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148 NAW 5/6 nr. 2 juni 2005 The Problem B.S. Yadav

B.S. Yadav Indian Society for History of Mathematics Department of Mathematics Dehli 110 007 India [email protected]

Beroemde problemen The Invariant Subspace Problem

Heeft elke begrensde lineaire , wer- problem. It apparently arose after Beurl- Next, suppose H is infinite-dimensional kend op een Hilbert ruimte, een niet-triviale ing [1] published his fundamental paper but not separable. Let T be a bound- invariante deelruimte? Het antwoord is posi- in Acta Mathematica in 1949 on invariant ed operator on H. Take a non-zero vec- tief voor zowel eindig-dimensionale ruimtes subspaces of simple shifts, or after von tor x and consider the closed subspace M als voor niet-separabele ruimtes. Het onop- Neumann’s unpublished result on com- generated by the vectors {x, Tx, T2x,...}. geloste probleem voor het geval daar tus- pact operators which we shall discuss in Then M is invariant under T and obvious- senin, dus voor separabele Hilbert ruimtes the sequel. ly M = {0}. Moreover, M does not coin- staat bekend als het invariante deelruimte cide with H as this would contradict that probleem. A history of the problem H is non-separable. Thus every operator Professor B.S. Yadav van de Indian Soci- Let H be any complex and T T on a non-separable infinite-dimensional ety for History of Mathematics in Delhi heeft a on H. An eigenvalue complex Hilbert space H has a non-trivial de afgelopen 25 jaar veel over het onder- λ of T clearly yields an invariant subspace invariant subspace. werp gepubliceerd en geeft hier een histo- of T, namely the kernel of T − λ. So if T What remains to be examined is actu- risch overzicht. has an eigenvalue, the problem is solved ally the invariant subspace problem: does (the special case where T is multiplication every bounded operator T on an infinite- The invariant subspace problem is the by λ being trivial). However, not every dimensional separable complex Hilbert simple question: “Does every bounded bounded operator T on a complex Hilbert space H have a non-trivial invariant sub- operator T on a separable Hilbert space space has an eigenvalue. For example, the space? H over C have a non-trivial invariant sub- shift operator T on ℓ2, the Hilbert space space?” Here non-trivial subspace means of all square-summable of com- The solution for Banach spaces a closed subspace of H different from {0} plex numbers, defined by During the annual meeting of the Amer- and different from H. Invariant means ican Mathematical Society in Toronto in that the operator T maps it to itself. The 1976, the young Swedish mathematician Tx =(0, x0, x1,...) problem is easy to state, however, it is still Per Enflo announced the existence of a 2 open. The answer is ‘no’ in general for for each vector x = (x0, x1,...) ∈ ℓ , does and a bounded linear op- (separable) complex Banach spaces. For not have any eigenvalue. However, if H is erator on it without any non-trivial in- certain classes of bounded linear opera- finite-dimensional, then of course every T variant subspace. Enflo was visiting the tors on complex Hilbert spaces, the prob- on H has an eigenvalue, so the problem is University of California at Berkeley at lem has an affirmative answer. solved for finite dimensional complex vec- that time. However, nothing appeared in It seems unknown who first stated the tor spaces. print for several years and it was only in B.S. Yadav The Invariant Subspace Problem NAW 5/6 nr. 2 juni 2005 149

1981 that he finally submitted a paper for facilities as they did to Read. Beauzamy’s publication in Acta Mathematica. Unfor- paper appeared later in June 1985 in Inte- tunately the paper remained unrefereed gral Equations and . with the referees for more than five years, The ℓ1-example of [6] was further sim- though its manuscript had a world-wide plified by A.M. Davie, as can be found in circulation amongst mathematicians. This Beauzamy’s book (1988). happened, as they say, because the pa- One should not get the impression that per was quite difficult and not well writ- all counterexamples which have been pro- ten. The paper was ultimately accepted duced so far are based directly or indirect- in 1985 and it actually appeared in 1987 ly on the techniques developed by Enflo. with only minor changes: [4]. However, As a matter of fact, a series of papers writ- he had announced his construction of the ten by Read himself after his first paper counterexample earlier in the “Seminaire in 1984 makes a further significant con- Maurey-Schwarz (1975–76)” and subse- tribution to the subject. For example, the 1 quently in the “Institute Mittag-Leffler Re- counterexample that he constructed on ℓ Per Enflo port 9 (1980)”; see [2], [3]. in 1985 is characteristically different from In the meantime, C.J. Read, following and simpler than Enflo’s, and could be The Lomonosov technique the ideas of Enflo, also constructed a coun- counted as a major achievement. Again, in The result of Arveson and Feldman was, terexample and submitted it for publica- yet another paper in 1988, Read construct- in a sense, the climax of the line of ac- tion in the Bulletin of the London Math- ed a bounded linear operator on ℓ1 which tion initiated by von Neumann. Howev- ematical Society. The paper was quickly has no invariant closed sets (let alone in- er, operator theorists were stunned in 1973 refereed and it appeared in July 1984 [5] variant subspaces) other than the trivial when the young Russian mathematician breaking the queue of backlog for publica- ones. Not only is this a stronger result, V. Lomonosov obtained a more general re- tion. A shorter version of this proof was it also gives rise to a new situation: sup- sult: published again by Read in 1986. He also pose that the invariant subspace problem If a non-scalar bounded operator T on a constructed in 1985 [6] a bounded linear is solved in the negative one day (as in the Banach space commutes with a non-zero com- operator on the Banach space ℓ1 without case of Banach spaces), one would ask a pact operator, then T has a non-trivial hyper- non-trivial invariant subspaces. next question: “Does every bounded op- The temptation on the part of Read to erator have a non-trivial invariant closed have precedence over Enflo for solving set?” Normed linear spaces the problem was considered profession- Building on his earlier work, Read pub- A X over the field R(C) ally unethical by many mathematicians. lished in 1997 an example of a quasinilpo- of real (complex) numbers is called Particularly, because his work was essen- tent bounded operator (i.e., lim ||Tn||1/n = a normed linear space if each vector tially based on ideas of Enflo. For ex- 0) on a Banach space without a non-trivial x ∈ X has a ‘’ ||x|| ∈ R, such ample, the French mathematician Bernard invariant subspace. The same result is that ||x|| ≥ 0 and ||x|| = 0 if and on- Beauzamy also sharpened the techniques nicely described in [8]. ly if x = 0, ||αx|| = |α| ||x|| for each of Enflo and produced a counterexam- scalar α and ||x + y|| ≤ ||x|| + ||y|| ple. He presented it at the - Von Neumann’s unpublished result for all x, y in X. Every normed lin- al Analysis Seminar, University of John von Neumann (unpublished) showed ear space X is a with the (VI-VII) in February, 1984. But he de- that every on a Hilbert metric defined by d(x, y) = ||x − y||. clined to publish his result in the Bulletin space has a non-trivial invariant subspace. A Banach space is a normed linear of the London Mathematical Society, al- The first proof of this result was pub- space which is complete (as a met- though the Editors offered him the same lished by Aronszajn and Smith in 1954. ric space). A Hilbert space is a Ba- The result was extended to polynomial- nach space endowed with the addi- ly compact operators by A.R. Bernstein tional structure of an inner product Cyclic vectors and A. Robinson in 1966 using techniques < x, y > such that the norm is related A vector x in H is called a cyclic vec- from non-standard analysis due to Robin- to the inner product by the equality tor of a bounded operator T on H son. Halmos translated their proof into < x, x >= ||x||2. By a bounded oper- if the closure of the span of all Tnx standard analysis. Interestingly, his paper ator on a Banach space X one means equals H. The operator T has no non- appeared in the same issue of Pacific Jour- a linear transformation of X to itself trivial invariant subspaces if and on- nal of Mathematics, just after theirs. In such that there exists a constant K > ly if every non-zero vector is a cyclic 1967, Arveson and Feldman transformed 0 for which ||Tx|| ≤ K||x|| for all x ∈ vector of T: if a vector x is non-cyclic, the result in a still more general form by X. The operator norm of a bounded then the closure of the span of all Tnx essentially chiselling the technique of Hal- operator T, denoted ||T||, is by defi- is a non-trivial invariant subspace of mos: if T is a quasinilpotent operator such nition ||T|| := sup{||Tx||/||x|| ; x = T. And if M is a non-trivial invariant that the uniformly closed algebra generat- 0}. A normed space X is called sepa- subspace, then every non-zero vector ed by T contains a non-zero compact op- rable if it has a countable dense sub- in M is non-cyclic. erator, then T has a non-trivial invariant set. subspace. 150 NAW 5/6 nr. 2 juni 2005 The Invariant Subspace Problem B.S. Yadav

theorem might lead to a solution of the In 1987 Brown, extending his techniques general ‘invariant subspace problem’ and using descriptions of hyponormal op- in the affirmative. However, seven erators due to M. Putinar (1984), proved years after his result, in 1980, Hadvin- that every hyponormal operator with the Nordgren-Radjavi-Rosenthal gave an spectrum having a non-empty interior has example of an operator that does not a non-trivial invariant subspace. commute with any non-zero compact Lastly we mention yet another signifi- operator. cant result in this direction due to Brown, VI. A number of extensions and applica- Chevreau and Pearcy: every contraction tions of Lomonosov’s theorem have whose spectrum contains the unit circle been obtained by several mathemati- has a non-trivial invariant subspace. cians. Heritages of the problem C.J. Read Normal-like non-normal operators For an operator T one denotes by LatT the A bounded operator T on a Hilbert space lattice of all invariant subspaces of T, with invariant subspace (this means, a subspace H is called ‘normal’ if it commutes with set-inclusion as partial order. For a general which is invariant under every operator its adjoint T∗. It is called ‘subnormal’ if operator T, it is extremely difficult to de- that commutes with T). it is the restriction of a normal operator to scribe LatT, particularly when we do not This theorem was quite exciting for an invariant subspace, and ‘hyponormal’ know whether there exists a bounded op- many reasons: if ||T∗x|| ≤ ||Tx|| for all x ∈ H. It is not dif- erator T for which LatT is isomorphic to I. Lomonosov used a brand-new tech- ficult to see that normality ⇒ subnormal- the lattice {0, 1} (this is the invariant sub- nique (namely, an ingenious use of ity ⇒ hyponormality, but the converse is space problem!). However, for certain spe- Schauder’s fixed point theorem), en- true in neither case. An important result in cial operators T, namely the shifts and the tirely different from the line of action operator theory, known as Fuglede’s theo- Volterra operators, the structure of LatT is followed hitherto by other mathemati- rem, states that if T is a normal operator completely known. We now describe this, cians. and S ∈ B(H) is such that TS = ST, then and discuss the role of shifts and their in- II. His result was much stronger than T∗ S = ST∗. variant subspaces in the structure theory what was known so far: every poly- Fuglede’s theorem implies that every of operators, as initiated by G.-C. Rota. nomially compact operator has a non- non-scalar normal operator on a Hilbert ∞ Let {en}n=0 be an orthonormal trivial invariant subspace. space has a non-trivial hyperinvariant for H. The operator U on H such that III. His theorem highlighted another, subspace. To show the existence of Uen = en+1, n = 0, 1, 2 . . . is called the stronger, form of the ‘invariant sub- non-trivial invariant (hyperinvariant) sub- (forward) shift operator. A simple calcu- space problem’: “Does every bound- spaces of non-normal operators satisfying lation shows that its adjoint S is the back- ed linear operator on a Hilbert space certain nice conditions has been a fascinat- ward shift, given by Se0 = 0 and Sen = have a non-trivial hyperinvariant sub- ing subject for operator theorists. One of en−1 for n ≥ 1. We shall be concerned with space?” the most striking results in this direction IV. Many mathematician tried to find al- was due to Scot Brown who showed in ternative proofs of Lomonosov’s the- 1978 that every subnormal operator has a orem, say, by replacing the use of non-trivial invariant subspace. J.E. Thom- Compact operators Schauder’s fixed-point theorem by the son (1986) found a simple and elegant An operator T on a Banach space X Banach contraction principle, but the proof of Brown’s result. Consider the is called ‘compact’ (completely con- theorem stands as it was even to- Hilbert space L2(µ), where µ is a suit- tinuous) if for every bounded subset day. M. Hilden, however, succeeded able positive Borel measure with compact A ⊂ X, the closure T(A) of its im- in proving its special case that every support in the complex . Thomson age is compact in X. An operator non-zero compact operator has a non- makes a decisive use of the fact that a T is called ‘polynomially compact’ if trivial hyperinvariant subspace with- cyclic subnormal operator can be mod- there exists a polynomial p such that out using any fixed-point theorem. elled as a multiplication by z on the clo- the operator p(T) is compact. Every In fact, Hilden assumed without sure of the space of all polynomials in compact operator is obviously poly- any loss of generality a non-zero com- L2(µ). (A bounded operator T on the nomially compact, but the converse pact operator also to be quasinilpo- Hilbert space H is called cyclic if there ex- is not true; examples can be found in tent: if a non-zero compact oper- ists x ∈ H such that the closure of the span ’ A Hilbert space problem ator is not quasinilpotent, then it n of {Tx ; n ≥ 0} equals H.) As a matter book (1967). must have a non-zero eigenvalue, and of fact, Thomson’s method gives rise to a A bounded operator T is ‘quasinilpo- n 1/n hence the eigenspace corresponding more general result: tent’ if limn→∞ ||T || = 0. ∞ to this eigenvalue is a non-trivial hy- Let A be a subalgebra of L (µ) contain- We say that a subalgebra of the alge- perinvariant subspace. Hilden ex- ing z and let H be a subspace of L2(µ). If H bra of bounded operators on X is uni- ploits the quasinilpotence of the com- contains constants and is invariant for A, then formly closed if it is closed with respect pact operator to finish his proof. there is a non-trivial subspace of H that is A- to the operator norm. V. Initially it was felt that Lomonosov’s invariant. B.S. Yadav The Invariant Subspace Problem NAW 5/6 nr. 2 juni 2005 151

the following concrete representations of numerous ramifications both in harmonic U and S. analysis and . Mainly Let L2 = L2(C, µ) be the Hilbert space there have been three directions: of all square-integrable functions defined I. Replacing the Hardy space of scalar- on the unit circle C, where µ is the normal- valued functions by the Hardy space ized Lebesgue measure on C(i.e.µ(C) = of vector-valued functions; n 1). If for each integer n, en = en(z) = z , II. Extending Beurling’s characterization ∞ then {en}n=−∞ is an orthonormal basis to the Hardy space of scalar-valued of L2. The Hardy space H2 is the closed functions on the torus; subspace of L2 generated by the vectors III. Viewing (i) and (ii) in the sense of de {e0, e1, e2,...}. We see that the multipli- Branges, which puts Beurling’s theo- 2 cation by e1(z)= z on H is U. rem as well as its vector-valued gen- As a second example, let ℓ2 be the eralizations due to Halmos (1961) and Hilbert space of all square-summable com- others in a more general setting. ∞ plex sequences x = (xn)n=0. Then U and 2 S on ℓ appear as Ux =(0, x0, x1,...) and Weighted shifts Sx =(x1, x2, x3,...). Shifts form an important class of oper- ators. They have been rightly called Beurling’s theorem and its ramifications the ‘Building Blocks’ of operator theory. In 1949, A. Beurling characterized the in- Many important operators are, in a sense, The work of the Swedish mathemati- variant subspaces of the shift operator on ‘made up’ of shifts, for example, every cian Arne Beurling (1905–1986) has the Hardy space H2 on the unit circle. His pure isometry is a direct sum of shifts and been pace-setting in many directions result is: every contraction with powers strongly in abstract harmonic analysis, func- If M is an invariant subspace of the shift tending to zero is a ‘part’ of a backward tional analysis and operator theory. operator on the Hardy space H2 on the unit shift. When conscripted in 1931, he recon- circle C, then there exists an inner function More importantly, shifts serve as an un- structed Swedish cryptology ingeni- φ on C (this means that φ is measurable and ending source of counterexamples. Read ously leading to information vital for |φ(z)| = 1 almost everywhere on C), such uses a shift to construct his counterexam- the survival of during the that M = φH2. ple of a bounded operator on a Banach World War II. Although appointed If both φ1 and φ2 are such functions, then space without a non-trivial invariant sub- professor at the Institute of Advanced φ1/φ2 is equal to a constant function almost space. Study, Princeton in 1954, he always everywhere. Let H be a Hilbert space with an or- missed the right social environment ∞ As Beurling’s theorem showed an inter- thonormal basis {en}n=0 and let w = and would not even apply for a Green ∞ play between the theory of functions and {wn}n=1 be a of non-zero com- Card. A review of this book appeared the operator theory, it has naturally had plex numbers. Consider the weighted for- in September 2003 in the Notices of ward shift Tw: the AMS. Codebreakers: Arne Beurling and the Swedish Crypto Program during World War II, Bengt Beckman, AMS 2002, ISBN 0-8218-2889-4. Twen = wn+1en+1, n = 0, 1, 2, . . .

and the corresponding weighted back- ward shift Sw: can show that LatSw consists of Mn’s on- S e = 0, ∞ w 0 ly: if a weight sequence w = {wn}n=1 is Swen = w¯nen−1, n = 1, 2, 3, . . . such that {|wn|} is monotonically decreas- ing and A weighted forward shift is the adjoint of a weighted backward shift and vice- ∞ ∑ |w |2 < ∞ versa. Note that a subspace M is invari- n , n=0 ant under an operator T if and only if its orthogonal complement M⊥ is invariant ∗ then every non-trivial invariant subspace under T . Hence determining LatSw is in LatSw is some Mn. equivalent to determining LatTw. This result is due to N.K. Nikolskii Let Mn denote the closed subspace −n (1965). The case wn = 2 was obtained spanned by in 1957 by W.F. Donoghue.

{e0, e1,... en}. Volterra integral operators Then Mn ∈ LatSw for all n. Under certain Consider the Volterra integral operator V John von Neumann (1903–1957) conditions on the weight sequence w, one defined on L2(0, 1) by 152 NAW 5/6 nr. 2 juni 2005 The Invariant Subspace Problem B.S. Yadav

x (V f )(x)= f (t)dt, 0 ≤ x ≤ 1, case may be used to obtain a functional- the image M of A is closed and Sw A = AT. Z0 analytic proof of the famous classical This implies M is an invariant subspace of Titchmarsh convolution theorem (G.K. Ka- Sw and T is similar to Sw|M. 2 for all f ∈ L (0, 1). This operator is an- lisch, 1962). If the spectral radius, other one whose invariant subspaces have been characterized. For each α ∈[0, 1], let Rota’s models of linear operators 1/n r(T) := lim ||Tn||2 , 2 By a part of an operator T on a Hilbert n→∞ Mα = { f ∈ L (0, 1) : f = 0   space H, we mean the restriction T|M of almost everywhere on[0,α]}. T to an invariant subspace M of T. of T is less than 1, then the conditions of Let l2(H) denote the Hilbert space the above result are satisfied for the con- Obviously, Mα ∈ LatV for all α ∈ [0, 1]. In of all square-summable sequences x = stant sequence w = 1. This observation fact, n (x0, x1,..., xn,...) in H. Take a bounded leads to the result of G.-C. Rota (1960): LatV = {Mα : α ∈ [0, 1]}. sequence w = (wn) of positive real num- If a bounded operator T on a Hilbert space 2 bers. The backward shift Sw on l (H) is H has spectral radius r(T) < 1, then T is sim- This was proven by J. Dixmier (1949) in given by ilar to a part of the standard backward shift on case of the real space L2(0, 1). W.F. Donog- l2(H). In particular, this holds for a strict con- hue and M.S. Brodskii independently set- S x =(w x , w x ,..., w x ,...). traction T, i.e. if ||T|| < 1. tled it in 1957 for the complex space w 1 1 2 2 n+1 n+1 Since any bounded operator can be L2(0, 1). Put β = 1 and β = w w ... w , for ‘scaled’ so as to be a strict contraction, Ro- These results have been extended to in- 0 n 1 2 n n ≥ 1. One has the following result. ta’s work yields a reformulation of the in- tegral operators K on L2(0, 1) defined by Suppose T is a bounded operator on H and variant subspace problem: Are the mini- x ∞ −2 n 2 < ∞ mal non-zero invariant subspaces of backward (K f )(x)= k(x, y) f (y)dy, 0 ≤ x ≤ 1, ∑ βn ||T || . Z0 n=0 shifts one-dimensional? 2 Then T is similar to a part of Sw on l (H) More details on this work initiated by for all f ∈ L2(0, 1), where k(x, y) is in the following sense: define A : H → l2(H) Rota may be found in [7]. k a square-integrable function on [0, 1] × by Ax = {β−1x, β−1Tx, β−1T2x,...}, then [0, 1]. The characterization of LatK in this 0 1 2

References 1 A. Beurling, ‘On two problems concern- 4 P. Enflo, ‘On the invariant subspace prob- 7 B.S. Yadav and R. Bansal, ‘On Rota’s mod- ing linear transformations in Hilbert space’, lem for Banach spaces’, Acta Math. 158 els of linear operators’, Rocky Mountain J. of Acta Math. 81 (1949), 239–255. (1987), p. 213–313. Math. 13 (1983), p. 253–256. 2 P. Enflo, ‘On the invariant subspace prob- 5 C.J. Read, ‘A solution to the invariant sub- 8 P. Rosenthal, ‘Featured Review of C.J. lem for Banach spaces’, Seminaire Maurey- space problem’, Bull. London Math. Soc. 16 Read’s Quasinilpotent Operators and the In- Schwarz (1975–1978). (1984), p. 337–401. variant Subspace Problem’, MR 98m:47004. 3 P. Enflo, ‘On the invariant subspace prob- 6 C.J. Read, ‘A solution to the invariant sub- lem for Banach spaces’, Institute Mittag- space problem on the space ℓ1’, Bull. London Leffler, Report 9 (1980). Math. Soc. 17 (1985), p. 305–317.