, has been defined. This paper develops the usual theorems concerning projections, which may be found for Hilbert Space in M. H. Stoner-~ The method of proof consists essentially in the use of an interspace inner product, similar to that of A. D. Michal and D. H. Hyers for real Banach SpaCeSi''-iH~ If f is an element of a complex Banach Space B, and Fan element of the space (B) of all complex valued linear functionals defined on B, then (B) can be shown to be a space of the same ty~e as B, the norm of F, II F/1 being defined a s t he bound of the functional F(f), ( F(f)( ~ /IFlf /If/I. The interspace inner product [F ,f] f € B, F E. (B) has certain demonstrable properties, see II below. The first paragraph is devoted to an existence proof for corn;;ilex nwnber valued linear functionals defined on a complex Bana ch Space. Be- fore the theory of projections is discussed, t wo par agr aphs discuss some of the properties of the inner product and of closed linear manifolds in Ban ach Space s . The theory of projections developed in the central portion of the paper is applied in the last t wo par agraphs; firstly to show that * E. R. Lorch. A Calculus of Operators in Reflexive Vector Spaces. ~ -lB* Trans. A.M.S. 45: 217-234 (1909). M. H. Stone. Linear Transformations in Hilbert Space. A. D. Michal and D. H. Hyers. General Differential Geometries with Coordinat e Interspace Inner Product. The Tohoku Math. Jour. 46: 309-318 (1940). -1- -2- [F, !~ f, f P.' f] = [F ,Pf], A being a linear operator and P a projection; C,,"!./ secondly to developing a theory of reducibility of linear operators defined on B. This latter is essentially an examination of the set of points in B which a linear operator A leaves invariant. I. Linear Functionals It is the purpose of this par a.graph to demonstrate the existence and to develop certain of the properties of complex valued linear functionals defined on a complex Banach Space. Let Ebe a complex vector space, that is a linear space with complex number multipliers. Let F be a complex number valued functional defined on E, which is additive F( f + g) = F( f) + F( g) , f, g E E, and homogeneous F(c:rf) = .,.F(f) , Theorem 1.1. a complex number. er - If there exists a functional p(f) on E to the complex num- bers Ip (cr f) \ = 16"1 \p ( f )\ l p(f + g)I ~ lp(f)I + lp( g)I and if there exists an additive and homogeneous functional G(f) defined on a complex linear subset E, c E, such .t hat I G( f ) l ~ I p(f)\ for all f EE,, then there exists an additive and homogeneous fw1ctional F(f), such that IF(f)I Proof. ~ F(f) = G(f) for all feE 1 • / p(f)I for all fc:E, If E, = E t he theorem is trivi al. \ G( f 1 ) Let f 0 c: E - E ,. I f f, , f 2 e E 1 , t hen G( fe ) I = I G( f, - f 2. ) \ ~ I P ( f 1 - f z. ) I - ~ /p ( f , + f.)I + IP ( - / (/G(f,)/ - IG(fe)I)/ ~ lp( f , + f.,)\ + IP( - fz. - f e. - Then, since l ( I G( f I ) I - IG( fz. ) I ) I ~ I G( f I ) I - I G( f e. ) ]G(f,)I - IG(f,)\ ~ lp(f , + f 0 )l + \p( and I ' fe. - f 0 )\ f.)/ f~)J -4- Thus there exi st s some complex constant fo, such t hat for all fEE 1 (1) -I p ( - f - fo)I - IG(f)j ~ ipol !:: l p ( f + f 0 ) \ - \G(f)/. Consider all elements of the form ( 2) g==f+O-f0 where f € E" f 0 ~ E - E 1 , er is complex number, and designate by E 0 t h e cla s s of all such elements. Evidently E 0 is a complex vector space. = G( f ) + er p. ( 3) ¢ (g) ( 3 ') 1¢ (g)j ~ /G(f)/ + fo fol Let then Since f 0 EE - E I and f EE 1 , all g 1:. Eo admit ex2.ctly one r epre sentrttion of Thus t he functional 1 ( g) is defined in E O in a unfo,ue the form ( 2). manner. In the inequ ality lfol~/ p (f + f .)j - IG(f)I write !,. in place of f, t h en l«rpol~lp(f +cr- f .)/ - IG( f )/ • Substituting thi s in ( 3 1) \¢( g)/ ~ l p (f+6""fo)I =- l p ( g)I (4) /9 ( g )[ ~ l p (g)I for gEEo• The functional¢ ( g) i s evidently additive and homogeneous. When fo = O, the inequ ality (1) shows that lfol = 0 since p(O) = 0 is a particula r c ase. ( 5) t ( g) Thus = G( g) , g c E 1 • Thus ¢(f) i s the r equired function al for the spa ce E0 , (the space E 1 to which have been added tho s e elements which con sist of a complex number times a f ixed element f 0 in E - E1 ). Rep eating t he above construction f or E - E 0 and a fi x ed el ement r: E E - E 0 & furthe r extension of G(f) may -5be obtained. Whence by transfinite induction, the func tion F(f) exists satisfying the condition of the theorem. Consicier the class of all complex valued linea r functionals defined on a complex Banach Space B, f ,g e: B. bound of F( f), IF( f) I ~ I/ F II l!f //. Then if the norm of F ,ll F II, is the The class of linea r functionals so de- fined form a Banach Space (B) of the same type as B. Theorem 1. 2. Given a linear functional G( f) defined on a complex vector space E c B, there exists a linea r functional F(f) defined on B and s atisfying the conditions F( f) = G( f) for f c E and II F II = I GIE:. /G/c is the bound of the functional G( f ) on E.* Proof. In Theorem 1.1 substitute p (f) = /GIE 1/fll. fies the conditions placed on p(f). It evidently satis- Then as a consequence of Theorem 1.1 there exists a functional defined on B such that F(f) = G(f) for all f E..E. Since the two functionals are e --1u&l for all f c.. E, their bounds a re equal, IIFII = IGIE. Theorem 1.3. For ea ch fo i::B there exists a linear functional F(f.,) defined on B, such that IF( f Proof. O ) I = II f. II ' II F /( = 1. In Theorem 1. 2 let E be t he set of elements of the form 'T f O, and substituting IG(f)I = ITll/f./1; !F(f,:r)I =l'rlllf.11.; l'F( f~)I =Hf.II Since by 1'heorem 1.2 II FIi = IGIE; ll'T FIi =IT/ 1:J11d II FIi = 1. The above theorem shows that the set. of complex number v&.lued linear -r.- Bohnenblust and Sobcyk have given a di f ferent proof of t his gener1-1lization of the Hahn - Banach Theorem. Bull. A. M. S. 44:91-93 (1938). -6- f1mctionals defined on a complex Banach Space is non-null. II. The Inter-Space Inner Product Using Theorem 1.3, it is possible to demonstrate the existence of an inter-space inner product with the following properties. If Bi s a com- plex Banach Space and (B) the set of complex number valued linear functionals defined on B to the complex numbers, then for feB, FE.(B) (1) [F,f] is bilinear on (B), B to the complex nwnbers, ( 2) If [F ,f] = 0 for all f then F = o, ( 3) If [F,f] - 0 for all F then f = o. The first two properties are an immediate re sult of the correspondence F(f) = [F, f]. The third property follows from Theorem 1.3, since we can write for each £~ eB lrF,fo]I = I/foll Thus [F,f] = 0 for all F implies f = O. Definition 2.1. An operator is an additive and continuous or linear trans- formation whose range and d.omain are both in B or both in (B). The letters A, B, P, ... ere used to represent linear operators; I Al, IBI, IPI , ... , their bounds, if they exist. The space (B) is said to be the adjoint or conjugate space to B. Definition 2.2. The operator A1 defined in Band the operator Ae defined in (B) are s&id to be adjoint, if [A 2 F,f] = [F,A,f] for all f in the dome.in of ci.efinition of A1 , and all F in the domain of definition of Ae. The adjoint of an operator A i s written A➔~. -7III. Linear Manifolds in Complex Banach Spaces Definition 3.1. A set of elements M such tha.t for f,gcM, f+gt:.M, pfc.. M is said to be a line ar manifold . Manifolds contained in B a re designated by or-t, 3'i., ... those contained in (B) by (~), ( o'<.), Definition 3.2. The elements Fe.(B), fe.B are said to be orthogonal if [F ,f] = O. Definition 3. 3. A linear manifold a]'( is said to be clo sed if the limit point f of any sequence {f"} , f.:dTI (i = 1, ... n, ... ) is also contained in lfl!i. In tha t which follows, a clo sed line ar manifold will be designated by the initia ls c.l.m. Definition 3.4'. The orthogona.l complement of a c.l.m. 10'{ c B is the set of all elements FE (B) such tha t [F ,f] = 0 for all f e ~. The orthogonal ~ ~ ~.L complement of irri. is designated Jo\, and the orthogonal complement of on,~. It may be shown that the orthogonal complement of a c.l.m. is a c.l.m • .1-.L Lemma 3 .1. 11'{ = ti. , for rt c B• .L .L Proof. J. Clea rly3"L .::> ri . Assume there exists an fe. ol , f¢ '6i.. Then there exists an element Fc(B), such that Fis orthogon.s.l to ti., F1-ri, [F,f] f- O. But [F ,f] must e ,1u al zero for F.l.ri. . This contradicts the assu..rnption . .L .L Therefore t{ = t\ . J..L. Lemma 3 . 2. Proof. (~) = (~) for (tl) c (B). Clea rly (rt r .:> ( ~). Assume there exists an FE. (oi. ; \ F ¢ (o'{). Then there exists an element f c.. b, f 1( irt.. ) such tha t [F, f] i O. But [F, r] must e lJ.UE•l zero for fl(~). Th erefore (6'() .t..L Definition 3.4. This contradicts the as sumption. Two closed linear manifolds '6o'i. and o't. = (~) . ((art) anci. (ol)) will be said to be disjoint i f there exists a constant lo O (lbO) such that for every f e ot1. and every g di (IIF + G/1? lcllFll). ( for every F E ( l!t{) and every G c. (t\)), (\ f+gl\ ~ kUf/1, -8- This definition does not ciifferenticcte bet ween the .u1anifolds, since 11 !If+ gl/? kl/fl/, i mplies II f + g/li) k llgl/ , Since Hf+ g/1 H g/I - llf/1 211d it is given that II f + gll~ kl!f/1, multiplying the first of the se ine,iualitie s by k and adding them, ( 1 + k) )I f + gll ~ k I/ g II. Thus 11 (If + gll ?- k tl gll, where k" = k/(k + 1). In an entirely am1.l ogous manner it may be shovm that IIF + GIi~ k'IIFII, im.pl..i e s II F + GI I ~ kIll JIG/ I , where k 1/1 = k' / ( k / + 1 ) • Theorem 5.1. Two clo sed linear manifolds ool. and r( are disjoint if and only if they satisfy the following conditions: (1) The manifolds have only the zero element in common, ( 2) The set of all elements of the form f + g, fc m , gt: o'{ is a clo sed linear manifold. Proof of (1). Let fE 'tr(. rt., then, since if rri and oi are linea r manifolds, m-rt is a line ar manifold, -fc: ~ . By Definition 6 . 4 O = llf - f/1~ kllf/1 and since k:>O, llfl\ = O, f = O. Proof of ( 2). The set of elements of the form f + g, f.e oo1. , g E oi are readily shown to be line ar. h~➔h. Leth"= fn + g" (n = 1, 2, ... ) , anci let Then by the definition of disjointnes s , ru1d since llh" - h.,,il ➔ O, /If., - f..,11 ➔ O. Ilg,. - g,,,1\ ..... O. 0 iriceITT and iN Similarly, since 1/h.., h..,/1~ 1t11 g., - g ,,,I{, are clo sed f.- f, g.,- g , thus h = f + g and the set of all element s of the form f + g form a closed linear manifol d. -9- Proof of the Converse. Since m and o( have only the zero element in common, elements of B can be expressed in only one way in the form f + g, ft:a-(, gE.ol. There exists a tr~msform2.tion A such that, A(f + g) tributive. The conditions h,, that g"➔ g e o'( , where h = f, which is dis- = f~ + g,. (n = 1, 2, ... ) , h,,➔ h, f"➔ f, imply = f + g. Thus A is closed, and a closed distrib- utive operator is by definition line ar. Let k> 0 be a constant such that IIAhtl ~ llhll/k, then l\ A(f + g)II = II f 11 ~ 1/kllf + gll. Thus II f + g n~ k/lfll, which is the definition of disjointne s s, completing the proof of the theorem. The following theorem may be proved in an entirely analogou s manner to tha t of Theorem 3.1. Theorem 3.2. Two closed linear manifolds (m) and (3"t) are disjoint if and only if they satisfy the following conditions: (1) They have only the zero element in common. ( 2) The set of all elements of the form F + G, FE.(~), GE..(~) i s a closed linear manifold. IV. The Theory of Projections Definition 4.1. An additive and continuous, or line ar, opera tor Pis c alled a projection if P~ = P. The op erations+ and• referred to manifolds will be interpreted a s the set sum and set intersection respectively. Th eore,!!l__jd• If P is any projection in B, and ai c B the set of elements f for which Pf = f and 3'1- c B the set of elements g for which Pg = O, then -10- 6o1 and ot are di s joint clo s ed linear manifold s such t he.t o3't + 3"i. = B. Con- ver sely, i f ITT and rt are di s joint closed linea r mani fo l d s for which m + 3"t = B, there exi s ts a uni q_ue projection P which satisfies the equ ations Pf= f, fc:rrt; Pg= O, ge~. Proof. If f,ge.ITT, then it follows fro m the equ ations Ppf = pf and P(f + g) = f + g, that 3rl is a. linea r manifold. Since P is continuous, oat ~imilarly ~ is a c.l. m. is clo s ed . For f E. B, f = Pf+ (f - Pf). Since Pa = P, Pf1:.rrt, f - Pfc.tt . Thus 33"\ + ol = B. Since II Pfll ~ I Pl tt fll = I Pl II Pf + (f - Pf)/1, and f - Pf = g E. rt, Pf= f1:..m, it follows that ll f + g II ~ k !If 11. This shows that ~ and ti. a re disjoint, by Definition 3 .4. m and o'( be disjoint c.l.m. such tha.t ~ + t( = B. Proof of the Converse. Let Leth c B, h = f + g , fE. m, gd't . Then the op erator P for which Ph ·= f is distributive and closed, hence linear. of Theorem 3 .1 was proved. This may be proved a s the conver s e Also, P~h = P(Ph) =Pf= f ;Ph, thus Pe= P. This completes the proof of the theorem . .Theorem 4 .2. If P is a projection, then the adjoint P~- (cf. Definition 2 . 2) is also a projection. If 5tf , rt. and (o1'0, (t() a re the manifold s a s soci ated .I. with P and P* re spectively, then (~ ) = tt and ( ot) Proof. --oo{J.. From Definition 2.2 \y➔~F,f] = [F,Pf] and since Pis a projection [P➔~F ,Pf] = [F ,Paf] = [F ,Pf]. Thus [P-r--z. F, f] = [P➔~F,f], which shows tha.t 2 pl(- = ~, and pl(- is thus a projection. -11Let GE. (m), then [P➔~G,f] and fc. m.. = [G,f], but [P➔~G,f] = [G,Pf]. If (f➔~G,f] = O, then for GE.. (rrt), [G,f] Pf = 0 in this ca se. = O. Thus Pf =f Then f t:.~ sin ce .1. Thus (m).L o(_ or (cWl) c o1.. Let G be any element orthogonal to ti , and let f c.B. Then = [G,Pf + (f - Pf)] = [G,Pf] + [G,f - Pf] but f - Pfc. IN. , therefore [G,f - Pf] = 0. Thus [G,f] [G,f] = [P* G,f] and GE. (3o"l), and it follows that (tl'tt) =tl . .1. It may be proved in an entirely analogou s manner tha t ( ili) Theorem 4. 3. .1.. =6ot. If P 1 and P.a a re projections and ~., cft., ; 0Dt2., oi2. t h eir a.s soci- e.ted manifolci s, then (1) P, P2. is a projection except if m, and o6'1.2. have only the zero element in common. ( 2) P, + Pe is a projection if and only if P, P~ = O. Proof of (1). If m,,3olz= O, then t he op erator P, Pe. is t h e null operator for all f EB. This will be seen to be a degenerate ca se. Let i f ofC• otlz. and B - m,.mz. a re disjoint c.l.m; there exists a projection (cf . Theorem 4.1) P3 which h a s lftL,-~z.. and B - m,. ~2. a s its associated manifold s. It i s thus re quired to show that m,-~z. and B - m,. me a re disjoint closed linea r manifold s . They are evidently clo s ed and linea r. In oreer to prove them disjoint' let f C m,. m2. and g C- B - oat., . irrt.L' then since f' g E. B \If + gl\ ~ llfll - llgll. Operating on both sides of this inequ ality with P, P2. -12- II P, Pz. (f + g)ll ? I( fU Also since P 1 and Pa are by definition bounded where k? 0 is the bound of P, Pe. • Thus II f + g ~ ?= 1/k Hf JI. Then by Definition 3.4 ~,• rn2.. and B - 001 1•$t.e. are disjoint closed linear manifolds. = ottz.• 15Ji.. and P, Pz = P Since at,· otf.e. 1 3 = Pa, the projections If 3'lt· ool2.. = 0 then and P.._ P, as here defined are necessarily commutative. Proof of (2). If P, Pz. = P P, = O, then 2 2 ( P, + Pe ) = P, + P, P~ + Pz. I: thus P1 + Pe.. is a projection. thus P, Pe. + Pe. P, Thus P, Pe. = P, + Pz. , Conversely, if P, + P2. i ::; a projection, From this P, (Pl Pe. + P2..P,) = (P, Pz. + P,P2.P, )P, pl =0 P2. + P, P2..P, = P Pz.P, + P,P P, = 0 2 1 = Pz. P = 0 completing the proof of the theorem. 1 Theorem 4.4. Proof. = O. + P~ If P1 and P2. b.re projections, P1 - P2.. is a projection if If P is a projection, then I - P is also a projection, since e . (I - P) = I - 2P + P = I - P. Thus if P, - P2 is a projection, I - (P, I - (P, - Pe.) - Pe.) is also a projection. = (I - P, ) + Pz. By Theorem 4.3 the sum of t wo operators is a projection if (I - P, )Pa_ or if P2 (I - P, ) = O, thus P P2.. = P, or P P, = Pe.. 1 2 =0 -13- = Pe. implie s P P, = P2. and Conver sely, since by Theorem 4.3 P, Pe. 2 2. - Pe.) = P, (P, - P, P.a. - P2. P, + P2. =P, - Pe. • This completes the proof of the theorem. V. Infinite Systems of Projections If !Sot and 3"< are disjoint Md if there exists an f(fO) E. B, Lemma 5.1. ff m + t{ , then there exists an F(fO) E. ~ o't.L. ot.L are ciisjoint,rri. + at. Proof. .L. Conversely, if rn and = B. Let fE. B, ¢- ~ + 'rt ; then there exists an Fe. (B) such that .1 [F,f] = O, thus F1- c'Jti. + ot. • .L Thus F(fO)e~ .L and Fe.~. .L ..L Thus FE ooC , f( ~ .L If &l a11d at are disjoint then there exist s no element out si de the zero element satisfying the condition [F,f] = O. Then the zero element is the only element .L m + ~ • = B. Theorem 5.1. Thus m + rt Let fPn} be a sequence of projections for which P"< P,,.. 1 (i.e. the cha r a cteristic manifold of P.,, m~ is contained in the chara.cteristic manifold of P••1, ~ 1), IP.,\ ~ k, (n = 1, 2, ..• ) • of P.., be pt; and let rri", fl.; ( (}o'(t , ( o't)"' Let the ao. joint denote the manifolds a ssociated - with P., anci Pt re spec ti vely. (Si.) =l; (o't )o<. • Then (1) a'i. and o{_ are disjoint; (cITT.) and (1ft.) are disjoint . .L .l. .l .l. ( 2) (m) =if'(, (ot) = aol; ITT= (o'<) , o'i. = (crtt) • ( 3) ~ + at. = B; (~) + (tt) = (B). Proof of (l}. that f.,--.f. Let fc: m, g E ~ • Then there exi s t elements f.,c. M'C., such Thus II P"' ( f.,, + g) II = II f .,II~ k I f., + g II. I/ f n + gl/~ 1/k//f.,1/, K Since f + gij ?-1/kl/fll, oo{ and N. are by Definition 3 . 4 disjoint manifolds. -14The case IP"I = 0 is trivial, it has thus been assumed that k> O. From Thus IPtl =IP.,\~ k. [Pt F,f] = [F,P.,f] it follows that IP-l;;I [F,f] = /P.,\ [F,f]. Proceeding similarly F.,E. (rrt).,, FE. (ITT), GE.(~), II~ (F"' + G) II = II F.,I\ ~ kllF., + G II Since F.,--,. .F, and k > o, I) F + GIi ~ 1/kllFn. Thus (ITT) and (r-t.) are ciisjoint manifolds. IJince by Theorem 4. 2 (r-1. )., = Proof of ( 2). m~, ~.L (r-t) c (oi )., . Hence .L 00 ob Let F..Llm., thus Fe.(tt).,, Fe:.lf (ot.).._, (tl.)1-~~ =6l'1 or (r{)cat. J.. Thus (oi.) = rrt, proving the second statement of part (2). F 1:.(ai). J...L 1.. J..L ..L. Since by Lemma 3.1 ffi = im, from (ol) =ool; (ol) = oft = n't, which proves the third statement of pa rt (2). Since by Theorem 4. 2 (oot)a.1 of.., ; (oat),, .L o'l (n = l, 2, ... ) since o( c al" , ( n = 1, 2, ... ) . Th en o<> = thus (oot) .:) ~. oi = (i53i) . I J.. .L L I. (ITT)... l. o(_ ' Let f c (ITTJ , then f ..L (m).,, ft:.. (rt)., , f E ( o'i). ..L..L By Lemma 3. 2 (oo'{) = ( m) . Hence J. Thus rt = (wt) which COIDflletes the proof of part ( 2) of the theorem . Proof of (3). m ru1d al are disjoint, since by part ( 2) of this theorem .L .L .L ..L. (ITT) = oi and (oi) = ro1. (~'O + (at) = (B). Theorem 5.2. Thus by Lemma 5.1 m + tt = B. Similarly, 'l 'his completes the proof of the theorem. If {P.,\ is a sey_uence of projections for which /P..,l ~ k, P.,<P.,.,, (n = 1, 2, ... ), then there exists a projection P having the following properti es: (1)a1ip = i lfi1i~' (2) IPI ~ k. rt.p = T[ H.... -15- (3) For any f e B, ll(P - P,,) fll-o. ( 4) P;:, Pn (n = 1, 2, ... ) . If Q is a projection such tha t Q;, P., (n = 1, 2, ... ) then Q > P. (5) If P" is permutable with a linear operator A, then P is permut a ble with A. Proof of (1). Pis by definition the projection whose associ ated mani- folds are mP and N.r. By Theorem 3.1 ~r and oip are disjoint and ~r+ rtP = B. By Theorem 4.1 Pis uniquely defined. Proof of ( 2). Since IP.,I ~ k (n = 1, 2, ... ) , it is evident from the def- inition of P that lP\ ~ k. Proof of (3). If fe.ttpthen Pf= 0 and P,..f = O, thus ll(P - P.,)fll-o. If fe. ~ P , then Pfc.~pand. there exists a g"1:.m"(n = 1, 2, ... ) such that g"-" Pf. Now Pf - P" f = Pf - g,, + P,,(g" + f - Pf) - P" f and II Pf - P"fll ~ /I Pf - g.11 + /IP.,( g ., - Pf)/1 ➔ 0 since I P"I ~ k. Proof of ( 4) . That P ;,- P" follows immecii ately f rom its definition and the asso ciated manifolds. If Q;,- P,, (n = 1, 2, ..• ) thenma.~~,,_(n = 1, 2, ... ) , hence m~::, ~ P and similarly f'lQ. c: tlr . Proof of (5). Thus P < Q. Since P.,Af = AP 11 f ➔ APf and P,,_ Af ➔ PAf , PAf = APf. This completes the proof of the theorem. VI. The Mean of the Itera tion of Linear Operators It is the purpose of this para graph to show that the mean of the iteration of a linear operator approa ches a linea r 09erator which is a -16- projection. Definition 6.1. lf ..,1, f, E.B (i = 1, 2, ... ) , is said to be A sequence sl I , 1,ly convergent towa rd hi:. B if n 0 II f" - h ll <. c , for n ~ n O ( exists such that E ), E an arbitra ry positive number. Definition 6.2. The complex number ;t is said to be a proper value for the operator A, if Af =). f f or some f e B. Definition 6.3. II A.,fll ~ Mllf/1, i Lemma 6.1. The operators An a re said to be equi-bounded if = 1, 2, ... , n. If 1 is not a. proper value for A, and i f the opera tors A" are equi-bounded , then for any f e B Proof. For a fixed element f let + A f j then Thus n·H Ag., - g"' = 1 I n (A f - A f) and 1 II Ag., - g.,11 = 1/n II l't- f - A f II and by t he properties of the norm Since A" 's are equi-bounded f II Ag"' - gri II ~ 1/n Mij fl! + Mf!f/1} -17Thu s i n t he limit, as n --7' o<1 fl Ag"' - g.,I[ = O .Ag., = g l'\ n~ o0. But 1 i s not a proper v a lue fo r A, t hus g., =- 0 as n ➔ c,0, and J 1/n { .hf + P.' f + . . . + A" f ~ 0 Theorem 6 .1. as n - oc ,, I f A is a linear operator on B, and A is equi-bouncieci and oTi. t he set of all elements f E. B, such that Af = f, and if P i s the pro jection who se characteristic manifold is oot., then 1/n (Af + p._2 f + ... + .A " f) ~ Pf Proof . as n - """ Let 1rr1 be the se t of element s such t hat Af = f, o'f... t he set of element s di sj oint to ITT. = B anci. P i s the Then by Theorem 4. 1 ITT+ If( projection s uch that Pf = f, f ~ ~ ; Pg = O, g e ti . Since ITT i s the set of all f c B such tha t Af - f, for the set of elements 3'i., A doe s not have 1 as a proper value. Le t f 1 = Pf an6 f 2. = f - Pf, then f. e. rr-< and f z. E. o't . The operator A transforms oft into itself since for all f, E. att., Af, = f 1 " A f = A f, + A f~ = f, + A f Yl r, t"I 2 It has been shovm that a.11 elements f c. B may be represented in the form f = f, + fe, f,Eol'\, f2.1::.t'i where ~ and t1. are disjoint c.l.m. (cf. Theorem 3 .1). By Lemma 6 . 1 since for f 2 E ~ A . a oe s not have 1 as a proper v&.lue, 2 1/n ( Afz. + A f2.+ .•• + l ' f 2 ) - O, a s n _. =. Substituting A"f = f, + A~f2 , f, = Pf 1/n (Af + p.,_2 f + + .A" f ) -+ Pf This complete s the proo f of the t heorem . as n -4 = -18- VII. Reducibility It is the purpose of this paragraph to characterize a linea r operator in terms of the manifolds which it leaves invariant. Definition 7.1. Let A be a bounded linear operator, or( a closed linear manifold and P the projection whose characteristic manifold isoo{. Got is said to reduce A if A and P are ·permutable in the following sense: whenever f is in the domain of A, Pf is also in the domain of A and APf = PAf. Theorem 7 .1. If the closed linear manifold oO't reduces the linear oper- ator A, then A leaves ITT invariant in the sense that it carries every element common to its domain and the manifold 10'-t into an element of ooi.. Proof. Let fE or{ and in the domain of A. Then since Pis the pro- jection whose cha.racteristic manifold isdft, Pf= f and APf = Af. Since do'( reduces A, by Definition 7 .1 Af = APf = PAf Af is some element of B, thus P.Af c. m or PAf = O, the latter is true only in the trivial case, A is the null-operator. Thsu A leaves ool invs.riant in the sense of the theorem. Theorem 7 .2. If of(, and me. are non-intersecting closed linear mani- folds which both reduce A then ~, + rN.z.. reduces A. Proof. Let P 1 &nd Pz.. be the projections whose characteristic manifolds are oot. 1 and rr{e. respectively. Since m_,. oo/.z.. = O, P, Pe. is a projection ( cf. Theorem 4. 3) . = 0 and P, + Pe.. Thus for f e. m, + mt.. and f contained in the domain of A. (P 1 + Pe.)f = f and Af = A(P, + Pa)f = (P, + Pa)Af: Thus art.,+ ml- reduces A if both J'o't, and (o'te. reduce A. -19- Theorem 7.3. If m1 andMte. S ()li 1 are c.l.m. both of which r educe A, then m, - oJi 2 reduces A. Proof. are Let P1 and Pe.. be the projections whose characteri s tic manifolds m, andirote.respectively. Since the condition P-1..P, =Pe.of Theorem 4.4 is satisfied, P, - Pe. is a })ro jection . Af Thus = A(P, - Pe.) f = (P, - Pe..)Af and since aat , - mt. is the cha r a cteristic manifold. of P, - Pe., ooi 1 - m2. reduces A. Evidently Band null set in B reduce any line ar oper a.tor which is defined throughout B or the null set. Definition 7.2. A bounded linear operator A is said to be irreducible if it is reduced by no c.l.m. other than the entire space o r the null set. Theorem 7. 4. If .1.:. is a bounded linear operator with domain Band A* is its adjoint, and d'ot a c.l.m. such that if f e m, Af E~, then oo( reduces A and (3'M.) reduce s A*, where (oo{) = of. J_ in the notation of Theorem 4.2. Proof. If A has Bas it s domain, then A* has (B) as its domain. Let P be the projection whose chare.cteri stic manifold is ITT, and P➔~ it s adj oint. .).. Then by Theorem 4.2 (rrt) = ft is the characteristic manifold of P*, for fe. oat, Pf = f and APf = f. [ F, APf - PAf] Consid er the exp ression = [F,Af - P.H] = f!,P Af - PeAf] for FE.(B), fc. rr-tc:B, thus since P.e. = P (! ,APf - PAf] = 0 and AP f = P Af. By Definition 7 .1 oY'i reduces A. [F ,APf] = [F ,PAf], Since -20by the definition of the B.djoint ..1. Thus for FE ( ~) :::o'{, As a result, if FE(c/o1), A-~FE (Mi); F-l~A-l~F == A*F. if FE:(ol ) then P-)(-A*F == O, and A*F is zero for all F e.(ot) . Thus (oat) reduce s A3~, where (oot) == oiJ... Theorem 7.5. Let {PJ be a bounded sequence of projections for which P., < P. ...1; let m..,, oot., .. 1 be their associated rncnifolds; and let A be a linear operator which is reduced by eol 11 (n == 1, 2, ... ) ; then if ooi == I oJ'1.t., then I oo/.. reduces A. 00 Proof. By Theorem 5.2, there exists a pro jection P such tha,too't p == LI ooi p the cha r a cteristic manifold of P. P 11 Af == AP,,, f For f c oo( .., (n == 1, 2, ... ) since oo("' ( n == 1, 2, ..• ) reduces A. In order to prove the theorem it is necesimry to show thc.t PAf == APf • Since there is no element f c o51, which is not contained in one of OOt,.. (n == 1, 2, ... ) n == 1, 2, oat.., PAf f- APf would mean that for some ft: ?rot wd some ... Thus P.Af == APf completing the proof of the theorem . -21Bibliography 1. S. Banach. 2. M. H. Stone. Theorie des Operations LinE3aires. Linear Transformations in Hilbert Space. 3. A. D. Michal and D. H. Hyers. General Differential Geometries with Coordinate Interspace Inner Product. The Tohoku Mathematical Journal 46:309-318 (1940). 4. E. R. Lorch. A Calculus of Operators in Reflexive Vector Spaces. Transactions of the American Mathematic&J. Society 45:217-234 (1939). 5. F. H. Bohnenblust and A. Sobczyk. Complex Linear Spaces. Extensions of Functionals on Bulletin of the American Mathematical Society 44:91-93 (1938). 6. C. Visser. On the Iteration of Linear Opera.tors in Hilbert Space. Koninklijke NederlBndsche Akademie van Wettenscha.ppen (Proceedings) 41:487-495 (1938).