The numerical radius of a nilpotent operator on a Hilbert space

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The numerical radius of a nilpotent operator on a Hilbert space

HAAGERUP, Uffe, DE LA HARPE, Pierre

Abstract

ABSTRACTL.e t T be a bounded linear operatoro f norm 1 on a Hilbert space H such that T" = 0 for some n > 2. Then its numerical radius satisfies w (T) < cos (n'n+ 1) and this bound is sharp. Moreover, if there exists a unit vector s E H such that I( TIg) l= cos (n+'+1 ) then T has a reducings ubspace of dimension n on which T is the usual n-shift. The proofs show that these facts are related to the following result of Fejer: if a trigonometric polynomial f(6) = Zk-!n+i fkeikO is positive, one has If, I < fo cos (n+); moroever, there is essentially one polynomial for which equality holds.

HAAGERUP, Uffe, DE LA HARPE, Pierre. The numerical radius of a nilpotent operator on a Hilbert space. Proceedings of the American Mathematical Society, 1992, vol. 115, no. 2, p. 371-379

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The Numerical Radius of a Nilpotent Operator on a Hilbert Space Author(s): Uffe Haagerup and Pierre De La Harpe

Source: Proceedings of the American Mathematical Society, Vol. 115, No. 2 (Jun., 1992), pp.

371-379

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PROCEEDINGS OF THE

AMERICAN MATHEMATICAL SOCIETY Volume 11 5, Number 2, June 1992

THE NUMERICAL RADIUS OF A NILPOTENT OPERATOR ON A HILBERT SPACE

UFFE HAAGERUP AND PIERRE DE LA HARPE (Communicated by Palle E. T. Jorgensen)

ABSTRACT. Let T be a bounded linear operator of norm 1 on a Hilbert space H such that T" = 0 for some n > 2. Then its numerical radius satisfies w (T) < cos _(n+1) 'n and this bound is sharp. Moreover, if there exists a unit vector s E H such that I (TIg) l= cos _(n+1) '+ then T has a reducing subspace of dimension n on which T is the usual n-shift. The proofs show that these facts are related to the following result of Fejer: if a trigonometric polynomial f(6) = Zk-!n+i fkeikO is positive, one has If, I ^< fo cos (n+); moroever, there is essentially one polynomial for which equality holds.

Let H be a complex Hilbert space, possibly finite dimensional, and let T be a (bounded linear) operator on H. The numerical radius of T is

w(T) = Sup{I(T4I4)I: ⁴ e HI}

where H, denotes the unit sphere in H. It is well known that r(T) < w(T) < II TI

where r(T) and IITIJ denote respectively the spectral radius and the norm of T. In particular w(T) = JITIJ whenever T is normal. Also, it follows from polarization that

w(T) > ITIH.

For all this and much more, see [Hal,

?

1TIH cos

4+,

n-i

f(O)=

Z

k=-n+l

with fk e C. Such a polynomial is positive if f(O) > 0 for all 0 e JR. The com- putation of the numerical radius for the n-shift implies that fi ? < fo cos nI

(cf. Theorem 2).

Received by the editors September 7, 1990.

1991 Mathematics Subject Classification. Primary 47A12; Secondary 15A45, 42A05.

Key words and phrases. Numerical radius, nilpotent operator, positive trigonometric polynomial.

() 1992 American Mathematical Society 0002-9939/92 $1.00 + $.25 per page 371

372 UFFE HAAGERUP AND PIERRE DE LA HARPE

Theorem 1. Let T be an operator on H such that Tn = 0 for some n > 2.

(1) One has

w(T) < 11 TIT cos

and equality holds when T is the n-dimensional shift on the standard hermitian space Cn .

(2) Suppose morover that IITIH = 1. Suppose that there exists a unit vector 4 e Hi ^with I(T4I4)j ^{= cos} 4+* (If dimH < oc, this holds if and only if w(T) = cos41.) Let Ve be the linear span of {f, T4, ...,Tn-1'}. Then VJ

is an n-dimensional subspace of H which is reducing for T, and the restriction of T to Ve is unitarily equivalent to the n-dimensional shift on Cn .

The proof uses inequalities on positive matrices (see Proposition 3), namely on matrices IB = (i,1j)1<i?j<n e _Mn(C) such that (fl4j4) > 0 for all 4 e Cn;

for such a matrix, we write ,B > 0.

From now on, n is an integer, n > 2. For the proof of Theorem 1 we will need the following computation. (This is probably standard; see in particular [DH].)

Proposition 1. Let S be the n-dimensional shift on Cn, given by the matrix O 0 0 ...0 0\

1 0 0 ... 0 O 0 1 0 ...0 0O

: : : : 1:

I

O 0 0 ... 1 0/

and set Xo= ((2)12 sin kt) e C'. Then

(1) iiSii ⁼ 1,

(2) w(S) = cos ⁿ

(3) for 4 E C'n with

1

=eiO40 for some X E R.

Proof. The first claim is obvious.

For any operator T on any Hilbert space H, one has

w(T) = sup{f R(e'6(T4Il))j: 0 e R and 4 e H1}

I sup{I((e'T + e-io T*)I)I: 0 e R and e H1}

2 I sup{ lei'T + e-ioT*H1: 0 e RI}.

Set now

A0 1 0 ... O

1 o 1 ... o ol o 1 o ... o ol

A=S+S*=~ . . . .. 1 .

THE NUMERICAL RADIUS OF A NILPOTENT OPERATOR ON A HILBERT SPACE 373

If D(O) denotes the unitary diagonal matrix with entries ^{ei6, e216,..., eniC,} one has

D(O)*(eiOS + e-iOS*)D(O) = A for all 0 E R. Consequently w(S) ^{= 1} IIA I

For k = 1, ^{.. ,} n, let ^{Xk e Cn} be the vector with coordinates

nk s 2kn sin ^nk7r

n +1, n+1 ^' n +1

One has

A4k = 2 cos ( 7

+ k ,

k =1...,n .

(This computation appears already in [Lag, Nos. 19-20].) Consequently

w(S)=1 IAII=1 sup 2 cos +1)COS ,

21 <k<n n + \f+ 1 n

so that Claim (2) holds. Observe that

112 = E 2 (k)_ = n ⁺ 1

ii~1iiL~Ssnkf+j 2

k=i

so that

o=

Choose now 4 E C'n with 11411 = 1. As (4kIHkII-)1<k<n is an orthonormal basis of C'n, there are complex numbers cl, ..., cn with E Ik 2 = 1 and

= ZckXk XkK1 -. Assume that (SXI1) = cos n -. Then cos _{- =-(AXI4)} ^E _IckI2 ^cos

n + 1

2k=i n +

This implies that Ic ⁼ 1 and C2 = = = 0, so that ⁴ = cj4o. This shows (3). FO

Notation. If

IB

Lemma 1. Consider an operator T on H such that Tn = 0 and II T I < 1, and a unit vector 4 e H1. Define

fi,j = (Ti-lTj-)) 1,j > 1.

Then the matrix (8ih,j -

is positive.

Proof. Consider complex numbers cl, ^{... ,} cn and set

n

1=Z CkTk-l4 k=1

One has 1 - TT* > 0 , because TH < 1 , and thus n

((1 ^- T*T)jtI) =

c - (fi,- _fj fli++l,j+l) > 0.

i,j=l

As this holds for any choice of cl, ... , cn, the lemma follows. O

374 UFFE HAAGERUP AND PIERRE DE LA HARPE

Definition. With S and 40 as in Proposition 1, we define a matrix a = (a,,;) 1<i,j<n by

ai,j= (Si-= 4sj-'XO).

Observe that al, ^{1 =} jjXo ^{12 =} 1, and that

7E al,2 = a2,1 ^{= COS} n + 1

by Proposition 1.3. Observe also that (ax,j - aj+I j+i)i<i,j<n > 0 by Lemma 1.

Proposition 2.

Let

Mn(C) be a matrix such that

Assume that the matrix y ^e Mn(C) defined by

Yij= f,j - fhi+l,j+i is positive. Then

(1) Ifil,21 < al,2 = COS 7n+)

(2) If moreover Ih,,2 = ^a1,2 then B = a.

Proof. From the definition of y, one has

AJi,j = Yi,j + Yi+l,j+l + + Yi+k,j+k with k = min{n - i, n -

j}.

(I) trace(y) = fli, I=1

and _fl1, 2 = Y1, 2 + Y2, 3 +* + Yn- 1, n . As y is positive, yi, i is real nonnegative for i = 1, ..., n and Iy,, j12 < y, yj,jfor i, j= 1,..., n. Consequently

(II) Ifil,1 < (yl 1) 12 (Y2,2)1/2 + ..+ (Yn-l,n-I )1/2 (Yn, n) 1I

Let I ^{e Cn} be the vector with coordinates (yl,i)1/2, ..., (Yn,n)1/2 Then

11H112 ⁼ 1 by (I) and Ifil,21 < (Sqlt) by (II), so that Claim (1) follows by Proposition 1.

Recall that a state on Mn (C) is a linear form w on Mn (C) such that w(l) = 1 and w(T) > 0 whenever T > 0. Each positive matrix 3 e Mn(C) with trace(J) = 1 defines a state wc5 by wc5 (T) = trace(To5) . In particular, the matrix y defines a state wjy and one has

n n

wjy

(S) = Si, jYi, ⁼ j Yj, j+

i,j=l j=l

Given vectors Cl, C2 e Cn , we denote by

Cl

C2

An

n

Y = Z)vqv 01 7v v=l

and n

Sy= E _AvSqv 0 tv.

v=l

THE NUMERICAL RADIUS OF A NILPOTENT OPERATOR ON A HILBERT SPACE 375

The hypothesis for Claim (2) reads now

n

wJy (S) = trace(Sy) =

E

n+

v=l

As En=l Av = trace(y) = 1, it follows from Proposition 1 that Al ^{= 1,} 2 ⁼ A*n = O

and that t1 = _eiOo for some 0 e R . This can be expressed as

Y = 0o 9 Xo0 -

Denote by 4 ^{... n} j the coordinates of 40. One has

j= i + Xi+l i+l +* + Xi+kX +k

where k = min{n - i, n - j} = n - max{i, j}. But 4p = Xn+l-P for all p E{1,.., n} and

. ._j n+1-ijn+1-j + +n-iXn-, + + Xmax{i,j}-i+lmax{i,j}-j+l n

- Z (Si-lo)k(Si-l1o)k

k=max{i, j}

(S=- =X oSi- 1o ) a,1 forall i, je{1,...,n}. O

Lemma 2. Let T be an operator on H such that ITI ⁼ 1 and Tn = 0, and assume that there exists a unit vector E e HI such that

(T4) = cos n+1

Define VE to be the subspace of H spanned by X, T4,..., Tn-l . Then (1) (T*)k- 'X E VX for k = 1,1 ... , n.

(2) VE is a reducing subspace for T.

Proof. Lemma 1 and Proposition 2 imply that

(III) (T'-lXl IV_1

) =

o)

for all i, j e

{

Define ,B e _{Mn(C) by} fli/, ⁼ ((T* )1l (T*)i-l). Lemma 1 and Proposi- tion 2 imply also that

(IV) ⁶ = a

and in particular that 1(T*)i-l = (ai, ^)1/2 for i = 1, ^, n.

Let P be the orthogonal projection from H onto VJ'. Consider some k e {1, ... , n}, and let cl, ^..., cn be the (uniquely defined) complex numbers such that

P(T*)kl =C1 + c2TX + + CnTn-l.

376 UFFE HAAGERUP AND PIERRE DE LA HARPE

As (Tv-l4)i<v<n is a basis of VJ (though not an orthonormal one!) these cv 's can be written as functions of the numbers

((T*)k-lXlT-ll4) and (T'-1clT'IV-), i, j= 1, ^..., n.

But (IV) shows that

((T* )k-l~iTi-l4) = ((T*)k+i-2I1) = ((S*)k-l4 jSi-1 0) for i=1,...,n and (III) reads

(Tl- XlTI- (Si-'olsj-'10o)

for i, j ⁼ 1, ^..., n. It follows that the coordinates of (S*)k-10 with respect to the non-orthogonal basis 40, So, ^...S, S'- Io of C'n are also cl, c2, ^... , Cn, i.e.

(S*)k l 0 ⁼ c10 + C2SR0 ^{+ *} +CnSn lO

hence

IIP(T*)kl1i11 ⁼ jj(S*)k-1XoH1 = (ak, k)/2 =-I(T*)k-1411

Consequently (T* )k- 1l E Im(P) = VX and this proves (1).

Define W< to be the linear span of 4, T*4, ... , (T*)n-l4. One checks as for VJ that WX has dimension n, and Claim (1) shows that WX c V,:. Hence WX = V~. As V, [respectively W] is obviously invariant by T [resp. T*], it follows that VX is invariant by T and T* . O

Proof of Theorem 1. Let T be an operator on H such that Tn = 0. One may assume that II T I ⁼ 1 .

(1) Choose a vector 4 e H1 and define

,B

,8i,j = (Ti-l'Ij-'4).

By Lemma 1 and Proposition 2.1, one has 1,1,21 < cos . As this holds for all 4 E HI, this shows that w(T) < cos n+-1

(2) By hypothesis, there exist 4 e H1 and 0 e JR such that (TIl) ⁼ eio COS ^7. Let VX be the linear span of 4, T4,..., Tn-i .

Define T' = e-i T. Then VJ is a reducing subspace for T' by Lemma 2 and the restriction of T' to VX is unitarily equivalent to the n-dimensional shift S by Proposition 2. Hence V<: is reducing for T and the restriction of T to VX is unitarily equivalent to eiOS. But D(O)eiOSD(O)* = S if D(O) is the unitary diagonal matrix defined in the proof of Proposition 1, so that the proof is complete. O

From Theorem 1, it is easy to deduce the following 1915 result of L. Fejer (see [PS, Problem VI.52]). We are grateful to J. Steinig who brought Fejer's result to our attention.

Theorem 2. Consider a positive integer n ^> 2 and a positive trigonometric poly- nomial

n-i

f(0)=

S

of degree at most n - 1 . Assume that f

$&

THE NUMERICAL RADIUS OF A NILPOTENT OPERATOR ON A HILBERT SPACE 377

(1) One has _Ifi ^< fo cos

4+

(2) Suppose moreover that _Ifi I ⁼ fo cos n Then f(O) ⁼ folg(O + V)12

for all

g(o)

=

and where qi = arg(f1).

Proof. As f

$&

>

+j

and there is no loss of generality if we assume that fo

= 1.

n-I

f(o) = lg(0)12 = E (gg )ikO

k=-n+l

for all 0 E R, where the second summation is over p, q E {0, 1,..., n - 1}

with p - q = k. One has in particular

n-I

(V) _1gp2 = 1 and fi = (SI I ¹⁾

p=O

where I ^E Cn is the unit vector with coordinates

go,

w(S) ⁼ cos

n

Suppose now that

fi

f

f(0

A)

e-iOg,

Remarks. (a) Let T be as in Theorem 1 and such that IITIJ ⁼ 1. For each eH1 and

0 E

J

f;,kk (TkCIC) if k > 0,

1 (CITIkIC) if

k <

and

n-I

Z ~kikO.

fC(0) ⁼ fC,kk=e k=-n+l

378 UFFE HAAGERUP AND PIERRE DE LA HARPE

Set also

X = 1 +

Z((ei0

k>1

= (1 - e'0T)-1 + (1 - e-iOT*)-l - 1

= (1 - e-i0T*)-l{l - e-'OT* + 1 - e'8T

- (1 - e iOT*)(1 - e'0T)}(l ^- eiFT)-

= (1 -e-iOT*)-1{l - T*T}(l -ei0T)-

and

= (1 - ei0T)-1C.

Then

fi(0) = (XCIC) = ((1 ^- T*T)nl) > 0 because

TI TiI

J(TCJC)J

for all ^' E H1 . This provides a new proof of the inequality in Theorem 1.1.

(b) Assume now that n > 3. For each t e [0, 1], set 0 O O ... OO'

1 0 0 ... 0 0 O t O ... O O

S(t) = . . . . . . E MnM(C)

0 0 0

0 0

o O O ...

t ^O

and let w(t) be the numerical radius of S(t). One has jjS(t)JJ = 1 for all

t

.

[-,

On the other hand, let T be an operator of norm 1 on a finite dimensional Hilbert space H (or more generally on a Hilbert space in which there exists a unit vector 4 such that

IT4II

For n > 3 let In be the set of those positive real numbers t for which there exists an operator T acting on a finite dimensional Hilbert space with the following properties: t = w(T), 11Til = 1, Tn = 0, Tn- 1

7

(c) Let S(t) and w(t) be as above, now with 0 < t < 1 . It is easy to check that w (t) < cos n+E . Let (t,),>I be a strictly increasing sequence of positive numbers converging to 1. Consider an operator on an infinite dimensional Hilbert space which is the orthogonal sum of the S(t,)'s. Such an operator shows that the hypothesis that T attains its numerical radius cannot be omitted in Claim (2) of Theorem 1.

THE NUMERICAL RADIUS OF A NILPOTENT OPERATOR ON A HILBERT SPACE 379

(d) We have illustrated one more appearance of the ubiquitous sequence

V n+l)n>2

Its recent popularity is due to the work of V. Jones on index for subfactors and all that [Jon], but it appears in many other domains, such as the elementary geometry of regular polygons [Cox, Formula 2.84], graph theory or Fuchsian groups [GHJ], to mention but a few.

The second author wishes to note that the original idea of this paper is due to the first author. The paper itself was written up while we were both enjoying the hospitality of the Mittag-Leffler Institute during the autumn 1988. We are grateful to A. Sinclair and J. Steinig for their useful comments.

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INSTITUT FOR MATEMATIK OG DATALOGI, ODENSE UNIVERSITET, DK-5230 ODENSE M, DENMARK.

SECTION DE MATHEMATIQUES, UNIVERSITt DE GENEVE, C.P. 240, CH-121 1 GENEVE 24,

SWITZERLAND.