Invariant Subspaces for Certain Commuting Operators on Hilbert Space

Annals of Mathematics

Invariant Subspaces for Certain Commuting Operators on Hilbert SpaceAuthor(s): Allen R. BernsteinSource: Annals of Mathematics, Second Series, Vol. 95, No. 2 (Mar., 1972), pp. 253-260Published by: Annals of MathematicsStable URL: http://www.jstor.org/stable/1970799 .

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Invariant subspaces for certain commuting operators on Hilbert space

By ALLEN R. BERNSTEIN1

1. Introduction

It is the purpose of this paper to establish the existence of common invariant subspaces for certain commuting bounded linear operators on Hilbert space. Of course for a single operator on Hilbert space (or even Banach space) the property of being compact or just polynomially compact guarantees that the operator possesses an invariant subspace ([1], [2], [5]). However the problem of establishing common invariant subspaces for commuting operators has remained intractable even, say, when both operators are compact. The main result presented here (Corollary 4.1) does make progress on the problem of pro- viding such common invariant subspaces although the conditions necessary to guarantee their existence are stronger than compactness. However these conditions are required only of one of the operators.

The method of proof here is similar to those presented in [1], [2], and [5]. All of those proofs proceeded by approximating the given operator by finite- dimensional operators and then using the fact that linear operators on finite- dimensional space have a whole chain of invariant subspaces. The latter two proofs were set in the theory of Non-Standard Analysis where the approxi- mation of the given operator was obtained by restricting it to a suitable subspace whose dimension was an infinite integer.

For the commuting operator case dealt with here, of course an equally strong result holds in the finite-dimensional case, that is there is a whole chain of common invariant subspaces. However if an attempt is made to apply this result to assist in the infinite-dimensional case a difficulty presents itself immediately. If one starts with commuting operators A, B on Hilbert space and approximates them by finite dimensional operators A', B' then there is no guarantee that A' and B' will still commute. The best one can hope for in general is that A' and B' nearly commute, that is to say that A'B' - B'A' I is small.

Therefore it becomes necessary to first obtain a result for finite dimensional operators which almost commute. This is done in Section 2 where it is

1 Financial support received from National Science Foundation, GP 11263.



254 ALLEN R. BERNSTEIN

shown that such operators have a chain of common almost invariant subspaces. Once this is done the remainder of the proof uses the theory of Non- Standard Analysis following a line similar to that in [5]. For Sections 3 and 4 we shall presume the reader is familiar with either [2], [5] or Chapter VII of [6]. The last reference also contains a complete treatment of the theory of Non-Standard Analysis. It is worth remarking that while the theory of Non-Standard Analysis provides a natural setting for this problem, it would be possible to adapt the proof given here so that only standard techniques appear.

2. Almost commuting matrices

We denote by C the set of complex numbers and by He an n-dimensional inner product space over C. For any subset X of Ha, V X denotes the linear subspace generated by X, and X- denotes the set of elements of He orthogonal to everything in X. For linear operators A on He we denote by I I A I I the norm of A, IAII = sup {1IAx I xII = 1}. If s > 0 then an element z of He is called an s-eigenvector for A if I I Az - Xz ? < s I I zI for some X e C. Then z is a common s-eigenvector for two operators A and B if there are X, , e C such that IIAz - XZ < se zII and IIBz - zII < s Iz I. It is worth pointing out that if A and B have a common s-eigenvector of norm 1 then it does not necessarily follow that A and B have eigenvectors of norm 1 which are close to each other (cf. [4]). A linear subspace M of He is called s-invariant under A if given x e M, Ax-zz ?< s for some z e M.2 M is a common s-invariant subspace for A and B if it is s-invariant under both A and B.

In order to prove the basic result of this section (Theorem 2.2) we make use of the following theorem whose proof may be found in [4].

THEOREM 2.1. Let A and B be linear operators on Ha, n > 2, with Bl <

l and let 0<s<1. Suppose IIAB-BAII <?fl/(l + S + 62?+ + ?n-2). Then A and B have a common s-eigenvector of norm 1.

THEOREM 2.2.3 Let A and B be linear operators on a complex finite-di- 2 The definition of M being an e-invariant subspace can also be formulated as follows: If P is

the projection with range M, then II AP - PAP I1 < e. Then an e-eigenvector is one whose span is e-invariant.

3As the referee has pointed out, the principal new accomplishment of Theorem 2.2 is the quantitative one; qualitatively it is an immediate consequence of known facts. That is: since "almost commuting matrices are near commuting matrices," there is some function of C and n with the property that as soon as 11 AB - BA I is less than it, then there exist commuting con- tractions Ao and Bo with A - Ao I I1, 11 B-Bo - B s. Find a chain of common invariant subspaces for Ao and Bo, and observe they are 3&-invariant for A and B. Since Theorem 2.1 is qualitatively (if not quantitatively) a special case of Theorem 2.2, this comment applies to Theorem 2.1 also. Similar comments can be made about Theorem 4.1 and its corollary.



INVARIANT SUBSPACES FOR COMMUTING OPERATORS 255

mensional inner-product space Ho of dimension n with A ?ll < 1, I B < 1. Let 0 < s < 1 and suppose that AB - BA ? < (s/121f)". Then there exists a chain of common s-invariant subspaces for A and B, E0 c E1 c * c En, where for each i, Ei is i-dimensional.

Proof of Theorem 2.2. The proof is by induction on n. For n 1 the theorem is trivial. Now suppose the theorem is true for n. Let H,~1 be an (n + l)-dimensional inner-product space and let A and B be linear operators on H,+,, AII < 1, IIBII < 1, with AB - BAl < (,s/12fl+1)(fl+1 '. For ease of notation, given any positive integer lo and any e C we denote by dk($) the quantity (e/12 )k. Thus

/ S (n+1)!

_ (n+l) 2 HAB - BA~ < dl+(s) (= +)

_ ____ ((;n+)fl l1 n -6 !

12n* 6 12'n 6n|

(dn( ))2 (21)(dn(2)) ( 6 ? 2n+L

- 1 + Us + . .. + en-1)\ 3) Hence we may apply Theorem 2.1 with s replaced by (dn(s/2))/3 to obtain an element eeHn+1, leH = 1, such that for some X, feC, HAe - Xe < (dn(s/2))/3 and Be - e-e ? (dn(s/2))/3.

Now let G = {e}l' and let P be the projection of H, onto G. Then let At and B' be the restrictions of PA and PB to G. Thus A' and B' are linear op r- ators on the n-dimensional space G. We wish to show that I I A'B' - B'A' < dn(s/2). To this end let x be an element of G of norm 1. Then let

Bx = ble + y Ibll < 1, y I e, Ilyll < 1 Ay =ale + z jal< 1, z I e, IIzII< 1

Then B'x = y and A'B'x z. On the other hand ABx = blAe + Ay - blAe + ale + z. But I I blAe -ble b1 < I b, I I I Ae - Xe I I < (dn(s/2))/3. Therefore

ABx = (blX + al)e + z + blAe - biXe = (blx + al)e + z + X7

where I 11 < (dn(s/2))/3. Thus using the fact P is the linear projection onto {e}l' we have




PABx = P((bX + al)e) + P(z) + P(rl) = O + z + P(Y).

Recalling that A'B'x = z we obtain

PABx - A'B'xH H =lz + P(r)-z1I = zH P(rl) II

<_ 11PII 11410 ' 1.11ul0 <_ 2

-3

An identical argument shows that I I PBAx - B'A'x I I _ (d,(s/2))/3. To get the third inequality needed, we use the hypothesis of the theorem, obtaining

d? (-

JJPABx-PBAxIP < IJPII IIAB-BAII IxJI < dn+1(s) < 2

Thus putting these together we get

IA'B'x - B'A'x ? I I I A'B'x - PABx I I + I I PABx - PBAx I + I I PBAx-B'A'x I 1

dn ___ d,(j- ?d__ < (2) + ( 2 ) dn( 2)

It follows from the last inequality that we may apply the induction hypothesis to conclude that there is a chain of subspaces of G, FO c F1 c * * * cF, where for each i Fi is an i-dimensional common s/2-invariant subspace for A' and B'. Now define E = {O} and Ek = VWe, Fk-1} for 1 ? k ? n + 1. Then E0 c F, c * cEE,+E and each Ek is a #-dimensional Hn+1. To complete the proof we wish to show that each Ek is s-invariant under A and B. Excluding the trivial case kl=O, let 1< k<c +?1 and let xGEk, IIx I=1. Then we may write

x = ae +x2 Jal < 1, lIx2hI _ 1, X2eFk--.

Thus Ax = aAe + Ax2. Since P is the projection of H,+, onto {el' we may write Ax2 = ,e + PAx2 = 1e + A'x2 for some a C C, so

Ax = aAe + Se + A'x2 -

Now I Ae - Xe I _ (dn(s/2))/3 < s/2 and since Jal ? 1, 1 Ae - aXe ? ,S/2. Fur- thermore since Fkl is s/2-invariant under A', IIA'x2 -Y2 < 's/2 for some ?2 e Fkl c Ek. If these last two inequalities are applied to the preceding ex- pression for Ax we obtain




Ax-(axe + e+y2) ?y a < 11aAe-aXeJ + HJA'x2-y,1

= 2 2

where (a\e + fe + Y2) ? Ek. Thus Ek is s-invariant under A. An identical argument shows that Ek is -invariant also under B which completes the proof.

3. Non-standard Hilbert space

Hereafter we shall use mostly the same notation as in [5] (alternatively [2] or [6]). Thus H will denote a separable Hilbert space over the complex numbers C; R and Z denote the sat of reals and the positive integers, respec- tively. We then select a suitable higher order language L and take a non- standard extension relative to L. Thus we wind up with a non-standard Hilbert space *Hover the non-standard complex numbers *C; similarly we obtain *R and *Z. For x, y E *H, we write x y y when 11 x - y is infinitesimal and for a near-standard point x E *H we denote by ox its standard part. Let {en}, n E Z, be an orthonormal basis for Hand this extends to a non-standard sequence *{en} whose vth term is denoted by e, for any v E *Z. Then for v *Z - Z, H, denotes the subspace of *H spanned by {e,, e2, . . , e,} and Pp denotes the projection of *H onto H,. Then x - P~x for near-standard x E *H. Let T be a bounded linear operator on H. By an invariant subspace M for T we shall mean a proper (i.e., #z {0}, 7 H) closed linear subspace M of H such that Tx E M whenever x E M. We denote by * T the extension of T to *H and by T, the restriction of P* T to H,.

Now let E be a linear subspace of H, so that it has as non-standard dimension some integer (finite or infinite) d(E) E *Z. We associate with E the subspace 0E of H defined by E -- {x e HI x - y for some y E}. Then if PE is the projection of *H onto H,, x ? 0E if and only if x - PEX. The proofs of the following two theorems may be found in [5] or [6].

THEOREM 3.1. If E is a linear subspace of H, then 0E is a closed linear subspace of H.

THEOREM 3.2. If E and E' are linear subs paces of H, with d(E') = d(E) + 1 then any two points of 0E' are dependent modulo 0E.

Next we make the following definition. If E is a subspace of H, we say E is almost invariant under T, if given any x E E there is a y E E such that Tx - y.

THEOREM 3.3. If E is almost invariant under T, then 0E is invariant under T.




Proof. Let z e OE so z - z'e E. Then T~z' - y' for some y'e E. But * Tz' - * Tz = Tz since * T has finite norm so applying P, to the near-standard *Tz' we get y' ~- T~z' = P,*Tz' - *Tz' - Tz. Therefore Tz E 0E wrhich shows 0E is invariant under T.

4. The main result

Let H be a separable Hilbert space over the complex numbers C. (Ob- serve that if H is non-separable and if A and B are bounded linear operators on Hthen foranyxe H, I I x O, the set {T1T2. * TnxI Ti = A or Ti = B each i} generates a proper closed common invariant subspace for A and B.) Let {en} be an orthonormal basis for H and for each n E Z let HII = V {el, * * , en} with Pa the projection of H onto He. For any linear operator T on Hand any n E Z we denote by T, the restriction of PnT to HIe. T, is called the compression of Tto H.

We can now state-the main theorem.

THEOREM 4.1. Let A and B be bounded linear operators on Hwith A compact. If IIAB - BAII/(nI)! -+0 as n o then A and B have a common proper invariant subspace.

Proof. Clearly we may suppose that IAH _ 1, IBH < 1. Then for each n, H1A, _ I P. I AII < 1 and similarly I Bn I < 1. For each n let An =

I I AnBn- BnAn 1 Thus we are assuming that 1/?1n+' ! 0 as n - . Then by means of the equality

12

we see that 12n`ln' 0 as n oo. Now let v E *Z - Z; thus 12' = 4 V 0. Then if we look at the operators A, = PA and B, = PB restricted to H, we have IIAB, - BA, = a, = (C/12') '.

Therefore we may apply Theorem 2.2 to conclude that there is a chain of subspaces of H,, E, c E1 c * * * c E, where for each j, d(Ej) = j and Ej is C- invariant under both A, and B,. Since C is infinitesimal this says Ei is almost invariant under both A, and B,. Thus by Theorems 3.1 and 3.3 it follows that each 0Ej is a common invariant closed linear subspace for A and B. The only remaining task is to show that for some j, 0Ej is non-trivial, i.e., unequal to {0} or H. The argument used is like that in [5].

Let Qj be the projection of *Honto Ej, j = 0, 1, ..., , so Q,-= P. Choose any E ? H, 11 II = 1, such that Ad # 0. Denote by A' the operator P,*A on *H so that A, is A' restricted to H,. Then A'd = P,*Ad = PA: A: # 0 so I A': 11 > r for some standard positive r. Consider the expressions




rj = I]JA' - A'Qj~ll, j = .0 1, 2, . . .e, 2)

and note that rj < I I A' I I I 1 -Qj: I 1. Since Q0 is the projection onto {0} we have ro = IIA' II so ro > r. Also since d is standard I - Q- I = II -P>:I is infinitesimal, hence r, < r/2. It follows that there exists a smallest positive integer ,[ which may be finite or infinite such that r, < r/2 but r,_1 > r/2.

Now 0E,,_1 cannot coincide with H, in particular it cannot include d. For if it did then II -Q,: I would be infinitesimal so ru1 which is bounded by I A' I I I 1 - Q,1I I would be infinitesimal contrary to the choice of 1a.

On the other hand 0EP cannot reduce to {0}. Consider the point C = A'Qd. A is compact and Qua has finite norm so *AQUA is near-standard (cf. Theorem 3.2 in [5]). Then C = P,*AQp - *AQy: so C is near-standard and possesses a standard point 06. Now Qua e E. and E, is almost invariant under A. Thus C is infinitesimally close to an element of E:, which means o6 E 0E,. If 0? = 0 then we would have

r = II A'S - A'QPd I I > II A'. II-I I A'Q I I > r -1 C

where IICII is infinitesimal. This would imply that r. > r/2 contrary to the choice of 1a. We conclude that 0E, contains a point different from 0, namely

Both 0E,_1 and 0E, are invariant for A and B. If neither were proper invariant subspaces we would have E,_1 {}= 0 and 0E, = H. But this contra- dicts Theorem 3.2 which completes the proof of the theorem.

For commuting operators the burden of the additional hypotheses may be shifted to one of the operators as in the following corollary.

COROLLARY 4.1. Let A and B be commuting bounded linear operators on H. Suppose A is compact and IIPnAPn, - AHl(?)+-" GO as n o o. Then A and B have a common proper invariant subspace.

Proof. Clearly we may suppose II A II < 1, BI < 1. Let n e Z and let x e Ha, IlxH = 1. Then AnBnx = PnAPnBx so

AnBnx - P-PABx I (PnAPn - PnA)Bx |= P(PnAP,, - A)Bx < 11P.11 H1P.AP - All IIBlJ llxHl =lPnAPn - All

Similarly BnAnx = PnBPnAx = PnBPnAPnx so

IBAnx - PnBAxll = HIPfB(PflAPI, - A)xJI ?< IIPnH IIBH IIP.AP - All IIIxHl = IHPnAPn - All

But since AB = BA, PnAB = PBA. Therefore




l(A.B.- B.A.)x I I A.Bnx - PnABxH + lPnBAx - BnAx

211P.AP. - All

Thus

IIABn - BnAnll < 211PAPn - All

This last inequality shows that the hpyotheses of the corollary imply the hypotheses of the theorem. Hence A and B have a common proper invariant subspace.

UNIVERSITY OF MARYLAND, COLLEGE PARK, MARYLAND

REFERENCES

[1] N. ARONSZAJN and K. SMITH, Invariant subspaces of completely continuous operators, Ann. of Math. 60 (1954), 345-350.

[2] A. BERNSTEIN, Invariant subspaces of polynomially compact operators on Banach space, Pac. J. of Math. 21 (1967), 445-463.

[3] , Almost commuting matrices and invariant subspaces in Hilbert space, Notices of the Amer. Math. Soc. 17 (1970), 439-440.

[4] , Almost eigenvectors for almost commuting matrices. S.I.A.M. J. on Appl. Math. to appear.

[5] and A. ROBINSON, Solution of an invariant subspace problem of K. T. Smith and P. R. Halmos, Pac. J. of Math. 16 (1966), 421-431.

[6] A. ROBINSON, Non-Standard Analysis, North Holland, Amsterdam, 1966.

(Received June 16, 1971)



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Invariant Subspaces for Certain Commuting Operators on Hilbert Space