PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 126, Number 2, February 1998, Pages 553–560 S 0002-9939(98)04168-9

A REPRESENTATION THEOREM FOR SCHAUDER BASES IN HILBERT SPACE

STEPHANE JAFFARD AND ROBERT M. YOUNG

(Communicated by Dale Alspach)

Abstract. A sequence of vectors f1,f2,f3,... in a separable Hilbert space { } H is said to be a Schauder basis for H if every element f H has a unique norm-convergent expansion ∈

f = cnfn. If, in addition, there exist positive constantsX A and B such that 2 2 2 A cn cnfn B cn , | | ≤ ≤ | |

then we call f1,f2,fX3,... a Riesz X basis. In theX first half of this paper, we { } show that every Schauder basis for H can be obtained from an orthonormal basis by means of a (possibly unbounded) one-to-one positive self adjoint oper- ator. In the second half, we use this result to extend and clarify a remarkable theorem due to Duffin and Eachus characterizing the class of Riesz bases in Hilbert space.

1. The main theorem

A sequence of vectors f1,f2,f3,... in a separable Hilbert space H is said to be a Schauder basis for H{ if every element} f H has a unique norm-convergent expansion ∈

f = cnfn. If, in addition, there exist positive constantsX A and B such that 2 A c 2 c f B c 2, | n| ≤ n n ≤ | n|

then we call f1,f2,f3,...X a Riesz basis.X Riesz basesX have been studied intensively ever since Paley{ and Wiener} first recognized the possibility of nonharmonic Fourier expansions

iλnt f = cne for functions f in L2( π, π) (see, for example,X [2] and [5] and the references therein). In the first half of− this paper, we show that every Schauder basis for H can be obtained from some orthonormal basis by means of a (possibly unbounded) one-to- one positive self adjoint operator. In the second half, we use this result to extend

Received by the editors April 17, 1996 and, in revised form, August 22, 1996. 1991 Mathematics Subject Classification. Primary 46B15; Secondary 47A55. Key words and phrases. Schauder basis, Riesz basis.

c 1998 American Mathematical Society

553

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and clarify a remarkable theorem due to Duffin and Eachus [1] characterizing the class of Riesz bases in Hilbert space. Theorem 1. Let H be a separable Hilbert space and f a Schauder basis for H. { n} Then there exists an orthonormal basis en and a one-to-one positive self adjoint transformation T of the Hilbert space such{ that} T : e f for n =1,2,3,... . n → n Proof. Define a linear transformation S on H by setting

∞ S(x)= (x, fn)fn n=1 X for those points x in H for which the series converges weakly.If fn forms a conditional basis for the Hilbert space, then S may not be bounded{ and} may not even be defined for all elements of the Hilbert space. Let us use the symbols D(T ) and R(T ) to designate the domain and range, respectively, of a linear transformation T . We are going to show that S is self adjoint, positive, and one-to-one. In order to define the adjoint transformation S∗, we must first show that D(S)isevery- where dense in H.Let gn be the system in H biorthogonal to fn ,sothat (f,g )=δ .ThenS:{g }f (n=1,2,3,...) and therefore every{ finite} linear n m nm n→ n combination c1g1 + +cngn also belongs to D(S). Since the system biorthogonal to a Schauder basis··· is itself a Schauder basis (see [5], p. 29), the set of all such combinations forms a dense subspace of H, and the adjoint of S can be defined. It follows at once from the definition that S is symmetric, (Sx,y)=(x, Sy)

for every x and y in D(S), and hence that D(S∗) D(S). Now the domain of S∗ consists, by definition, of all y for which (Sx,y) is⊇ continuous for x in D(S). Since D(S) is dense, there is a uniquely determined point y∗ in H such that (Sx,y)= (x, y∗) for every x in D(S). Accordingly,

∞ (x, fn)(fn,y)=(x, y∗) n=1 X whenever x D(S)andy D(S∗). It is crucial to show that the series on the ∈ ∈ left converges for all x in the Hilbert space. For this purpose, take x = gn in the expansion above, so that (f ,y)=(g ,y∗)forn=1,2,3,....Since f is a n n {n} Schauder basis, the biorthogonal expansion y∗ = (y∗,gn)fn is valid in norm, and hence weakly. Thus, the series (y∗,gn)(fn,x) is convergent for every x in H and P y in D(S∗) and hence for these values of x and y, P ∞ (y,fn)(fn,x) n=1 X is also convergent. This shows that (y,f )f converges weakly whenever y n n ∈ D(S∗), and hence that D(S∗) D(S). Thus, D(S)=D(S∗)andSis self adjoint. To see that S is one-to-one,⊆ supposeP that S(x) = 0, so that

∞ (x, fn)fn = 0 (weakly). n=1 X

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Taking the inner product of both sides with gn (n =1,2,3,...), we see that (x, fn)=0forn=1,2,3,..., and, since the fn are complete, x must be zero. Thus, S is one-to-one. That S is positive follows at once since

∞ (Sx,x)= (x, f ) 2 0. | n | ≥ n=1 X Now, the null space of an operator is the orthogonal complement of the range of its adjoint, and since S is one-to-one and self adjoint, its range is dense in H. 1 Therefore, S− is also one-to-one, positive, and self adjoint. It follows that there exists a unique one-to-one, positive, self adjoint transformation A such that

2 1 A = S− (see [3], p. 265). It is to be shown that the system e defined by { n} en = A(fn)(n=1,2,3,...) forms an orthonormal basis for H. That the en are well-defined is clear, since

1 f R(S)=D(S− ) D(A). n ∈ ⊂ Orthonormality follows at once from

2 (en,em)=(Afn,Afm)=(fn,A fm) 1 =(fn,S− fm)=(fn,gm)=δnm.

1 To establish completeness, let x be a fixed but arbitrary element in D(S− ). 1 1 Then S− (x) R(S− )=D(S), and hence we can write ∈ ∞ 1 1 x = S(S− x)= (S− x, fn)fn 1 X ∞ ∞ = (Ax, Afn)fn = (Ax, en)fn 1 1 X X where each of the series is weakly convergent to x.Ifxndenotes the nth partial sum of the last series, then

n Ax = (Ax, e )e P (Ax), n m m → 1 X where P denotes the orthogonal projection onto the subspace generated by the en (n =1,2,3,...). Since every self adjoint transformation is closed, the relations x x weakly, n → Ax P (Ax) n → imply that P (Ax)=Ax.Thus,Preduces to the identity transformation on a dense subspace of H and hence on all of H. This shows that the system en is complete and so forms an orthonormal basis for H. The required transformation{ } T 1 is obtained by taking T = A− . This completes the proof of the theorem.

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2. On a classic characterization of Riesz bases Let us now turn our attention to the class of Riesz bases in a separable Hilbert space H.If f is such a basis, then for any orthonormal basis e , the mapping {n} { n} T : e f (n =1,2,3,...) n → n can be extended to a bounded invertible operator on all of H. Riesz bases constitute the largest and most tractible class of bases known. It is extremely difficult to exhibit at least one bounded basis for a Hilbert space that is not equivalent to an orthonormal basis. An example in L2( π, π) was discovered by Babenko (see [4], p. 351). − The simplest and perhaps the most obvious way of ensuring that a bounded linear transformation T is invertible is by showing that it is “close” to the identity transformation I in the sense that I T < 1. This is a stringent requirement to place on a linear operator and onek − mightk well suppose that the class of Riesz bases is very small. A remarkable theorem due to Duffin and Eachus [1] shows, surprisingly, that just the opposite is true: Every Riesz basis for H is obtained in essentially this way. Theorem 2. Let H be a separable Hilbert space and f a Riesz basis for H. { n} Then there exists an orthonormal basis en and a positive number ρ such that the mapping { } T : e f (n =1,2,3,...) n → n can be extended to a bounded invertible operator on all of H such that I ρT < 1. k − k Let us briefly review the construction of the orthonormal basis e . { n} Start with any orthonormal basis ϕn for H and let V be the unique bounded invertible{ operator} on H that maps ϕn fn. Factorize V by writing V = PU,whereP is a positive→ operator and U is a unitary operator. Then en = U(ϕn),n=1,2,3,..., is the desired basis. We are going to show that the basis e so constructed is independent of the { n} choice of ϕn and that it is, in fact, the same basis guaranteed by Theorem 1. We have{ }

V (x)= (x, ϕn)fn and X

V ∗(x)= (x, fn)ϕn, and hence X

VV∗(x)= (x, fm)(ϕm,ϕn)fn n m XX = (x, fn)fn = S(x), n X where S is the same operator used in the proof of Theorem 1. We shall continue 1 to use the symbol A to designate the positive square root of S− .Since fn is a Riesz basis, both A and S are bounded invertible operators defined on all{ of }H.

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1 Using the canonical factorization V = PU, we find V ∗ = U ∗P ∗ = U − P ,and hence 2 S = VV∗ =P . Thus, 1 1/2 P − = S− = A and 1 U = P − V = AV. Since V : ϕ f , it follows that n → n U(ϕn)=A(fn)(n=1,2,3,...). Thus the two algorithms of Theorems 1 and 2 generate the same orthonormal basis. The next theorem shows that this distinguished basis is “closest” to fn in a certain precise sense. { } Theorem 3. Let f be a Riesz basis for a separable Hilbert space H and let { n} en be the distinguished orthonormal basis guaranteed by Theorems 1 and 2. If T denotes{ } the operator on H for which T : e f (n =1,2,3,...), n → n then there exists a positive number ρ with the following property: If hn is any other orthonormal basis for H and G the operator for which { } G: h f (n =1,2,3,...), n → n then I λG I ρT k − k≥k − k for any positive real number λ. Proof. Since f is a Riesz basis, we know from Theorem 1 that the mapping { n} ∞ S(x)= (x, fn)fn n=1 X defines a positive self adjoint invertible operator on all of H and that T is the 1 positive square root of S.Ifα=1/ S− and β = S ,then k k k k αI S βI. ≤ ≤ Here, as usual, the order relation A B between symmetric operators A and B means that (Ax, x) (Bx,x) for all elements≤ x of H. We define ≤ 2 ρ = . √α + √β Select an arbitrary orthonormal basis h for H and let G be the operator for { n} which G(hn)=fn (n=1,2,3,...). Then for any x in H, x = (x, hn)hn and hence P ∞ G(x)= (x, hn)fn n=1 X

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and

∞ G∗(x)= (x, fn)hn, n=1 X so that

GG∗ = (x, fm)(hm,hn)fn = (x, fn)fn = S(x). n m n X X X 1 We can now calculate G and G− .Wehave k k k k 2 2 G = G∗ =sup(G∗x, G∗x) k k k k x=1 kk =sup(GG∗x, x)= sup(Sx,x)= S , x=1 x =1 k k kk k k 1 1 and hence G = √β. A similar calculation applied to G− shows that G− = 1/√α. k k k k Let ε be an arbitrary positive number and select unit vectors x1 and x2 such that β ε G(x ) β − ≤k 1 k≤ and p p √α G(x ) √α+ε. ≤k 2 k≤ For any positive number λ,seta= (I λG)(x ) and b = (I λG)(x ) .Then k − 1 k k − 2 k a x λG(x ) 1 λ β λε, ≥|k 1k−k 1 k| ≥ | − |− and p b x λG(x ) 1 λ√α λε, ≥|k 2k−k 2 k| ≥ | − |− and hence I λG max a, b max 1 λ√α , 1 λ β 2λε. k − k≥ { }≥ {| − | | − |} − By inspecting the graphs of y = 1 λ√α and 1 λ√β , wep see that the smallest possible value of the second maximum| − occurs| when| − | 1 λ√α = λ β 1, − − in other words, when p 2 λ = = ρ. √α + √β Accordingly, √β √α I λG I ρG − 2λε, k − k≥k − k≥√β+√α− and, since ε was arbitrary, √β √α I λG − k − k≥√β+√α for any positive λ. It remains only to show that this lower bound is in fact attained when G = T . Since αI S βI ≤ ≤

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and T = S1/2, it follows that √αI T βI. ≤ ≤ Therefore, p √β √α √β √α − I I ρT − I, − √β + √α ≤ − ≤ √β + √α     which in turn implies that √β √α I ρT − k − k≤√β+√α (see [4], p. 262). This shows that I ρT =(√β √α)/(√β+√α) and the proof is complete. k − k −

Remark. It is worth pointing out that an orthonormal basis en with this minimum 4 { } norm property is not unique. To see this, take H = R and let e1,e2,e3,e4 be the standard orthonormal basis for H.Choose { }

f1=e1/2,f2=e2,f3=e3,f4=3e4/2.

Then the mapping T : en fn extends to a positive self adjoint operator on H. The minimum value of I →λT occurs when λ =1and k − k I T =1/2. k − k Consider now the orthonormal basis h ,h ,h ,h defined by { 1 2 3 4} h1 = e1,

h2 =(cosθ)e2+(sinθ)e3, h =( sin θ)e +(cosθ)e , 3 − 2 3 h4 =e4 where θ is a small positive number. If G: h f , then the matrix of I G is n → n − 1/20 0 0 01cos θ sin θ 0  0sin−θ1− cos θ 0  −  00 01/2   and   I G =1/2 k − k provided that θ is sufficiently small. This shows that en is not the only orthonor- mal basis with the minimum norm property described{ in} Theorem 3.

References

1. R. J. Duffin and J. J. Eachus, Some notes on an expansion theorem of Paley and Wiener, Bull. Amer. Math. Soc. 48 (1942), 850–855. MR 4:97e 2. S. V. Khrushchev, N. K. Nikolskii and B. S. Pavlov, Unconditional bases of exponentials and reproducing kernels,in“Complex Analysis and Spectral Theory” (V. P. Havin and N. K. Nikolskii, editors), pp. 214–335, Lecture Notes in Mathematics, vol. 864, Springer-Verlag, Berlin/Heidelberg (1981). MR 84k:46019 3. F. Riesz and B. Sz.-Nagy, Functional Analysis, Frederick Ungar Publ. Co., New York (1955). MR 17:175i

License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 560 STEPHANE JAFFARD AND ROBERT M. YOUNG

4. I. Singer, Bases in Banach spaces I, Springer-Verlag, New York/Heidelberg (1970). MR 45:7451 5. R. M. Young, An Introduction to Nonharmonic Fourier Series, Academic Press, New York (1980). MR 81m:42027

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