(X) On invariant subspaces oï invertible linear operators

(1)

A N N A LES SO C IETA T IS M ATH EM ATICAE PO LO N AE Series I : COM M ENTATIONES M ATH EM ATICAE X V I I I (1974) RO C ZN IK I PO LS K IE G O T O W A R Z YST W A M ATEM ATYCZNEGO

Séria I : P R A C E M ATEM ATYCZN E X V I I I (1974)

W.

Wo j t y n s k i

(Warszawa)

On invariant subspaces oï invertible linear operators

The purpose of this note is to give a relatively simple proof of the following theorem of J . Wermer [3]:

Th e o r e m.

I f A : X -> X is an automorphism o f a Banach space X such that

where G and Tc are constants independent on n, then A has a non-trivial invariant subspace (i. e. there is a closed linear subspace Y o f X such that О ф Y Ф X and T (Y ) = Y).

P ro o f. Let C°°(T) be the algebra of complex G°° functions defined on the unit circle T = {ze C : \z\ = 1} with the pointwise algebra opera

tions and with the topology of uniform convergence of functions together with all their derivatives. For f e G°°(T), xe X , we let

where f(n ) = (2тг)

¹

J f(t)t nclt, the n -th Fourier coefficient o f /. B y (

¹

) and the elementary properties of the Fourier transform, the A f (x) are well-defined, Af : X X is a continuous linear operator, and the map /-► Af is a continuous representation of the algebra C°°(T) into the algebra* of bounded linear operators on X . In particular

(3) A f f2 — A f oA f2 and A e = A, where e is the identity on T .

»

Let x 0e X be an arbitrary non-zero element. Clearly,

is a closed ideal in the algebra C°°(T) and I Ф G°°(T) (since A is an auto

morphism, О Ф A (xf) = A e(x0), i. e. e i l ) . Using this fact we easily conclude that

(X ) ||Awj| < G \n\k fo r n —

0

, ± 1 , ±

2

, . . . ,

(

2

)

T

I = { f € G°°(T): A f (xo) =

0

}

Zj = {te T : f(t) = 0 for all f e 1}

is a closed non-empty subset of T.

(2)

122 W . W о j t y ii s к i

1

° Assume that Z j has at least two distinct points, say tj_ and tz.

Let C°°(T) be such that Ф 0 for i = 1, 2, but / i

^-/2

= 0.

Since f t 4 I , we have A f .(œf) Ф 0, i. e. Af . Ф 0. By (3), Af oA f = 0. Hence the kernel of at least one of the operators Af . (i =

¹

,

²

) is a non-trivial invariant subspace for each operator Af , in particular for A — A e .

2° Assume that Z7 = {I0}, a one-point set.

Under this assumption the ideal I is of one of the forms:

( 4 J I = { f € C°°(T) : f k)(t0) = 0 for all 0 < к < n },

where n = 1 , 2 , . . . , сю (/(0) = /). An elementary proof of this fact can be left to the reader.

Let

W = {A f(x0): f e C°°(T)}.

B y (3), W is invariant for each operator Af with f e G°°(T). For each /e C°°(T), let B ( f ) = A f (x0). Clearly B : C°°(T) X is a continuous linear operator, I = K e rB and W = B(C°°(T)). Let E = C°°(T)jI and В : E -> X , the quotient operator induced by B . In the case where I is of form (4n) with n < oo, we have dim B = n, and therefore W = B (E ) is a finite

dimensional, hence proper and closed.

If I is of the form (4^), then by a result of E . Borel (cf. [2], p. 120), E is isomorphic to s, the product of copies of the complex plane. I t is known (cf. [1]) that each continuous linear operator from s into a Banach space has finite dimensional range. Thus, also in this case, W = B (E ) is a closed and proper subspace of X .

A c k n o w l e d g e m e n t . The author is indebted to Dr. S. Kwapien for stimulating discussions.

References

[1] C. B e s s a g a , and A. P e lc z y iis k i, On a class of B 0 spaces, Bull. Polon. Acad.

Sci. vol. 5, No. 4 (1957), p. 375-377.

[2] B. S. M itia g in , Aproximative dimension and bases in nuclear spaces (russian) U. M. N. XVI 4 (1961), p. 63 -132.

[3] J. W e rm e r, Existence of invariant subspaces, Duke Matli. J. 19 (1952), p. 615-622.